548 results on '"Dehghan, Mehdi"'
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2. Iterative method for constrained systems of conjugate transpose matrix equations.
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Shirilord, Akbar and Dehghan, Mehdi
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CONJUGATE gradient methods , *COMPLEX matrices , *EQUATIONS , *LINEAR dynamical systems , *MARKOVIAN jump linear systems , *MATRICES (Mathematics) - Abstract
This study presents some new iterative algorithms based on the gradient method to solve general constrained systems of conjugate transpose matrix equations for both real and complex matrices. In addition, we analyze the convergence properties of these methods and provide numerical techniques to determine the solutions. The effectiveness of the proposed iterative methods is demonstrated through various numerical examples employed in this study. [ABSTRACT FROM AUTHOR]
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- 2024
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3. Solving 2D damped Kuramoto-Sivashinsky with multiple relaxation time lattice Boltzmann method.
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MohammadiArani, Reza, Dehghan, Mehdi, and Abbaszadeh, Mostafa
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LATTICE Boltzmann methods , *DISTRIBUTION (Probability theory) , *FLUID flow - Abstract
Lattice Boltzmann method (LBM) is a powerful fluid flow solver. Using this method to solve other PDEs might be a difficult task. The first challenge is to find a suitable local equilibrium distribution function (EDF) capable of recovering the desired PDE. The next difficulty arises from the explicit nature of LBM. The conditional stability of the LBM algorithm affects the numerical solution accuracy. Damped Kuramoto–Sivashinsky (DKS) equation is a fourth–order PDE that recently challenged many numerical methods' abilities. This equation is highly sensitive to a parameter that causes three states of solutions in a small interval of freedom. In this paper, we challenged LBM to solve the two–dimensional DKS equation by finding EDF using the Chapman–Enskog analysis up to the fourth–order. Also, the von Neumann analysis and a simple genetic algorithm are applied to find reliable values for the free parameters. Furthermore, a modification on image-based ghost node method is proposed for implementation of the boundary conditions in the complex geometries. [ABSTRACT FROM AUTHOR]
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- 2024
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4. Error estimates of divergence-free generalized moving least squares (Div-Free GMLS) derivatives approximations in Sobolev spaces.
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Mohammadi, Vahid and Dehghan, Mehdi
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SOBOLEV spaces , *LEAST squares , *INCOMPRESSIBLE flow , *FLUID flow , *VECTOR fields - Abstract
The divergence-free generalized moving least squares (Div-Free GMLS) approximation has recently been utilized to solve some incompressible fluid flows problems. In our recent work (Mohammadi and Dehghan (2021) [28]), we have presented its formulation more precisely, and also the error estimates of derivatives have been carried out in L ∞ (Ω) , where Ω ⊂ R d is a bounded set satisfying an interior cone condition. However, the error estimates of this vector-valued approximation in Sobolev spaces are not done. So, in this paper we make the error estimates of Div-Free GMLS derivatives approximations in L q (Ω) , where 1 ≤ q ≤ ∞ , using a stable local divergence-free polynomial reproduction property. Note that, the method is a direct approximants of exact derivatives of a divergence-free vector field, which possesses the optimal rates of convergence. This vector-valued technique can also be developed to find the numerical solution of the incompressible fluid flows problems easier than the other available mesh-dependent methods. Finally, we have shown how the proposed approximation can recover the velocity field variable of the well-known Darcy's problem in a two-dimensional space. • The error analysis of the div-free GMLS derivatives approximations has been released on the Sobolev spaces. • The method can approximate the vector field satisfying the div-free property with the order of O (h m + 1 − | α |). • The developed vector-valued approximation satisfies analytically the div-free property for all polynomials belong to P m. • We have shown how the div-free GMLS approximation can recover the velocity field variable related to the Darcy's problem. [ABSTRACT FROM AUTHOR]
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- 2023
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5. A reduced-order model based on integrated radial basis functions with partition of unity method for option pricing under jump–diffusion models.
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Ebrahimijahan, Ali, Dehghan, Mehdi, and Abbaszadeh, Mostafa
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PARTITION of unity method , *RADIAL basis functions , *PARTITION functions , *PROPER orthogonal decomposition , *PRICES , *ADVECTION-diffusion equations , *REDUCED-order models - Abstract
The current research aims to develop a fast, stable and efficient numerical procedure for solving option pricing problems in jump–diffusion models. A backward partial integro-differential equation (PIDE) with diffusion and advection terms was investigated. Up to the best knowledge of the authors, some special numerical methods and strategies must be selected to solve advection–diffusion problems with reliable stability and accuracy. For the mentioned aims, the first- and second-order derivatives are approximated by integrated radial basis function based on partition of unity method. The IRBF-PU method is local mesh-free method that provides high order accurate result and is flexible for PDEs problems with sufficiently smooth initial conditions and also has a moderate condition number. In particular, we highlight European and American style put options, whose underlying asset follows a jump–diffusion model. For the distribution of the jumps, the Merton and Kou models are studied. Furthermore, the main model is classified in advection-x-diffusion category. As a result, we must increase the number of collocation points as well as the time steps to arrive at the final time. This procedure lengthens the execution time. To address this issue, we use the proper orthogonal decomposition (POD) method to reduce the size of the final algebraic system of equations. This numerical procedure is known as the proper orthogonal decomposition-IRBF-PU method (POD-IRBF-PU). The presented computational results (including the computation of option Greeks) and comparisons with other competing approaches suggest that the IRBF-PU and POD-IRBF-PU methods are efficient and reliable numerical methods to solve elliptic and parabolic PIDEs arising from applied areas such as financial engineering. [ABSTRACT FROM AUTHOR]
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- 2023
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6. Simulation of the coupled Schrödinger–Boussinesq equations through integrated radial basis functions-partition of unity method.
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Ebrahimijahan, Ali, Dehghan, Mehdi, and Abbaszadeh, Mostafa
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PARTITION of unity method , *CONCORD , *PARTITION functions , *EQUATIONS , *CONSERVED quantity , *RADIAL basis functions , *CONSERVATION laws (Mathematics) - Abstract
In this paper, integrated radial basis functions-partition of unity (IRBF-PU) method is presented for the numerical solution of the one- and two-dimensional coupled Schrödinger–Boussinesq equations. The IRBF-PU method is a local mesh-free method that prepares flexibility and high orders of accuracy for PDEs problems with adequately smooth initial conditions and also has a moderate condition number. First, the temporal direction is discretized using the Runge–Kutta method with non-decreasing abscissas and nine stages, that allows for greater flexibility in the temporal step width. The integrated radial basis function based on partition of unity method (IRBF-PUM) then makes an approximation of the spatial direction. Numerical simulations are also used to track conserved quantities to determine how well the suggested approach keeps them. IRBF-PU simulates solitary waves for an extended period of time while effectively preserving conservation laws, according to numerical experiments. To confirm the effectiveness and dependability of the suggested method, the obtained results are contrasted with those obtained using other methods found in the literature. [ABSTRACT FROM AUTHOR]
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- 2023
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7. Numerical solution of Allen–Cahn model on surfaces via an effective method based on generalized moving least squares (GMLS) approximation and the closest point approach.
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Zamani-Gharaghoshi, Hasan, Dehghan, Mehdi, and Abbaszadeh, Mostafa
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LEAST squares , *PARTIAL differential equations , *SURFACE structure - Abstract
A numerical method has been designed for solving the surface Allen–Cahn model. The proposed numerical procedure is based on the generalized moving least-squares approximation and the closest point method. This approach is free from the structure of the underlying surface. It only needs a set of arbitrarily distributed mesh-free points on the surface. [ABSTRACT FROM AUTHOR]
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- 2023
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8. Simulations of dendritic solidification via the diffuse approximate method.
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Najafi, Mahboubeh and Dehghan, Mehdi
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SOLIDIFICATION , *KRONECKER delta , *HEAT equation , *LINEAR systems - Abstract
The current work presents simulations of two-dimensional dendritic solidification via the meshless Diffuse Approximate Method (DAM). The presumed Stefan problem is studied through the phase-field model. Isotropic and anisotropic materials are considered for comparisons with the benchmark tests. Investigations on the change of some constants are carried out to discover their effects on the obtained patterns. The nodal DAM (with the Kronecker delta property of its shape functions) for the spatial discretization and the forward Euler temporal discretization for the coupled phase-field and heat equations provide results that comply with the patterns of previous works. The significant feature of the proposed numerical method manifests through the explicit time marching and the local property of the chosen meshless method that ensues solving small-size linear systems for each subdomain. [ABSTRACT FROM AUTHOR]
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- 2023
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9. Combined real and imaginary parts method for solving generalized Lyapunov matrix equation.
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Shirilord, Akbar and Dehghan, Mehdi
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EQUATIONS , *MATRICES (Mathematics) , *COMPLEX matrices , *LINEAR systems - Abstract
Based on the Combined Real part and Imaginary part (CRI) method for solving linear systems Wang et al. (2017) [50] , we solve generalized Lyapunov matrix equation A X + X A T + ∑ j = 1 m K j X K j T = C , where A , K j (j = 1 ,... , m) , C ∈ C n × n are known and X ∈ C n × n must be determined. The convergence of the CRI iteration method is proved and an upper bound on the convergence rate is derived and then minimized. Numerical results show that the CRI iteration method is efficient for this matrix equation. [ABSTRACT FROM AUTHOR]
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- 2022
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10. A unified analysis of fully mixed virtual element method for wormhole propagation arising in the petroleum engineering.
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Dehghan, Mehdi and Gharibi, Zeinab
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PETROLEUM engineering , *FIXED point theory , *CARBONATE reservoirs , *GAS reservoirs , *CRANK-nicolson method , *PETROLEUM reservoirs , *TRANSPORT equation , *GAS condensate reservoirs - Abstract
Wormhole propagation, arising in petroleum engineering, is used to describe the distribution of acid and the increase of porosity in carbonate reservoir under the dissolution of injected acid and plays a very important role in the product enhancement of oil and gas reservoirs. In this paper, a fully mixed virtual element method (VEM) is employed to discretize this problem, in which mixed VEM is used not only for the Darcy flow equations but also for approximation the concentration equation by introducing an auxiliary flux variable to guarantee full mass conservation. The stability, existence and uniqueness of solution of the associated mixed VEM are proved by fixed point theory. Also, we obtain unconditionally optimal error estimate for concentration and auxiliary flux variable of convection-diffusion equation, as well as for the velocity and pressure of Darcy equations in the L 2 norm. Finally, several numerical experiments are presented to support the theoretical analysis of convergence and to illustrate the applicability for solving actual problems. [ABSTRACT FROM AUTHOR]
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- 2022
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11. Application of direct meshless local Petrov–Galerkin method for numerical solution of stochastic elliptic interface problems.
- Author
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Abbaszadeh, Mostafa, Dehghan, Mehdi, Khodadadian, Amirreza, and Heitzinger, Clemens
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LEAST squares , *PROBLEM solving - Abstract
A truly meshless numerical procedure to simulate stochastic elliptic interface problems is developed. The meshless method is based on the generalized moving least squares approximation. This method can be implemented in a straightforward manner and has a very good accuracy for solving this kind of problems. Several realistic examples are presented to investigate the efficiency of the new procedure. Furthermore, compared with other meshless methods that have been developed, the present technique needs less CPU time. [ABSTRACT FROM AUTHOR]
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- 2022
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12. Unconditionally energy stable C0-virtual element scheme for solving generalized Swift-Hohenberg equation.
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Dehghan, Mehdi, Gharibi, Zeinab, and Eslahchi, Mohammad Reza
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PARTIAL differential equations , *NONLINEAR equations , *FINITE element method , *EQUATIONS , *DISCRETE systems , *CRANK-nicolson method , *MIXED reality - Abstract
The very recently introduced Virtual Element Method (VEM) is a numerical method for solving partial differential equations that was created out of the mimetic difference method, but was later reformulated into the Galerkin framework. It is a generalization of the standard Finite Element Method (FEM) to general meshes made up by arbitrary polyhedra. The greatest advantage of VEM is its ability to deal with very complex geometries, i.e., made up by elements of any number of edges not necessarily convex, hanging nodes, flat angles, collapsing nodes, etc., while retaining the same approximation properties of FEM. In this article, the C 0 -virtual element method is formulated and analyzed to solve generalized Swift-Hohenberg equation on polygonal meshes. The Swift-Hohenberg equation as a central nonlinear model in modern physics has a gradient flow structure. Here, we present the spatial VE discretization based on a mixed formulation for the nonlinear Swift-Hohenberg equation as a class of fourth-order gradient flow problems. For time discretization, we use Crank-Nicolson so that the resulting scheme is unconditionally stable and second-order in time. By following the algebraic implementation of the discrete system, we provide numerical tests validating the theoretical estimates and plotting two-dimensional pattern formation problems. [ABSTRACT FROM AUTHOR]
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- 2022
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13. Radial basis function partition of unity procedure combined with the reduced-order method for solving Zakharov–Rubenchik equations.
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Dehghan, Mehdi, Hooshyarfarzin, Baharak, and Abbaszadeh, Mostafa
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PARTITION functions , *PROPER orthogonal decomposition , *REDUCED-order models , *RADIAL basis functions , *EQUATIONS , *SOLITON collisions - Abstract
A meshless radial basis function based on partition of unity (RBF-PU) method is proposed to solve Zakharov–Rubenchik equations. In this local method, the domain is split into overlapping patches forming a covering of it and also, it provides accurate results for PDEs. Time discretization is performed using a second-order implicit explicit backward difference method (IMEX-BDF2). Although the proper orthogonal decomposition (POD) is applied to reduce the dimension of the governing model, the computational complexity of the reduced model for nonlinear terms still depends on the number of variables of the full model. To overcome this subject, we employ the discrete empirical interpolation method (DEIM). Two problems with different situations are solved by the proposed method and the comparison of numerical findings with the conservative compact difference scheme and RBF-FD method shows that the presented method provides accurate results at a low computing cost. [ABSTRACT FROM AUTHOR]
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- 2022
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14. PROPER ORTHOGONAL DECOMPOSITION--LATTICE BOLTZMANN METHOD: SIMULATING THE AIR POLLUTANT PROBLEM IN STREET CANYON AREAS.
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MOHAMMADIARANI, REZA, DEHGHAN, MEHDI, and ABBASZADEH, MOSTAFA
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LATTICE Boltzmann methods , *AIR pollutants , *PROPER orthogonal decomposition , *DISTRIBUTION (Probability theory) , *CANYONS - Abstract
Recently, the reduced order model (ROM) has been applied to reduce the computational cost of numerical methods. Merging this approach with the lattice Boltzmann method (LBM) as a powerful mesoscale fluid solver seems promising. Dealing with this idea requires a matrix form for the LBM. This matrix form must represent all the LBM's algorithm steps (including boundary condition implementation) to just one specific matrix-vector relation. Furthermore, the LBM contains some nonlinear parts inside the equilibrium distribution function (EDF), which must be treated with a nonlinear ROM like the discrete empirical interpolation method (DEIM). So, in this paper, we discuss proper orthogonal decomposition (POD)-ROM LBM in the general form, present an appropriate matrix form for the LBM as explained, and implement POD or DEIM on that matrix form. In the end, the ability of this scheme is challenged with solving the urban air pollutant transition problem over some canyons areas as a practical problem. [ABSTRACT FROM AUTHOR]
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- 2022
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15. Integrated radial basis functions to simulate modified anomalous sub‐diffusion equation.
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Dehghan, Mehdi, Ebrahimijahan, Ali, and Abbaszadeh, Mostafa
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FINITE difference method , *EQUATIONS - Abstract
This research work presents a newly truly meshless approach based upon the integrated radial basis functions (IRBFs) technique to study the fractional modified anomalous sub‐diffusion equation. First, the temporal direction is discretized by a finite difference method with second‐order accuracy. The stability analysis and convergence of the proposed time‐discrete formulation are investigated, theoretically. Then, the spatial direction is approximated by the IRBFs methodology. In this approach, the largest‐order derivative is approximated by a linear combination of RBFs and then the lower‐order derivative and also the unknown function are constructed by repeated integration. According to this procedure, the existing derivatives in the main fractional PDE are approximated smoothly. Furthermore, to show the efficiency of the developed numerical procedure, we employed some non‐rectangular computational domains to obtain the numerical results. On the other hand, the mentioned model is solved in one‐ two‐, and three dimensional cases. The numerical results confirm the ability and efficiency of the new numerical method for solving time fractional PDEs on complex computational domains. [ABSTRACT FROM AUTHOR]
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- 2022
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16. A high‐order weighted essentially nonoscillatory scheme based on exponential polynomials for nonlinear degenerate parabolic equations.
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Abedian, Rooholah and Dehghan, Mehdi
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DEGENERATE parabolic equations , *FINITE differences , *DEGENERATE differential equations , *POLYNOMIALS , *EXPONENTIAL functions , *FINITE difference method - Abstract
In this research the numerical solution of nonlinear degenerate parabolic equations is investigated by a new sixth‐order finite difference weighted essentially nonoscillatory (WENO) based on exponential polynomials. In smooth regions, the new scheme, named as EPWENO6, can achieve the maximal approximation order while in critical points it does not lose its accuracy. In order to better approximation near steep gradients without spurious oscillations, the EPWENO6 scheme is designed by the exponential polynomials that are incorporated into the WENO reconstruction. In design of nonlinear weights that play a very important role in WENO reconstructions, a global smoothness indicator using generalized undivided differences is introduced. To analyze the convergence order of the EPWENO6 method in full detail, the Lagrange‐type exponential functions have been used. By comparing the EPWENO6 scheme and other existing WENO schemes for solving degenerate parabolic equations, it can be seen that the new scheme offers better results while its computational cost is less. To illustrate the effectiveness of the proposed scheme, a number of numerical experiments are considered. [ABSTRACT FROM AUTHOR]
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- 2022
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17. A class of moving Kriging interpolation-based DQ methods to simulate multi-dimensional space Galilei invariant fractional advection-diffusion equation.
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Abbaszadeh, Mostafa and Dehghan, Mehdi
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ADVECTION-diffusion equations , *KRIGING , *FINITE differences , *FINITE difference method - Abstract
The current work concerns to develop a new local meshless procedure to simulate the one-, two- and three-dimensional space Galilei invariant fractional advection-diffusion (GI-FAD) equations. The fractional derivative is discretized by a second-order finite difference formula. The unconditional stability and rate of convergence of the time-discrete scheme are analytically studied. Then, we employ the shape functions of moving Kriging interpolation for the differential quadrature (DQ) method. According to this combination, we can derive a new local meshless technique. At the end, some examples are solved to confirm the theoretical results and efficiency of the proposed meshless method. [ABSTRACT FROM AUTHOR]
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- 2022
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18. A weighted combination of reproducing kernel particle shape functions with cardinal functions of scalable polyharmonic spline radial kernel utilized in Galerkin weak form of a mathematical model related to anti-angiogenic therapy.
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Narimani, Niusha, Dehghan, Mehdi, and Mohammadi, Vahid
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POLYHARMONIC functions , *MATHEMATICAL forms , *SPLINES , *NEOVASCULARIZATION inhibitors , *MATHEMATICAL models , *BIOLOGICAL mathematical modeling , *MESHFREE methods , *SPLINE theory - Abstract
In this manuscript, a new localized meshfree (meshless) approximation, i.e., a combination of the reproducing kernel particle (RKP) shape functions with the cardinal functions of the scalable polyharmonic spline radial kernel with polynomial augmentation (PHS+poly) is introduced. It is called the RKP+PHS+poly approximation, and its convergence rate is of order O (h m + 1) , where m is the total degree of polynomials. We apply this method to construct the spaces of trial and test functions in a Galerkin scheme of an extended version of the biological mathematical model in two dimensions describing the interactions between endothelial cells, fibronectin, angiogenic growth factors, and fasentin concentrations. By considering the row-sum method, the eigenvalue stability is also numerically carried out for the discrete equations corresponding to the obtained weak formulation. Accordingly, a semi-implicit form of the backward difference method of order 1 (SBDF1) has been utilized to approximate the weak form in time. We complete our numerical algorithm by solving the obtained full-discretized problem overtime via the biconjugate gradient stabilized (BiCGSTAB) solver with a proper preconditioner. Some simulation results are investigated by estimating the maximum velocity parameter of an enzymatic reaction from the experimental dose–response curve of fasentin to demonstrate the effect of using fasentin drug. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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19. Iterative algorithm for a generalized matrix equation with momentum acceleration approach and its convergence analysis.
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Shirilord, Akbar and Dehghan, Mehdi
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IMAGE reconstruction , *CONSTRAINED optimization , *EQUATIONS , *MATRICES (Mathematics) , *ALGORITHMS - Abstract
This article discusses a numerical gradient-based method for solving a generalized matrix equation. The iterative method mentioned includes two positive parameters, for which a range is determined to ensure the convergence of the introduced method. It has been shown that the optimal parameters of this method satisfy a constrained optimization problem. Then, specific solutions of this equation, such as the reflexive solution, are examined. Furthermore, to increase the rate of convergence of the proposed method, the idea of momentum is utilized, and a range for the momentum parameter is obtained to ensure the convergence. Finally, the efficiency of this method is investigated through numerical simulations. Additionally, in numerical results section, applications of mentioned matrix equation in image restoration and anti-linear systems are addressed. [ABSTRACT FROM AUTHOR]
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- 2024
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20. Stationary Landweber method with momentum acceleration for solving least squares problems.
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Shirilord, Akbar and Dehghan, Mehdi
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LEAST squares , *ACCELERATED life testing - Abstract
In this article, we proposed an enhancement to the convergence rate of Landweber's method by incorporating the concept of momentum acceleration. Landweber's method is commonly used to solve least squares problems of the form min x ‖ A x − b ‖. Our approach is based on Landweber's method, which is acknowledged as a particular case of the methodologies outlined in Ding and Chen (2006). Through optimizing the momentum parameter, we were able to demonstrate the superior performance of the momentum-accelerated Landweber method. Specifically, we established that when A is a nonsquare m × n matrix with Rank (A) = n and σ min (A) ≠ σ max (A) , the momentum-accelerated Landweber method with the optimal parameter consistently outperforms the standard Landweber method. Our numerical experiments have confirmed the theoretical findings, demonstrating a notable improvement in the convergence rate of the Landweber method. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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21. Solving a system of complex matrix equations using a gradient-based method and its application in image restoration.
- Author
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Shirilord, Akbar and Dehghan, Mehdi
- Abstract
This study presents some new iterative algorithms based on the gradient method to solve general constrained systems of conjugate transpose matrix equations for both real and complex matrices. In addition, we analyze the convergence properties of these methods and provide numerical techniques to determine the solutions. Then we prove that the optimal parameters of the new algorithm satisfy a constrained optimization problem. The effectiveness of the proposed iterative methods is demonstrated through various numerical examples employed in this study and compared the results by some existing algorithms. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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22. Extending matrix–vector framework on multiple relaxation time lattice Boltzmann method.
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MohammadiArani, Reza, Dehghan, Mehdi, and Abbaszadeh, Mostafa
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LATTICE Boltzmann methods , *COMPUTATIONAL fluid dynamics , *FLUID flow , *PARALLEL programming - Abstract
This paper introduces an extension of a fully discrete matrix–vector form (MVF) for the lattice Boltzmann method (LBM) to handle the multiple relaxation time parameter (MRT) LBM. The proposed approach offers a more efficient and practical framework for simulating fluid flows, with the added benefit of being able to handle complex geometries using the image-based ghost (IBG) method. The advantages and limitations of this approach are discussed, including its simplicity in linear algebraic extension, parallel computing capability, computational speed, and memory usage. The results of numerical experiments demonstrate the improved computational efficiency of the proposed method, highlighting its potential for future applications in various fields of computational fluid dynamics. [ABSTRACT FROM AUTHOR]
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- 2024
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23. A POD-RBF-FD scheme for simulating chemotaxis models on surfaces.
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Mohammadi, Vahid and Dehghan, Mehdi
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ORTHOGONAL decompositions , *CHEMOTAXIS , *SINGULAR value decomposition , *PROPER orthogonal decomposition , *NONLINEAR differential equations , *PARTIAL differential equations , *FINITE differences , *ORTHOGONAL functions - Abstract
The main aim of this paper is to develop a new framework of a meshless approximation for solving numerically three nonlinear partial differential equations in biology, i.e., the chemotaxis models defined on the smooth, closed manifolds embedded in R 3. The radial basis function-generated finite difference scheme is considered to deal with the spatial variables, which depends only on the location of nodes and the value of normal vector at each point per spherical cap. The robust artificial hyperviscosity formulation is derived for each model, which has been used for preventing the spurious growth modes in the numerical solution. An implicit–explicit time discretization is employed to deal with the time variable. The resulting fully discrete scheme is solved via the biconjugate gradient stabilized method with a zero-fill incomplete lower upper preconditioner per time step, where a positivity-preserving filter is used to prevent the negative sign of the cell density variable. Besides, to reduce the used central processor unit (CPU) time, the proper orthogonal decomposition is considered for constructing a set of new orthogonal basis vectors based on the singular value decomposition. The developed numerical method is called a proper orthogonal decomposition-radial basis function-generated finite difference (POD-RBF-FD) scheme. Finally, the ability of the proposed method is investigated by simulation results showing the blowing-up, pattern formulation (perforated stripe) and aggregations of bacteria on some surfaces. • Three chemotaxis mathematical models on the surfaces have been numerically solved. • An RBF-FD scheme is employed to deal with the space variables. • A new fully discrete scheme has been derived via an implicit–explicit time integration method. • A POD-RBF-FD scheme on the surfaces is introduced to reduce the elapsed CPU time. • Some numerical results showing blow-up, a pattern of formation, and aggregations of bacteria are given on different surfaces. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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24. Simulation of Maxwell equation based on an ADI approach and integrated radial basis function-generalized moving least squares (IRBF-GMLS) method with reduced order algorithm based on proper orthogonal decomposition.
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Ebrahimijahan, Ali, Dehghan, Mehdi, and Abbaszadeh, Mostafa
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PROPER orthogonal decomposition , *ORTHOGONAL decompositions , *MAXWELL equations , *RADIAL basis functions , *LEAST squares , *ALGEBRAIC equations - Abstract
Basic equations of electromagnetic are Maxwell equations. In this manuscript, ADI-IRBF-GMLS is employed for solving the time-dependent Maxwell equations in two-dimension. For approximating the time variable, we utilize alternative direction implicit (ADI) method and integrated radial basis function based on generalized moving least squares (IRBF-GMLS) method is used for space direction. Alternative Direct Implicit (ADI) technique includes two steps in each time stage, that their computations are simple. We have to increase the number of collocation points and also time steps to reach the final time. This procedure increases the used execution time. To overcome this issue, we employ the proper orthogonal decomposition (POD) method to reduce the size of the final algebraic system of equations. This numerical procedure can be called ADI-IRBF-GMLS-POD method. Numerical results are presented and they illustrate the accuracy and efficiency of the proposed method. The point to note is that the used time-discrete scheme i.e. ADI approach cannot be employed in the numerical simulations on non-rectangular computational domains for solving Maxwell equations. This is the main issue in this numerical approach for Maxwell equations. We also compare the ADI-IRBF-GMLS with POD with full model of presented method applied to solve Maxwell equations. [ABSTRACT FROM AUTHOR]
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- 2022
- Full Text
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25. Legendre spectral element method (LSEM) to simulate the two-dimensional system of nonlinear stochastic advection–reaction–diffusion models.
- Author
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Abbaszadeh, Mostafa, Dehghan, Mehdi, Khodadadian, Amirreza, and Wick, Thomas
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STOCHASTIC systems , *NONLINEAR systems , *STOCHASTIC models , *LEGENDRE'S functions , *RANDOM variables , *ADVECTION-diffusion equations , *SPECTRAL element method - Abstract
In this work, we develop a Legendre spectral element method (LSEM) for solving the stochastic nonlinear system of advection–reaction–diffusion models. The used basis functions are based on a class of Legendre functions such that their mass and diffuse matrices are tridiagonal and diagonal, respectively. The temporal variable is discretized by a Crank–Nicolson finite-difference formulation. In the stochastic direction, we also employ a random variable W based on the Q-Wiener process. We inspect the rate of convergence and the unconditional stability for the achieved semi-discrete formulation. Then, the Legendre spectral element technique is used to obtain a full-discrete scheme. The error estimation of the proposed numerical scheme is substantiated based upon the energy method. The numerical results confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]
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- 2022
- Full Text
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26. An asymptotic analysis and numerical simulation of a prostate tumor growth model via the generalized moving least squares approximation combined with semi-implicit time integration.
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Mohammadi, Vahid, Dehghan, Mehdi, Khodadadian, Amirreza, Noii, Nima, and Wick, Thomas
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LEAST squares , *TUMOR growth , *PROSTATE tumors , *NUMERICAL analysis , *KRYLOV subspace , *COMPUTER simulation , *CIRCLE - Abstract
• The prostate tumor growth model. • An asymptotic analysis. • A generalized moving least squares technique. • Experimental data. This paper focuses on presenting an asymptotic analysis and employing simulations of a model describing the growth of local prostate tumor (known as a moving interface problem) in two-dimensional spaces. We first demonstrate that the proposed mathematical model produces the diffuse interfaces with the hyperbolic tangent profile using an asymptotic analysis. Next, we apply a meshless method, namely the generalized moving least squares approximation for discretizing the model in the space variables. The main benefits of the developed meshless method are: First, it does not need a background mesh (or triangulation) for constructing the approximation. Second, the obtained numerical results suggest that the proposed meshless method does not need to be combined with any adaptive technique for capturing the evolving interface between the tumor and the neighboring healthy tissue with proper accuracy. A semi-implicit backward differentiation formula of order 1 is used to deal with the time variable. The resulting fully discrete scheme obtained here is a linear system of algebraic equations per time step solved by an iterative scheme based on a Krylov subspace, namely the generalized minimal residual method with zero–fill incomplete lower–upper preconditioner. Finally, some simulation results based on estimated and experimental data taken from the literature are presented to show the process of prostate tumor growth in two-dimensional tissues, i.e., a square, a circle and non-convex domains using both uniform and quasi-uniform points. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
27. Optimal convergence analysis of the energy-preserving immersed weak Galerkin method for second-order hyperbolic interface problems in inhomogeneous media.
- Author
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Dehghan, Mehdi and Gharibi, Zeinab
- Subjects
- *
INHOMOGENEOUS materials , *GALERKIN methods , *STANDING waves - Abstract
This article reports our explorations for solving second-order hyperbolic interface problems by immersed weak Galerkin (IWG) method on interface independent meshes. The method presented here uses IWG functions for the discretization in spatial variable. The study includes the Newmark algorithm which has been used extensively in applications. The stability analysis based on the energy method is presented for semi-discrete and fully-discrete schemes under some conditions on parameters of the Newmark algorithm. In this work, we carried out a convergence analysis and obtained optimal a priori error estimates in both energy and L 2 norms for the semi-discrete and fully-discrete schemes under piecewise H 2 regularity assumption in space and some conditions on parameters of the Newmark algorithm. We demonstrate that the maximal error in the L 2 -norm error over a finite time interval converges optimally as O (h 2 + τ r (γ)) , where r (γ) = 1 if γ ≠ 1 / 2 , r (1 / 2) = 2 , h and τ are the mesh size and the time step, respectively. Numerical examples are provided to confirm theoretical findings and illustrate the efficiency of the method for standing and traveling waves. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
28. Double parameter splitting (DPS) iteration method for solving complex symmetric linear systems.
- Author
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Shirilord, Akbar and Dehghan, Mehdi
- Subjects
- *
LINEAR systems , *POSITIVE systems , *DECOMPOSITION method , *LINEAR equations , *MATRIX decomposition - Abstract
Complex linear equation systems appear in many branches of mathematical sciences and have many applications in science and engineering. Therefore, there have been developed many methods to solve them. Here a new method based on decomposition of coefficient matrix for the solution of a positive symmetric linear system is studied. Our method uses two parameters. We obtain a domain on which this new method converges without any condition. We will also discuss how to calculate optimal parameters. Moreover we will prove that there exists a region in R 2 such that DPS method is faster than CRI method (Wang et al. (2017) [48]). Finally some test problems will be given and simulation results will be reported to support the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
29. On the Hermitian and skew-Hermitian splitting-like iteration approach for solving complex continuous-time algebraic Riccati matrix equation.
- Author
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Dehghan, Mehdi and Shirilord, Akbar
- Subjects
- *
RICCATI equation , *APPLIED mathematics , *ALGORITHMS , *HERMITIAN forms , *EQUATIONS - Abstract
Some matrix equations are important in application and studying in various fields of sciences. One of them is continuous-time algebraic Riccati matrix equation that is an interesting subject for many researchers in engineering and sciences specially applied mathematics. In this work we try to apply Hermitian and skew-Hermitian splitting (HSS) like method to solve this equation. Then we analyze the convergence of the new iterative method in detail. Also we try to determine the optimum parameter in our method which minimizes the upper bounds of absolute error. In the end we test the new algorithm by solving some numerical examples. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
30. Interpolating Stabilized Element Free Galerkin Method for Neutral Delay Fractional Damped Diffusion-Wave Equation.
- Author
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Abbaszadeh, Mostafa, Dehghan, Mehdi, Zaky, Mahmoud A., and Hendy, Ahmed S.
- Subjects
- *
GALERKIN methods , *PARTIAL differential equations , *FINITE differences , *EQUATIONS , *CAPUTO fractional derivatives , *DELAY differential equations - Abstract
A numerical solution for neutral delay fractional order partial differential equations involving the Caputo fractional derivative is constructed. In line with this goal, the drift term and the time Caputo fractional derivative are discretized by a finite difference approximation. The energy method is used to investigate the rate of convergence and unconditional stability of the temporal discretization. The interpolation of moving Kriging technique is then used to approximate the space derivative, yielding a meshless numerical formulation. We conclude with some numerical experiments that validate the theoretical findings. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
31. A Galerkin meshless reproducing kernel particle method for numerical solution of neutral delay time-space distributed-order fractional damped diffusion-wave equation.
- Author
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Abbaszadeh, Mostafa and Dehghan, Mehdi
- Subjects
- *
FUNCTIONAL equations , *FUNCTIONAL differential equations , *FINITE differences , *ALGORITHMS , *GALERKIN methods , *DELAY differential equations , *WAVE equation - Abstract
The delay PDEs are called partial functional differential equations as their unknown solutions are used in these equations as functional arguments. On the other hand, a neutral delay PDE is an especial case when the equation depends on the derivative(s) of the solution at some past stage(s). The current paper concerns to find an accurate and robust numerical solution for solving neutral delay time-space distributed-order fractional damped diffusion-wave equation based on the Galerkin meshless method. The test and trial functions for the used Galerkin method are constructed from the shape functions of reproducing kernel particle method (RKPM). For this aim, time derivative is approximated by a finite difference formula with convergence order O (τ 3 − α) where 1 < α < 2. The stability and convergence of the time-discrete formulation are studied. For the next stage and to derive a fully-discrete scheme, we employ the Galerkin RKPM method. Moreover, the error estimate of the full-discrete scheme of new technique is discussed. Finally, an example is examined to check the proposed algorithm. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
32. Fourth-order alternating direction implicit difference scheme to simulate the space-time Riesz tempered fractional diffusion equation.
- Author
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Abbaszadeh, Mostafa and Dehghan, Mehdi
- Subjects
- *
HEAT equation , *FINITE differences , *COMPACT operators , *FINITE difference method , *SPACETIME , *COMPUTATIONAL complexity - Abstract
The current paper proposes a new high-order finite difference scheme with low computational complexity to solve the space-time fractional tempered diffusion equation. At the first stage, the time derivative has been approximated by a difference scheme with second-order accuracy. Furthermore, in the next step, a compact operator has been employed to discretize the space derivative with fourth-order accuracy. After deriving the time-discrete scheme, its stability is analysed. So, a suitable term is added to the main difference scheme. By adding this term, we could construct the main ADI scheme. In the final stage, the convergence order of the full-discrete scheme based upon the ADI formulation is proved. The convergence order of the constructed technique is O ((h x α) 4 + (h y β) 4 + τ 2). The numerical results show the efficiency of the new technique. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
33. Proper orthogonal decomposition Pascal polynomial-based method for solving Sobolev equation.
- Author
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Dehghan, Mehdi, Hooshyarfarzin, Baharak, and Abbaszadeh, Mostafa
- Subjects
- *
PROPER orthogonal decomposition , *ORTHOGONAL decompositions , *NUMERICAL solutions to partial differential equations , *MESHFREE methods , *NUMERICAL solutions to equations , *POLYNOMIAL approximation , *EQUATIONS - Abstract
Purpose: This study aims to use the polynomial approximation method based on the Pascal polynomial basis for obtaining the numerical solutions of partial differential equations. Moreover, this method does not require establishing grids in the computational domain. Design/methodology/approach: In this study, the authors present a meshfree method based on Pascal polynomial expansion for the numerical solution of the Sobolev equation. In general, Sobolev-type equations have several applications in physics and mechanical engineering. Findings: The authors use the Crank-Nicolson scheme to discrete the time variable and the Pascal polynomial-based (PPB) method for discretizing the spatial variables. But it is clear that increasing the value of the final time or number of time steps, will bear a lot of costs during numerical simulations. An important purpose of this paper is to reduce the execution time for applying the PPB method. To reach this aim, the proper orthogonal decomposition technique has been combined with the PPB method. Originality/value: The developed procedure is tested on various examples of one-dimensional, two-dimensional and three-dimensional versions of the governed equation on the rectangular and irregular domains to check its accuracy and validity. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
34. The localized RBF interpolation with its modifications for solving the incompressible two-phase fluid flows: A conservative Allen–Cahn–Navier–Stokes system.
- Author
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Mohammadi, Vahid, Dehghan, Mehdi, and Mesgarani, Hamid
- Subjects
- *
RADIAL basis functions , *FINITE element method , *TWO-phase flow , *FLUID flow , *INTERPOLATION , *STOKES equations - Abstract
In this research work, we apply a numerical scheme based on the first-order time integration approach combined with the modifications of the meshless approximation for solving the conservative Allen–Cahn–Navier–Stokes equations. More precisely, we first utilize a first-order time discretization for the Navier–Stokes equations and the time-splitting technique of order one for the dynamics of the phase-field variable. Besides, we use the local interpolation based on the Matérn radial function for spatial discretization. We should solve a Poisson equation with the proper boundary conditions to have the divergence-free property during the numerical algorithm. Accordingly, the applied numerical procedure could not give a stable and accurate solution. Instead, we solve a regularization system in a discrete form. To prevent the instability of the numerical solution concerning the convection term, a biharmonic term with a small coefficient based on the high-order hyperviscosity formulation has been added, which has been approximated by a scalable interpolation based on the combination of polyharmonic spline with polynomials (known as the PHS+poly). The obtained full-discrete problem is solved using the biconjugate gradient stabilized method considering a proper preconditioner. We investigate the potency of the numerical scheme by presenting some simulations via uniform, hexagonal, and quasi-uniform nodes on rectangular and irregular domains. Besides, we have compared the proposed meshless method with the standard finite element method due to the used CPU time. • A new numerical scheme has been developed to solve the incompressible two-phase fluid flows, i.e., a conservative Allen–Cahn–Navier–Stokes system. • A first-order Navier–Stokes solver with a time-splitting approach of order 1 has been applied to deal with the time variable. • Two modifications of the full-discrete problem related to the applied localized meshless approximation have been carried out. • The high-order hyperviscosity operator has been approximated via the scalable PHS+poly approximation. • The method preserves the discrete mass in time. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
35. A local meshless procedure to determine the unknown control parameter in the multi-dimensional inverse problems.
- Author
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Dehghan, Mehdi, Shafieeabyaneh, Nasim, and Abbaszadeh, Mostafa
- Subjects
- *
RADIAL basis functions , *FINITE difference method , *PROBLEM solving , *INVERSE problems - Abstract
This article is devoted to applying a local meshless method for specifying an unknown control parameter in one- and multi-dimensional inverse problems which are considered with a temperature overspecification condition at a specific point or an energy overspecification condition over the computational domain. Finding the unknowns in inverse problems is a challenge because these problems are modeled as non-classical parabolic problems and also have a significant role in describing physical phenomena of the real world. In this study, a combination of the meshless method of radial basis functions and finite difference method (called radial basis function-finite difference method) is used to solve inverse problems because this method has two important features. First it does not require any mesh generation. Consequently, it can be exerted to handle the high-dimensional inverse problems. Secondly, since this method is local, at each time step, a system with a sparse coefficient matrix is solved. Hence, the computational time and cost will be much low. Various numerical examples are examined, and also the accuracy and computational time required are presented. The numerical results indicate that the mentioned procedure is appropriate for the identification of the unknown parameter of inverse problems. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
36. Numerical analysis of locally conservative weak Galerkin dual-mixed finite element method for the time-dependent Poisson–Nernst–Planck system.
- Author
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Gharibi, Zeinab, Dehghan, Mehdi, and Abbaszadeh, Mostafa
- Subjects
- *
FINITE element method , *NUMERICAL analysis , *PHYSICAL laws , *CONSERVATION of mass , *ENERGY dissipation - Abstract
In this study, a linearized locally conservative scheme, based on using a weak Galerkin (WG)-mixed finite element method (MFEM), is developed for the Poisson–Nernst–Planck (PNP) system. In the dual-mixed formulation of the PNP equation, in addition to the three unknowns of concentrations p , n and the potential ψ , their fluxes, namely, σ p = ∇ p + p σ ψ and σ n = ∇ n − n σ ψ and σ ψ = ∇ ψ are introduced. These fluxes have an essential role in specifying the Debye layer and computing the electric current. The WG-MFEM considered here uses discontinuous functions to construct the approximation space. Also, a linearization scheme is employed to treat nonlinear terms. In the proposed method, the important physical laws of mass conservation and free energy dissipation are preserved without any restriction on the time step. Error estimates are developed and analyzed for both semi- and fully discrete WG-MFEM schemes. Furthermore, optimal error estimates (under adequate regularity assumptions on the solution) are derived. Several numerical results are provided and they demonstrate the efficiency of the proposed method and validate the convergence theorems. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
37. Convergence analysis of weak Galerkin flux-based mixed finite element method for solving singularly perturbed convection-diffusion-reaction problem.
- Author
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Gharibi, Zeinab and Dehghan, Mehdi
- Subjects
- *
FINITE element method , *TRANSPORT equation , *NUMERICAL analysis , *SINGULAR perturbations , *GALERKIN methods - Abstract
This article is assigned to the numerical analysis of a new weak Galerkin mixed-type finite element method for the diffusion-convection-reaction problem with singular perturbation. The variational form of the considered method compared to the existing methods consists of a single variational equation, where flux is the only unknown. Hence it is sufficient to provide a weak Galerkin approximation for the flux variable, and immediately approximation of the primal unknown can be obtained by a post-processing strategy. We prove the well-posedness, stability and convergence of the weak Galerkin scheme. Eventually, some numerical experiments to support the nice efficiency of the technique and analytical results are presented. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
38. Simulation of plane elastostatic equations of anisotropic functionally graded materials by integrated radial basis function based on finite difference approach.
- Author
-
Ebrahimijahan, Ali, Dehghan, Mehdi, and Abbaszadeh, Mostafa
- Subjects
- *
FUNCTIONALLY gradient materials , *FINITE differences , *FINITE difference method , *BOUNDARY value problems , *RADIAL basis functions , *EQUATIONS - Abstract
We present a method based on integrated radial basis function-finite difference for numerical solution of plane elastostatic equations which is a boundary value problem. The two-dimensional version of the governed equation is solved by the proposed method on various geometries such as the rectangular and irregular domains. In the current paper, one of our goals is to present an improved integrated radial basis function method based on the finite difference technique to approximate the second-order mixed partial derivatives with respect to x and y to get more accurate numerical results. Several examples are solved by applying integrated radial basis function based on finite difference method to check its accuracy and validity. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
39. The meshless local Petrov–Galerkin method based on moving Taylor polynomial approximation to investigate unsteady diffusion–convection problems of anisotropic functionally graded materials related to incompressible flow.
- Author
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Abbaszadeh, Mostafa, Dehghan, Mehdi, and Azis, Mohammad Ivan
- Subjects
- *
TAYLOR'S series , *POLYNOMIAL approximation , *INCOMPRESSIBLE flow , *FUNCTIONALLY gradient materials , *CONTINUOUS functions - Abstract
This paper concerns to a meshless local Petrov–Galerkin (MLPG) method for studying the unsteady diffusion–convection problems of anisotropic functionally graded materials. A new version of MLPG method based on the moving Taylor polynomial approximation is developed to discrete the spatial variable. Then, we obtain a system of ODEs which depends to the time variable. A strong stability preserving (SSP) Runge–Kutta idea is provided to solve the final ODEs with enough accuracy and stability. Also, the grading function which defines the variable elastic coefficient can be any types of continuous functions. The developed numerical formulation is applied for different examples of non-rectangular domains to check its accuracy. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
40. A divergence-free generalized moving least squares approximation with its application.
- Author
-
Mohammadi, Vahid and Dehghan, Mehdi
- Subjects
- *
LEAST squares , *POLYNOMIAL approximation , *VECTOR valued functions - Abstract
An approximation based on moving least squares for vector-valued functions satisfying the divergence-free property was employed in [67] by Trask, Maxey, and Hu. They changed the traditional polynomial bases in moving least squares approximation to the vector-valued polynomial basis functions satisfying the divergence-free property for constructing a new vector-valued approximation. In this paper, we adopt [67] , but we consider another approach, i.e., a direct method based on the generalized moving least squares (GMLS) approximation for vector-valued functions, which satisfy the divergence-free property. This GMLS approximation produces diffuse or uncertain derivatives for each component of the proposed vector-valued approximation satisfying the divergence-free property. Using the new approach, the linear functionals such as the derivatives act only on each row of the vector-valued polynomial basis functions, which reduces the computational cost comparing with the divergence-free moving least squares approximation. The pointwise error estimates of the presented approximation are obtained for the bounded sub-domains in R d (d ≥ 2) with an interior cone condition. Some numerical results are provided to confirm the obtained theoretical results. As an application, the Cahn-Hilliard-Hele-Shaw (CHHS) equation is solved numerically via a stabilized semi-implicit scheme in time that holds the mass conservative and energy dissipative properties and also the GMLS approximation in space. Besides, the developed vector-valued approximation has been used to compute the Leray-Helmholtz projection of the advective velocity. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
41. The Crank‐Nicolson/interpolating stabilized element‐free Galerkin method to investigate the fractional Galilei invariant advection‐diffusion equation.
- Author
-
Abbaszadeh, Mostafa and Dehghan, Mehdi
- Subjects
- *
ADVECTION-diffusion equations , *GALERKIN methods , *PARTIAL differential equations , *FRACTIONAL differential equations - Abstract
Recently, finding a stable and convergent numerical procedure to simulate the fractional partial differential equations (PDEs) is one of the interesting topics. Meanwhile, the fractional advection‐diffusion equation is a challenge model numerically and analytically. This paper develops a new meshless numerical procedure to simulate the fractional Galilei invariant advection‐diffusion equation. The fractional derivative is the Riemann‐Liouville fractional derivative sense. At the first stage, a difference scheme with the second‐order accuracy has been employed to get a semi‐discrete plan. After this procedure, the unconditional stability has been investigated, analytically. At the second stage, a meshless weak form based upon the interpolating stabilized element‐free Galerkin (ISEFG) method has been used to achieve a full‐discrete scheme. As for the full‐discrete scheme, the order of convergence is O(τ2+rm+1). Two examples are studied, and simulation results are reported to verify the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
42. Numerical and analytical investigations for solving the inverse tempered fractional diffusion equation via interpolating element-free Galerkin (IEFG) method.
- Author
-
Abbaszadeh, Mostafa and Dehghan, Mehdi
- Subjects
- *
HEAT equation , *LEAST squares - Abstract
This manuscript is devoted to analysis of a novel meshless numerical procedure for solving the inverse tempered fractional diffusion equation. The employed numerical technique is based on a modification of element-free Galerkin (EFG) method, and the shape functions of interpolating moving least squares approximation are utilized for ingredients of the test and trial functions. At the first stage, the time derivative is discretized by a Crank–Nicolson idea to derive a semi-discrete scheme. In the next stage, the space variable is approximated by the EFG procedure. The convergence rate and stability of the time-discrete formulation are analyzed. Furthermore, the error estimate of the full-discrete plan is discussed in detail. In the end, some numerical experiments are investigated to check the theoretical results and the efficiency of the developed technique. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
43. RBF‐ENO/WENO schemes with Lax–Wendroff type time discretizations for Hamilton–Jacobi equations.
- Author
-
Abedian, Rooholah and Dehghan, Mehdi
- Subjects
- *
HAMILTON-Jacobi equations , *RADIAL basis functions , *FINITE difference method - Abstract
In this research, a class of radial basis functions (RBFs) ENO/WENO schemes with a Lax–Wendroff time discretization procedure, named as RENO/RWENO‐LW, for solving Hamilton–Jacobi (H–J) equations is designed. Particularly the multi‐quadratic RBFs are used. These schemes enhance the local accuracy and convergence by locally optimizing the shape parameters. Comparing with the original WENO with Lax–Wendroff time discretization schemes of Qiu for HJ equations, the new schemes provide more accurate reconstructions and sharper solution profiles near strong discontinuous derivative. Also, the RENO/RWENO‐LW schemes are easy to implement in the existing original ENO/WENO code. Extensive numerical experiments are considered to verify the capability of the new schemes. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
44. Optimal uniform error estimates for moving least‐squares collocation with application to option pricing under jump‐diffusion processes.
- Author
-
Shirzadi, Mohammad, Dehghan, Mehdi, and Bastani, Ali Foroush
- Subjects
- *
ELLIPTIC differential equations , *INTEGRO-differential equations , *COLLOCATION methods , *FINANCIAL engineering , *ESTIMATES - Abstract
In this study, we derive optimal uniform error bounds for moving least‐squares (MLS) mesh‐free point collocation (also called finite point method) when applied to solve second‐order elliptic partial integro‐differential equations (PIDEs). In the special case of elliptic partial differential equations (PDEs), we show that our estimate improves the results of Cheng and Cheng (Appl. Numer. Math. 58 (2008), no. 6, 884–898) both in terms of the used error norm (here the uniform norm and there the discrete vector norm) and the obtained order of convergence. We then present optimal convergence rate estimates for second‐order elliptic PIDEs. We proceed by some numerical experiments dealing with elliptic PDEs that confirm the obtained theoretical results. The article concludes with numerical approximation of the linear parabolic PIDE arising from European option pricing problem under Merton's and Kou's jump‐diffusion models. The presented computational results (including the computation of option Greeks) and comparisons with other competing approaches suggest that the MLS collocation scheme is an efficient and reliable numerical method to solve elliptic and parabolic PIDEs arising from applied areas such as financial engineering. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
45. Numerical and theoretical discussions for solving nonlinear generalized Benjamin–Bona–Mahony–Burgers equation based on the Legendre spectral element method.
- Author
-
Dehghan, Mehdi, Shafieeabyaneh, Nasim, and Abbaszadeh, Mostafa
- Subjects
- *
APPLIED sciences , *FINITE differences , *EQUATIONS , *SPECTRAL element method , *PLASMA physics - Abstract
This article is devoted to solving numerically the nonlinear generalized Benjamin–Bona–Mahony–Burgers (GBBMB) equation that has several applications in physics and applied sciences. First, the time derivative is approximated by using a finite difference formula. Afterward, the stability and convergence analyses of the obtained time semi‐discrete are proven by applying the energy method. Also, it has been demonstrated that the convergence order in the temporal direction is O(dt). Second, a fully discrete formula is acquired by approximating the spatial derivatives via Legendre spectral element method. This method uses Lagrange polynomial based on Gauss–Legendre–Lobatto points. An error estimation is also given in detail for full discretization scheme. Ultimately, the GBBMB equation in the one‐ and two‐dimension is solved by using the proposed method. Also, the calculated solutions are compared with theoretical solutions and results obtained from other techniques in the literature. The accuracy and efficiency of the mentioned procedure are revealed by numerical samples. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
46. A POD-based reduced-order Crank-Nicolson/fourth-order alternating direction implicit (ADI) finite difference scheme for solving the two-dimensional distributed-order Riesz space-fractional diffusion equation.
- Author
-
Abbaszadeh, Mostafa and Dehghan, Mehdi
- Subjects
- *
ORTHOGONAL decompositions , *FINITE differences , *HEAT equation , *FINITE difference method , *PROPER orthogonal decomposition , *NONLINEAR equations , *NUMERICAL integration - Abstract
This paper introduces a high-order numerical procedure to solve the two-dimensional distributed-order Riesz space-fractional diffusion equation. In the proposed technique, first, a second-order numerical integration rule is employed to estimate the integral of the distributed-order Riesz space-fractional derivative. Then, the time derivative is discretized by a second-order difference scheme. Finally, the spatial direction is approximated by a difference formulation with fourth-order accuracy. The stability of the semi-discrete scheme is analyzed. We conclude that the difference between two consecutive time steps i.e. U i , j n − U i , j n − 1 is nearly zero when n → ∞. So, a suitable term is added to the main difference scheme as by adding this term we could derive the main ADI scheme. Furthermore, to reduce the used CPU time, we combine the fourth-order ADI formulation with the proper orthogonal decomposition method and then we gain a POD based reduced-order compact ADI finite difference plane. In the next, the convergence order of the fully discrete formulation has been investigated. The numerical results show the efficiency of new technique. It must be noted that the finite difference method is an effective and robust numerical technique for solving nonlinear equations that the ADI approach can be combined with it to improve the numerical simulations. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
47. Application of spectral element method for solving Sobolev equations with error estimation.
- Author
-
Dehghan, Mehdi, Shafieeabyaneh, Nasim, and Abbaszadeh, Mostafa
- Subjects
- *
SPECTRAL element method , *EQUATIONS , *FINITE differences , *APPLIED mechanics - Abstract
This paper is dedicated to numerically solving the Sobolev equations that have several applications in physics and mechanical engineering. First, the temporal derivative is discretized by the Crank-Nicolson finite difference technique to obtain a semi-discrete scheme in the temporal direction. Afterward, the stability and convergence analysis of the time semi-discrete scheme are proven by applying the energy method. It also implies that the convergence order in the temporal direction is O (d t 2). Second, a fully discrete formula has been acquired by discretizing the spatial derivatives via Legendre spectral element method (LSEM). This method applies the Lagrange polynomial based on the Gauss-Legendre-Lobatto (GLL) points. Moreover, an error estimation is given for the obtained fully discrete scheme. Eventually, the two-dimensional Sobolev equations are solved by using the proposed procedure. The accuracy and efficiency of the mentioned procedure are demonstrated by several numerical examples. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
48. Matrix multisplitting Picard-iterative method for solving generalized absolute value matrix equation.
- Author
-
Dehghan, Mehdi and Shirilord, Akbar
- Subjects
- *
ABSOLUTE value , *APPLIED mathematics , *LINEAR complementarity problem , *OPERATIONS research , *EQUATIONS , *MATRICES (Mathematics) - Abstract
The absolute value equation appears in various fields of applied mathematics such as operational research. Here we consider its generalized version AX + B | X | = C , where A , B , C ∈ C n × n are given, | X | = (| x i , j |) and X ∈ C n × n is an unknown matrix that must be determined. In this investigation, based on the Picard matrix splitting iteration method, we applied a matrix splitting method for solving it. We will see that under the condition σ min (A) > n σ max (| B |) , this method is convergent, where σ max (| B |) denotes the largest singular value of matrix | B | and σ min (A) denotes the smallest singular value of matrix A. Then we give some convergence theorems for our new method and analyze this procedure in detail. Then we consider a p -step iteration method for solving this equation and analyze this procedure. Numerical experiment results show the efficiency of the method. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
49. The stability study of numerical solution of Fredholm integral equations of the first kind with emphasis on its application in boundary elements method.
- Author
-
Hosseinzadeh, Hossein, Dehghan, Mehdi, and Sedaghatjoo, Zeynab
- Subjects
- *
BOUNDARY element methods , *NUMERICAL solutions to integral equations , *DIFFERENTIAL forms , *RADIAL basis functions , *PARTIAL differential equations , *HELMHOLTZ equation , *FREDHOLM equations , *INTEGRAL operators - Abstract
In this paper stability of numerical solution of Fredholm integral equation of the first kind is studied for radial basis kernels which possess positive Fourier transform. As a result, the equivalence relation between strong and weak forms of partial differential equations (PDEs) is proved for some special radial test functions. Also the stability of boundary elements method (BEM) is proved analytically for Laplace and Helmholtz equations by obtaining Fourier transform of singular fundamental solutions applied in BEM. Analytical result presented in this paper is an extension of stability idea of radial basis functions (RBFs) used to interpolate scattered data described by Wendland in [51]. Similar to the interpolation, it is proved here mathematically that integral operators which have radial kernels with positive Fourier transform are strictly positive definite. Thanks to the stability idea presented in [51] , a positive lower bound for eigenvalues of these integral operators is found here, explicitly. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
50. Isogeometric collocation method to simulate phase-field crystal model.
- Author
-
Masoumzadeh, Reza, Abbaszadeh, Mostafa, and Dehghan, Mehdi
- Subjects
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JACOBIAN matrices , *CRYSTAL models , *DIRECTIONAL derivatives , *FINITE differences , *MATHEMATICAL models - Abstract
Purpose: The purpose of this study is to develop a new numerical algorithm to simulate the phase-field model. Design/methodology/approach: First, the derivative of the temporal direction is discretized by a second-order linearized finite difference scheme where it conserves the energy stability of the mathematical model. Then, the isogeometric collocation (IGC) method is used to approximate the derivative of spacial direction. The IGC procedure can be applied on irregular physical domains. The IGC method is constructed based upon the nonuniform rational B-splines (NURBS). Each curve and surface can be approximated by the NURBS. Also, a map will be defined to project the physical domain to a simple computational domain. In this procedure, the partial derivatives will be transformed to the new domain by the Jacobian and Hessian matrices. According to the mentioned procedure, the first- and second-order differential matrices are built. Furthermore, the pseudo-spectral algorithm is used to derive the first- and second-order nodal differential matrices. In the end, the Greville Abscissae points are used to the collocation method. Findings: In the numerical experiments, the efficiency and accuracy of the proposed method are assessed through two examples, demonstrating its performance on both rectangular and nonrectangular domains. Originality/value: This research work introduces the IGC method as a simulation technique for the phase-field crystal model. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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