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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]

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]

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]

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]

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]

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]

In this paper, we focus on the kernel-based solution of high-dimensional elliptic PDEs and propose an efficient algorithm to compute a trustable value of the shape parameter for a given positive definite kernel. The proposed strategy is based on the norm-minimal generalized interpolation problem coming from the optimality of radial basis function interpolation. Also, the effectiveness of the suggested algorithm is assessed by solving parabolic partial integro-differential equations arising in option pricing theory when the dynamic of the underlying asset is driven by a Lévy process. The performance of the proposed algorithm is evaluated by some high-dimensional PDEs on regular and irregular computational domains. Indeed, the numerical results obtained with European put options under Merton's model show the trustability of the suggested optimal shape parameter. [ABSTRACT FROM AUTHOR]

The coupled advection-dominated diffusion-reaction equations which arise in the prevention of groundwater contamination problem are approximated by the compact local integrated radial basis function (CLIRBF) method. To efficiently solve the resulting nonlinear system of advection-diffusion equations, we use the integrated radial basis function (IRBF) for discretizing the spatial variables. Afterwards, the system of ordinary differential equations (ODEs) obtained is discretized by the method of lines (MOL). MOL is a general way of viewing a partial differential equation (PDE) as a system of ordinary differential equations (ODE). The efficient fourth-order exponential time differencing Runge-Kutta (ETD-RK4) formula is utilized for solving this system. The main aim of this paper is to show that the integrated radial basis method based on the local form can be exerted for solving the coupled non-linear advection-diffusion-reaction system. The numerical tests are provided to illustrate its validity and accuracy. [ABSTRACT FROM AUTHOR]

Applying mathematical models to simulate dynamic biological processes has been a common practice for a long time. In recent decades, cancer research has also adopted this approach to understand how cancer cell populations grow and spread. This study focuses on a mathematical model that uses a system of PDEs to explain the time-dependent reaction–diffusion interaction among cancer cells, extracellular matrix, and matrix degradation enzymes. We use a computational method that involves the discrete Galerkin technique by employing local radial basis functions (LRBFs) as its basis to approximate the behavior of cancer cells as they grow and invade neighboring healthy tissues. This novel approach employs a smaller set of nodes to approximate the solution, instead of considering all data in the given domain of the cancer growth model. Utilizing locally supported radial basis functions, this method significantly reduces the computational volume required, in contrast to globally supported radial basis functions. Finally, we provide experimental examples to validate and illustrate the effectiveness of this new scheme in modeling the growth and behavior of cancer cells at different stages. [ABSTRACT FROM AUTHOR]

We apply a numerical scheme based on a meshless method in space and an explicit exponential Runge-Kutta in time for the solution of the damped Kuramoto–Sivashinsky equation in two-dimensional spaces. The proposed meshless method is radial basis function-generated finite difference, which approximates the derivatives of the unknown function with respect to the spatial variables by a linear combination of the function values at given points in the domain and weights. Also, in this approach there is no need a mesh or triangulation for approximation. For each point, the weights are computed separately in its local sub-domain by solving a small radial basis function interpolant. Besides, a numerical algorithm based on singular value decomposition of the local radial basis function interpolation matrix [59] is applied to find the suitable shape parameter for each interpolation problem. We also consider an explicit time discretization based on exponential Runge–Kutta scheme such that its stability region is bigger than the classical form of Runge-Kutta method. Some numerical simulations are provided on the square, circular and annular domains to show the capability of the numerical scheme proposed here. [ABSTRACT FROM AUTHOR]

The current paper is an improvement of the developed technique in Shu et al. (2005). The proposed improvement is to reduce the used CPU time for employing the local radial basis functions-differential quadrature (LRBF-DQ) method. To this end, the proper orthogonal decomposition technique has been combined with the LRBF-DQ technique. For checking the ability of the new procedure, the compressible Euler equation is solved. This equation has been classified in category of system of advection–diffusion equations. Moreover, several test problems are given that show the acceptable accuracy and efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

In the current investigation, we develop an efficient truly meshless technique for solving two models in optic and laser engineering i.e. Klein-Gordon-Zakharov and Schrödinger/Gross-Pitaevskii equations in one- two- and three-dimensional cases. The employed meshless is the upwind local radial basis functions-differential quadrature (LRBF-DQ) technique. The spacial direction is discretized using the LRBF-DQ method and also to obtain high-order numerical results, the fourth-order exponential time differencing Runge-Kutta method (ETDRK4) planned by Liang et al. [37] is applied to discrete the temporal direction. To show the efficiency of the proposed method, we solve the mentioned models on some complex shaped domains. Moreover, several examples are given and simulation results show the acceptable accuracy and efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this research, we investigate the numerical solution of nonlinear Schrödinger equations in two and three dimensions. The numerical meshless method which will be used here is RBF-FD technique. The main advantage of this method is the approximation of the required derivatives based on finite difference technique at each local-support domain as Ω i . At each Ω i , we require to solve a small linear system of algebraic equations with a conditionally positive definite matrix of order 1 (interpolation matrix). This scheme is efficient and its computational cost is same as the moving least squares (MLS) approximation. A challengeable issue is choosing suitable shape parameter for interpolation matrix in this way. In order to overcome this matter, an algorithm which was established by Sarra (2012), will be applied. This algorithm computes the condition number of the local interpolation matrix using the singular value decomposition (SVD) for obtaining the smallest and largest singular values of that matrix. Moreover, an explicit method based on Runge–Kutta formula of fourth-order accuracy will be applied for approximating the time variable. It also decreases the computational costs at each time step since we will not solve a nonlinear system. On the other hand, to compare RBF-FD method with another meshless technique, the moving kriging least squares (MKLS) approximation is considered for the studied model. Our results demonstrate the ability of the present approach for solving the applicable model which is investigated in the current research work. [ABSTRACT FROM AUTHOR]

The collocation technique based on the radial basis functions (RBFs) method is simple and efficient for solving a wide area of problems. But the mentioned technique is poor for solving problems that have shock (advection problems) or the discontinuous initial condition. The local RBFs collocation technique is a meshless method based on the strong form. The use of local collocation RBFs method overcomes the mentioned important issue. In the current paper, based on the proposed idea in Wang (2015) [54], we consider a linear combination of shape functions of local radial basis functions collocation method and moving Kriging interpolation technique. For showing the efficiency of new technique, some multi-dimensional problems such as Cahn-Hilliard, Swift-Hohenberg and phase field crystal equations have been chosen. Moreover, several test problems are given that show the acceptable accuracy and efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

The aim of this paper is to make a comparative study of some high order methods for the numerical solution of a non-classical one-dimensional two-phase Stefan problem. The moving boundary is captured explicitly via boundary immobilization method. The Chebyshev and Legendre spectral collocation methods as high order mesh-based techniques and some radial basis function (RBF) collocation techniques as high order meshless methods are used for spatial discretization. The considered Stefan problem has two stages: one before the extinction time ( 0 ≤ t ≤ t m ) and one after the extinction time ( t m ≤ t ) . For this particular model there exists a closed form solution for the former stage but there is no analytical solution for the latter one. Numerical results show that RBF-QR method can attain the accuracy of spectral methods when implemented on Chebyshev grid. The high order accuracy for the two stages shows the superiority of the proposed methods in comparison to the previous works. [ABSTRACT FROM AUTHOR]

Predicting the antitumor responses of cytotoxic T-lymphocytes (i.e., CD8 + T cells) can be accomplished by numerical simulations of a mathematical model of tumor-immune cell interactions to control tumor growth in cancer therapy. Cytotoxic T lymphocytes are considering as the main effector cells to target tumor cells that are in a dormant avascular state through the chemotactic mechanism and also during the cytotoxic activity of immune cells. We study a dynamical model that is formulated based on the nonlinear partial differential equations describing the interactions between tumor-infiltrating cytotoxic lymphocytes, tumor cells, TICL-tumor cell complexes, inactivated tumor-infiltrating cytotoxic lymphocytes, dead tumor cells, and chemokine. Here, a new meshless method, i.e., the direct radial basis function partition of unity (RBF-PU) technique for the spatial discretization and a semi-implicit backward differentiation formula of order 1 for the temporal discretization are employed. The proposed meshless technique is a generalized form of the original RBF-PU method such that the computational cost of computing the derivatives could be reduced in comparison with its standard form. Besides, the method depends only on the location of nodes located on the domain of the problem for the approximation, where both types of radial functions, i.e., a conditionally positive definite function, i.e., the polyharmonic splines and a positive definite function, namely Matérn function have been used The obtained fully discrete difference scheme based on the new spatial approximation is solved using an iterative method, i.e., the generalized minimal residual method with a zero-fill incomplete lower–upper preconditioner. Finally, we present some simulation results, in which we conclude that the tumor's growth entirely depends on CTLs activation, and this behavior can be predicted by simulating the proposed mathematical model, which may have a significant role in improving cancer immunotherapy. Besides, the developed numerical method is more accurate than the classical finite element method for solving the studied model as shown in one example. [ABSTRACT FROM AUTHOR]

Human immunodeficiency virus (HIV) infection may cause death by damaging some of the patient's vital organs through weakening the body's immune system, including CD 4 + T cells. In the meantime, mathematical models can be useful in dealing with this deadly virus by providing effective strategies based on the examination of different infection states. The major aim pursued in this paper is to present a computational algorithm for solving nonlinear systems of ordinary and partial differential equations resulting from the HIV infection models of CD 4 + T cells. The offered method is developed according to the use of local radial basis functions (LRBFs) as shape functions in the discrete collocation scheme. The new technique approximates the solution by a small set of nodes instead of all points located in the domain where the HIV mathematical model is given. Thus the presented method uses less computing volume compared to the globally supported radial basis functions, and as a result, its algorithm can be quickly run on a computer with relatively low memory. The computational efficiency of the scheme is studied by several test examples. [ABSTRACT FROM AUTHOR]

In this paper, a numerical technique is proposed for solving the stochastic advection–diffusion equations. Firstly, using the finite difference scheme, we transform the stochastic advection–diffusion equations into elliptic stochastic partial differential equations (SPDEs). Then the method of radial basis functions (RBFs) based on pseudospectral (PS) approach has been used to approximate the resulting elliptic SPDEs. In this study, we have used generalized inverse multiquadrics (GIMQ) RBFs, to approximate functions in the presented method. The main advantage of the proposed method over traditional numerical approaches is directly simulating the noise terms at the collocation points in each time step. To confirm the accuracy of the new approach and to show the performance of the selected RBFs, four examples are presented in one, two and three dimensions in regular and irregular domains. For test problems the statistical moments such as mean, variance and standard deviation are computed. [ABSTRACT FROM AUTHOR]

Radial basis function (RBF) approximation is an extremely powerful tool for solving various types of partial differential equations, since the method is meshless and can be spectrally accurate. A perceived practical obstacle is that the interpolation matrix becomes increasingly ill-conditioned as the RBF shape parameter becomes small, corresponding to flat RBFs. In this paper, the authors solve the one and two-dimensional time-dependent coupled sine-Gordon equations using RBFs collocation and RBF-QR methods and show how one can overcome the ill-conditioning of coefficient matrix for the small shape parameters using RBF-QR method. The main aim of the current paper is to show that the meshless techniques based on the collocation methods are also suitable for solving the system of coupled nonlinear equations especially sine-Gordon equation. Several test problems are employed and results of numerical experiments are presented and also are compared with analytical solutions. The obtained results confirm the acceptable accuracy of the new methods. [ABSTRACT FROM AUTHOR]

In this paper three numerical techniques are proposed for solving the nonlinear sinh-Gordon equation. Firstly, we obtain a time discrete scheme then we use the radial basis functions (RBFs) collocation based on Kansa׳s approach, RBF-pseudospectral (PS) technique and moving least squares (MLS) methods to approximate the spatial derivatives. The aim of this paper is to show that the meshless methods based on the RBFs using collocation approach and MLS are suitable for the treatment of the nonlinear partial differential equations and also we compare the mentioned methods in terms of condition number of coefficient matrix and absolute value of error. Also, several test problems are given that show the acceptable accuracy and efficiency of the proposed schemes. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the numerical solution of the Cahn–Hilliard (CH) equation in one, two and three-dimensions. We will apply two different meshless methods based on radial basis functions (RBFs). The first method is globally radial basis functions (GRBFs) and the second method is based on radial basis functions differential quadrature (RBFs-DQ) idea. In RBFs-DQ, the derivative value of function with respect to a point is directly approximated by a linear combination of all functional values in the global domain. The main aim of this method is the determination of weight coefficients. GRBFs replace the function approximation into the partial differential equation directly. Also, the coefficients matrix which arises from GRBFs is very ill-conditioned. The use of RBFs-DQ leads to the improvement of the ill-conditioning of interpolation matrix RBFs. The boundary conditions of the mentioned problem are Neumann. Thus, we use DQ method directly on the boundary conditions, which easily implements RBFs-DQ on the irregular points and regions. Here, we concentrate on Multiquadrics ( MQ ) as a radial function for approximating the solution of the mentioned equation. As we know this radial function depends on a constant parameter called shape parameter. The RBFs-DQ can be implemented in a parallel environment to reduce the computational time. Moreover, to obtain the error of two techniques with respect to the spatial domain, a predictor–corrector scheme will be applied. Finally, the numerical results show that the proposed methods are appropriate to solve the one, two and three-dimensional Cahn-Hilliard (CH) equations. [ABSTRACT FROM AUTHOR]

In this paper, we propose a numerical method for the solution of time fractional nonlinear sine-Gordon equation that appears extensively in classical lattice dynamics in the continuum media limit and Klein–Gordon equation which arises in physics. In this method we first approximate the time fractional derivative of the mentioned equations by a scheme of order O ( τ 3 − α ) , 1 < α < 2 then we will use the Kansa approach to approximate the spatial derivatives. We solve the two-dimensional version of these equations using the method presented in this paper on different domains such as rectangular and non-rectangular domains. Also, we prove the unconditional stability and convergence of the time discrete scheme. We show that convergence order of the time discrete scheme is O ( τ ) . We solve these fractional PDEs on different non-rectangular domains. The aim of this paper is to show that the meshless method based on the radial basis functions and collocation approach is also suitable for the treatment of the nonlinear time fractional PDEs. The results of numerical experiments are compared with analytical solutions to confirm the accuracy and efficiency of the presented scheme. [ABSTRACT FROM AUTHOR]