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The goal of this paper is to discover modern soliton solutions to long and short-wave interaction system by procedures called extended rational sine-cosine and rational sinh-cosh methods. We assume that the equation has a hypothetical soliton solutions. By reorganizing the resulting equations, we obtain a system of equations. Using Maple software, we get unknown coefficients in the system and writing them in the original equation, we obtain new solition solutions of the equation. The results show that the soliton solutions generated by the method for the long and short-wave interaction system are bright, kink type, bright periodic and dark solutions. We provided 3-D figures to illustrate the solutions. Computational results indicate that the method employed in this paper is superior than some other methods used in the literature to solve the same system equations.

The newest infection is a novel coronavirus named COVID-19, that initially appeared in December 2019, in Wuhan, China, and is still challenging to control. The main focus of this paper is to investigate a novel fractional-order mathematical model that explains the behavior of COVID-19 in Ethiopia. Within the proposed model, the entire population is divided into nine groups, each with its own set of parameters and initial values. A nonlinear system of fractional differential equations for the model is represented using Caputo fractional derivative. Legendre spectral collocation method is used to convert this system into an algebraic system of equations. An inexact Newton iterative method is used to solve the model system. The effective reproduction number ( R 0 ) is computed by the next-generation matrix approach. Positivity and boundedness, as well as the existence and uniqueness of solution, are all investigated. Both endemic and disease-free equilibrium points, as well as their stability, are carefully studied. We calculated the parameters and starting conditions (ICs) provided for our model using data from the Ethiopian Public Health Institute (EPHI) and the Ethiopian Ministry of Health from 22 June 2020 to 28 February 2021. The model parameters are determined using least squares curve fitting and MATLAB R2020a is used to run numerical results. The basic reproduction number is R 0 = 1.4575 . For this value, disease free equilibrium point is asymptotically unstable and endemic equilibrium point is asymptotically stable, both locally and globally.

In this work, an efficient novel technique, namely, the q -homotopy analysis transform method ( q -HATM) is applied to obtain analytical solutions for a system of time-fractional coupled nonlinear Schrodinger (TF-CNLS) equations with the time-fractional derivative taken in the Caputo sense. This system of equations incorporate nonlocality behaviours which cannot be modeled under the framework of classical calculus. With numerous important applications in nonlinear optics, it describes interactions between waves of different frequencies or the same frequency but belonging to different polarizations. We first establish existence and uniqueness of solutions for the considered time-fractional problem via a fixed point argument. To demonstrate the effectiveness and efficiency of the q − HATM, two cases each of two time-fractional problems are considered. One important feature of the q − HATM is that it provides reliable algorithms which can be used to generate easily computable solutions for the considered problems in the form of rapidly convergent series. Numerical simulation are provided to capture the behaviour of the state variables for distinct values of the fractional order parameter. The results demonstrate that the general response expression obtained by the q − HATM contains the fractional order parameter which can be varied to obtain other responses. Particularly, as this parameter approaches unity, the responses obtained for the considered fractional equations approaches that of the corresponding classical equations.

This work presents a novel scheme to couple the Discrete Element Method (DEM) and the Boundary Element Method (BEM) for the multi-scale modelling in the time domain. The DEM can model discontinuous material at micro scale very well, but it cannot represent infinite domains. Hence, coupling with the BEM is proposed. A formulation employing the DEM and BEM in different subdomains of the same body is presented. There is no overlap between the sub-domains, and the system of equations is derived based on strong equilibrium and compatibility conditions at the interface. The proposed coupling scheme is based on monolithic time integration. The conducted numerical experiments of one-dimensional wave propagation show excellent agreement with the analytical solution. Some spurious wave reflections are observed at the interface, but their effect is quantified and deemed not critical for infinite domains, which are of main interest. Even though the applications for one-dimensional wave propagation are of limited practical engineering interest, this work represents a significant theoretical breakthrough. This paper establishes a reference for future coupling schemes for two- and three-dimensional multi-scale analysis.

Chemotaxis is a fundamental mechanism of cells and organisms, which is responsible for attracting microbes to food, embryonic cells into developing tissues, or immune cells to infection sites. Mathematically chemotaxis is described by the Patlak–Keller–Segel model. This macroscopic system of equations is derived from the microscopic model when limiting behaviour is studied. However, on taking the limit and passing from the microscopic equations to the macroscopic equations, fluctuations are neglected. Perturbing the system by a Gaussian random field restitutes the inherent randomness of the system. This gives us the motivation to study the classical Patlak–Keller–Segel system perturbed by random processes. We study a stochastic version of the classical Patlak–Keller–Segel system under homogeneous Neumann boundary conditions on an interval O = [ 0 , 1 ] . In particular, let W 1 , W 2 be two time-homogeneous spatial Wiener processes over a filtered probability space A . Let u and v denote the cell density and concentration of the chemical signal. We investigate the coupled system { d u − ( r u Δ u − χ div ( u ∇ v ) ) d t = u ∘ d W 1 , d v − ( r v Δ v − α v ) d t = β u d t + v ∘ d W 2 , with initial conditions ( u ( 0 ) , v ( 0 ) ) = ( u 0 , v 0 ) . The positive terms r u and r v are the diffusivity of the cells and chemoattractant, respectively, the positive value χ is the chemotactic sensitivity, α ≥ 0 is the so-called damping constant. The noise is interpreted in the Stratonovich sense. Given T > 0 , we will prove the existence of a martingale solution on [ 0 , T ] .

The randomized extended Gauss-Seidel method is a popular representative among the iterative algorithm due to its simplicity for solving the inconsistent and consistent systems of linear equations, which builds the connection between the randomized Kaczmarz and Gauss-Seidel methods. In this work we develop a general version of the randomized extended Gauss-Seidel method, as well as some new iterative schemes. We prove that our algorithm can exponentially converge in expectation to the solutions of the consistent or inconsistent linear systems under two different sampling strategies. Numerical examples show that the proposed algorithm is feasible and effective, where the block method performs significantly better than the corresponding original form.

The Gauss Jordan method is used in this study to equalize chemical reactions using a system of linear equations. One of the most common topics in Chemistry is balancing chemical reaction equations. Students typically struggle with balancing the chemical equation and find it difficult to understand; sometimes teachers struggle as well to balance chemical equations. The results of the equation balancing comply with the law on the conservation of matter and confirm that the existing methods for balance of chemical equations do not contradict each other. To solve the mathematical problem, the Gauss-Jordan method was used. Any chemical reaction can be handled using this method by certain reactants and products. we can add several other features to the python application which can check whether the chemical equation is already balanced or not, we can also categorise the type of the chemical reaction.

The present article deals with the reflection and transmission of waves at an interface of piezoelectric (ALN & PZT-5T) half-spaces with microstructures. Unlike the classical piezoelectric material, in the considered media, five coupled waves (2 bulk and 3 surface) [namely the Quasi-longitudinal wave (QP), Quasi-transverse wave (QSV), Electric-acoustic wave (EA), P-type surface wave (SP) and S-type surface wave (SS)] are generated in response to an oblique incident Quasi plane wave. The main objective of this study is to investigate the influence of the characteristic length of microstructure, the inertial characteristic length and flexoelectric coefficients on the reflection and transmission coefficients. For this purpose, the linear algebraic equation, continuity conditions at the common interface are contracted. The linear system of equations comprising the reflection and transmission coefficients is derived, which is solved by using the Cramer's rule. The consequence of this electromechanical phenomenon may be advantageous in certain engineering applications that involve smart nano-composites.

International audience; Both compressible and incompressible porous medium models are used in the literature to describe the mechanical properties of living tissues. These two classes of models can be related using a stiff pressure law. In the incompressible limit, the compressible model generates a free boundary problem of Hele-Shaw type where incompressibility holds in the saturated phase. Here we consider the case with a nutrient. Then, a badly coupled system of equations describes the cell density number and the nutrient concentration. For that reason, the derivation of the free boundary (incompressible) limit was an open problem, in particular a difficulty is to establish the so-called complementarity relation which allows to recover the pressure using an elliptic equation. To establish the limit, we use two new ideas. The first idea, also used recently for related problems, is to extend the usual Aronson-Bénilan estimates in $L^\infty$ to an $L^2$ setting. The second idea is to derive a sharp uniform $L^4$ estimate on the pressure gradient, independently of space dimension.

In this paper, an effective numerical technique based on Lucas and Fibonacci polynomials coupled with finite differences is developed for the solution of nonlinear reaction–diffusion Brusselator system. The system arises in modeling of chemical processes such as enzymatic reactions, plasma and laser physics in multiple coupling between modes and in the formation of ozone by atomic oxygen via a triple collision. The proposed scheme first converts the problem to discrete form and then with a collocation approach to a system of linear equations which is easily solvable. Performance of the method is checked by solving one- and two-dimensional test problems. Validation of the results is examined in terms of L ∞ , L 2 and relative error L R norms. In case of no exact solution, quality of computed solution is examined for small value of diffusion coefficient η . It is observed that when 1 - ξ + β > 0 , the solutions converges to equilibrium points ( β , ξ / β ) .

K-nearest neighbor (KNN) based weighted multi-class twin support vector machines (KWMTSVM) is a novel multi-class classification method. In this paper, we propose a novel least squares version of KWMTSVM called LS-KWMTSVM by replacing the inequality constraints with equality constraints and minimized the slack variables using squares of 2-norm instead of conventional 1-norm. This simple modification leads to a very fast algorithm with much better results. The modified primal problems in the proposed LS-KWMTSVM solves only two systems of linear equations whereas two quadratic programming problems (QPPs) need to solve in KWMTSVM. The proposed LS-KWMTSVM, same as KWMTSVM, employed the weight matrix in the objective function to exploit the local information of the training samples. To exploit the inter class information, we use weight vectors in the constraints of the proposed LS-KWMTSVM. If any component of vectors is zero then the corresponding constraint is redundant and thus we can avoid it. Elimination of redundant constraints and solving a system of linear equations instead of QPPs makes the proposed LS-KWMTSVM more robust and faster than KWMTSVM. The proposed LS-KWMTSVM, commensurate as the KWMTSVM, all the training data points into a “1-versus-1-versus-rest” structure, and thus our LS-KWMTSVM generate ternary output {-1,0,+1} which helps to deal with imbalance datasets. Numerical experiments on several UCI and KEEL imbalance datasets(with high imbalance ratio) clearly indicate that the proposed LS-KWMTSVM has better classification accuracy compared with other baseline methods but with remarkably less computational time. This work was supported by Science and Engineering Research Board (SERB) under Early Career Research Award Grant No. ECR/2017/000053 and Ramanujan fellowship Grant No. SB/S2/ RJN-001/2016. This work was also supported by Council of Scientific & Industrial Research (CSIR), New Delhi, INDIA under Extra Mural Research (EMR) Scheme Grant No. 22(0751)/17/ EMR-II.

A meshfree method on fixed grids is devised for simulation of Poisson's equation on 3D image-based cellular materials. The non-boundary fitted discretization of such jagged voxel models of complex geometries is accomplished through embedding the micro-CT scan image in a Cartesian grid of nodes. The computational nodes inside the solid voxels are found by a simple point-in-membership test. Using a set of modified singular functions around the voids, along with the library-rational-exponential basis functions (EBFs) satisfying the governing differential equation, an enriched spatial solution is locally constructed on a generic computational cloud/cell (GCC) of nodes containing the voids. Within each GCC, the boundary conditions are satisfied through a weighted least-squares approximation. Finally, by establishing point-wise compatibility between the solutions of the GCCs a well-conditioned small linear system of equations is resulted. The results are compared with those of the finite element method (FEM) using extremely fine meshes with an excessive number of nodes.

A new fast multipole boundary element method (FM-BEM) is proposed to analyze 2-D elastostatic problems by using linear and three-node quadratic elements. The use of higher-order elements in BEM analysis results in more complex forms of the integrands, in which the direct Gaussian quadrature is difficult to calculate the singular and nearly singular integrals. Herein, the complex notation is first introduced to simplify all integral formulations (including the near-field integrals) in FM-BEM for 2-D elasticity. In direct evaluation of the near-field integrals, the nearly singular integrals on linear elements are calculated by the analytic scheme, and those on quadratic elements are evaluated by a robust semi-analytical algorithm. Numerical examples show that the present method possesses higher accuracy than the FM-BEM with constant elements. The computed efficiency of FM-BEM with higher order elements for analyzing large scale problems is still O(N), where N is the number of linear system of equations. In particular, the proposed FM-BEM is available for solving thin structures.

This manuscript brings the derivation of influence functions for a three-dimensional full-space under bilinearly-distributed time-harmonic loads. The differential equations describing the medium are decomposed in terms of uncoupled vector fields. A double Fourier transform allows the system of equations to be solved algebraically in the transformed space, where the bilinear-loading boundary conditions are imposed. The resulting displacement and stress solutions are presented in terms of double improper integrals to be evaluated numerically. The manuscript brings selected results from the evaluation of these solutions. These influence functions can be used in boundary element models of elastodynamic problems to yield computationally-efficient solutions and improved representation of sharply-varying contact traction fields.

In this paper, an improved node moving technique was developed for adaptive solution of some flow problems. Collocated Discrete Least Squares Meshless (CDLSM) method was applied for simulation. This method is a truly meshless method and also enjoys the symmetric and positive-definite property. Then, an improved node moving technique was developed based on the spring analogy. In this mechanism, each node was assumed to be connected to its neighboring nodes via some virtual springs. Then, higher values of stiffness were allocated to the springs located at higher error areas. In the previously published works, the corresponding system of equations was assumed to be linear. In this work, first, it was shown that this assumption may lead to inaccurate results in some cases and then, an improved method was proposed considering the nonlinear nature of the system of equations. Some numerical examples were used to illustrate the ability of the proposed node moving technique for some simple examples and benchmark problems in the Computational Fluid Dynamics (CFD) context. The Results of the study showed a considerable improvement in comparison with those of previously published works.

A meshless geometric multigrid (GMG) method based on a node-coarsening algorithm is proposed in the context of finite element method (FEM) with unstructured grids consisting of linear elements. Unlike the existing GMG methods, the present method does not require the generation of a sequence of coarse grids so that all the problems related to coarse-grid generation can be eliminated. Instead, only the sets of nodes in coarse levels are constructed for multigrid computation from the finest grid by using the node-coarsening algorithm that can be employed for any kind of a 2D/3D unstructured grid as well as a hybrid grid on the finest level. The implementation of the present coarsening algorithm is simple in the sense that the boundary information of the finest grid is not required. A searching algorithm to calculate the area/volume-shape function of the finite element method is also proposed to derive an operator for linear interpolation of multigrid computation. We have successfully validated the proposed method for various 2D/3D benchmark problems by showing that the elapsed time of the present GMG method is linearly proportional to the number of unknowns of a linear system of equations. We have also confirmed that the proposed method is nearly as efficient as the grid-based MG method based on a sequence of coarse grids in terms of CPU time. Lastly, we have successfully validated the meshless GMG method by solving an elliptic equation formulated by the finite volume method on a complicated 3D geometry filled with a hybrid unstructured mesh.

In this paper, we focus on solving a class of two-by-two block complex system of linear equations arising from finite element discretization of a distributed optimal control constrained by a time-harmonic parabolic equations. We propose a new iterative method for solving the obtained system. In each iteration of the method a two-by-two block system of linear equations with real coefficient matrix should be solved. We solve this system inexactly using the generalized minimal residual (GMRES) and the Chebyshev acceleration methods in conjunction with the real-valued preconditioned square block (PRESB) preconditioner. The convergence and spectral properties of the method are discussed. Numerical results in 2-dimensional case are presented to demonstrate the efficiency of the method.

Mesh refinement may give rise to variable dimension system of linear equations in adaptive analysis. In each adaptive step, the matrix decomposition methods are adopted to solve linear algebraic equations, which is obviously time-consuming and computationally expensive for large-scale problems. A new algorithm for solving a variable dimension system of linear equations is proposed in this paper, which combines matrix deformation with the Sherman-Morrison formula. This method not only retains the advantages of the Sherman-Morrison formula, but also is particularly suitable for adaptive analysis. Compared with the matrix decomposition method, our method has outstanding performance in promoting computational efficiency. Several numerical examples are used to illustrate the superiority of the proposed method when applying to solve 2D elasticity problems.

The localized method of fundamental solutions is a recent domain-type meshless collocation method with the fundamental solutions of governing equations as the radial basis functions. This approach forms a sparse system matrix and has a higher efficiency than the traditional method of fundamental solutions. In this paper, a modified version of the localized method of fundamental solutions is developed for bending analysis of simply supported and clamped thin elastic plates. Some auxiliary nodes on the boundary are firstly introduced to provide additional weight coefficients, which contribute to the construction of the determined system of equations and avoid the over-determined equation system of the localized method of fundamental solutions for bending analysis of thin elastic plates. Several numerical experiments with simply supported or clamped boundary conditions are provided, and numerical results are in good agreement with the analytical or COMSOL Multiphysics solutions.

For a block upper triangular matrix, a necessary and sufficient condition has been given to let it be the sum of block upper rectangular matrices satisfying certain rank constraints; see [12] . The proof involves elements from Integer Programming (totally unimodular systems of equations playing a role in particular) and employs Farkas' Lemma. The linear space of block upper triangular matrices can be viewed as being determined by a special pattern of zeros. The present paper is concerned with the question whether the decomposition result can be extended to situations where other, less restrictive, zero patterns play a role. It is shown that such generalizations do indeed hold for certain directed graphs determining the pattern of zeros. The graphs in question are what will be called L -free. This notion is akin to other graph theoretical concepts available in the literature, among them the one of being N -free in the sense of [16] .

Given the critical role played by natural gas in providing electricity, heat, and other essential services, better models are needed to understand the dynamics of natural gas networks during extreme events. This paper aims at establishing appropriate and fast simulation models to capture the slow dynamics of linepack depletion for ideal isothermal natural gas pipeline networks. Instead of solving partial differential equations (PDE) on a large scale, three alternative implicit ordinary differential equation (ODE) simulation techniques are derived and discussed. The first one is commonly used in the literature with a slack node assumption. We show that the system of equations associated with this model is degenerate when flux injections are controlled (i.e. specified) at all nodes. To recover regularity under such a condition, two novel implicit ODE models are proposed, both with different techniques for specifying boundary conditions. They are easy to derive and efficient to simulate with standard ODE solvers. More importantly, they present useful frameworks for analyzing how networks respond to system-wide mass flux imbalances. These techniques offer different alternatives for simulating system dynamics based on how sources and loads are chosen to be modeled, and they are all proven to be regular (non-degenerate) in tree-structured networks. These proposed techniques are all tested on the 20-node Belgium network. The simulation results show that the conventional model with the slack node assumption cannot effectively capture linepack depletion under long term system-wide mass flux imbalance, while the proposed models can characterize the network behavior until the linepack is completely depleted., Comment: 40 pages, accepted for publication in the Applied Mathematical Modeling (AMM) Journal

In this work, phase-field modeling of hydraulic fractures in porous media is extended towards a Global–Local approach. Therein, the failure behavior is solely analyzed in a (small) local domain. In the surrounding medium, a simplified and linearized system of equations is solved. Both domains are coupled with Robin-type interface conditions. The fractures inside the local domain are allowed to propagate and consequently, both subdomains change within time. Here, a predictor–corrector strategy is adopted, in which the local domain is dynamically adjusted to the current fracture pattern. The resulting framework is algorithmically described in detail and substantiated with some numerical tests.

The occurrence of buckling during cooling of an injection-molded fiber-reinforced plastic (FRP) object is a complex phenomenon which can result in unacceptably large deformations. The present work focuses on the thermal buckling of injection-molded thin disks, as they are representative for the local behavior in parts of a more complex object. An analytical approach based on the Foppl-von Karman theory for thin elastic plates is presented and used to analyze its buckling stability and initial post-buckling behavior. The system of equations is formulated in a coordinate-free approach for a thin anisotropic linear thermo-elastic body with residual thermal stresses. Large out-of-plane displacements are assumed. The particular case of a free disk with cylindrical orthotropy and uniform thermo-elastic properties through its thickness is considered. A general technique to study the bifurcation in the solutions (i.e., buckling) based on a perturbation approach and the Fourier-Galerkin method to determine post-buckling deflections is proposed. Experimental and numerical results available in the literature are used as validation cases. The results of the present model show qualitative agreement with the experimental data. It is found that the periodicity of the first buckling mode for orthotropic disks is fully determined by the ratio between the elastic moduli in the tangential and radial directions. The predictions are also in excellent quantitative agreement with the numerical results of FEM models for the same cases. The effect of the disk thickness on the buckling temperature and warpage magnitude is correctly captured.

The truncated singular value decomposition (TSVD) is a popular method for solving linear discrete ill-posed problems with a small to moderately sized matrix A. This method replaces the matrix A by the closest matrix A k of low rank k, and then computes the minimal norm solution of the linear system of equations with a rank-deficient matrix so obtained. The modified TSVD (MTSVD) method improves the TSVD method, by replacing A by a matrix that is closer to A than A k in a unitarily invariant matrix norm and has the same spectral condition number as A k . Approximations of the SVD of a large matrix A can be computed quite efficiently by using a randomized SVD (RSVD) method. This paper presents a novel modified truncated randomized singular value decomposition (MTRSVD) method for computing approximate solutions to large-scale linear discrete ill-posed problems. The rank, k, is determined with the aid of the discrepancy principle, but other techniques for selecting a suitable rank also can be used. Numerical examples illustrate the effectiveness of the proposed method and compare it to the truncated RSVD method.

This manuscript is related to compute approximate solutions for a class of fractional distributed order differential equations (FDODEs). The corresponding derivative of fractional order is taken in Caputo sense. The adopted numerical scheme is based on collocation method together with Haar wavelet. This is an efficient numerical algorithm which convert the consider problem to a some system of equations of algebraic type. Upon utilizing the Broyden tools the obtain nonlinear system is solved for the intended numerical results. Further in case of linear equations the obtained system is solved by Gauss elimination procedure. To strengthen our results we testify several problems corresponding to different collocation points. We present also maximum absolute and root mean square errors. The graphical presentations are also given.

This paper proposes two complex-valued zeroing neural network (Cv-ZNN) models for solving dynamic complex time-variant linear equations. The models involve two complex-valued nonlinear processing methods and adapt two real-valued activation functions . The convergence and robustness of the two Cv-ZNN models are analyzed comprehensively. Firstly, the convergence discussion shows that the corresponding stable error can rapidly converge to zero in finite time, with the upper limit of constriction time calculated. The upper limit of stable error is successfully obtained when the model implementation error is injected into the Cv-ZNN models, which displays their superior robustness. Besides, the results of numerical simulations reveal that the Cv-ZNN models effectively address the time-variant linear system of equations over complex field, even in noisy environments . The simulation results also demonstrate that two novel nonlinear activation functions perform much better than the Linear, Power, Bipolar Sigmoid Power-Sigmoid and Sign-bi-power activation functions.

The aim of this contribution is to develop a nonlinear formulation for modelling the large deformation of elastic curved beams made of fully incompressible hyperelastic materials. The basic idea is to apply the incompressibility constraint besides the plane stress assumption in the directions perpendicular to the centreline of the beam. Accordingly, a system of three nonlinear algebraic equations for the hydrostatic pressure as well as normal strain components in the beam cross section are obtained. By solving the system of equations, either analytically or numerically, it is possible to propose constitutive equations for the force and moment resultants present in the formulation. The main advantage of this strategy is that it allows for using three-dimensional incompressible constitutive equations. Due to highly nonlinear nature of the differential equations, a Total Lagrangian finite element formulation is developed. Performance and accuracy of the formulation are investigated through several numerical examples.

A multi-scale method based on a combination of the boundary element method (BEM) and peridynamics (PD) was developed to model crack propagation problems in two-dimensional (2D) elastic bodies. The special feature of this method is that it can take full advantage of both the BEM and PD to achieve a higher level of computational efficiency. Based on the scale of the structure and the crack location, the considered domain can be divided into non-cracked and cracked domains. The BEM is employed in the non-cracked domain, while the PD is applied in the cracked domain. This can reduce the dimension by one in the non-cracked domain for improving the modeling efficiency. A stiffness equation of the bond-based PD is established by using Taylor’s series expansion for the bond stretch and applied to simulate the cracked domain. The PD approach can automatically model the initiation and propagation of a crack. A coupling model using shared nodes is constructed by introducing the BEM nodes on the interface at the same location as the PD material points. With the continuity of displacements and equilibrium of tractions at the interface, a combined system of equations is obtained by merging the stiffness and force matrix from each domain. For test problems, the deformation and crack propagation in 2D elastic bodies subjected to quasi-static loads were analyzed. The numerical results clearly demonstrate the accuracy and efficiency of the proposed method for crack problems based on coupling the BEM and PD.

For solving nonlinear elliptic equations given in arbitrary plane domains, the meshless methods of radial-polynomial and Pascal-polynomial are easy to programming, which are employed as the bases to expand the solution. After a simple collocation technique, we can derive nonlinear equations to determine the expansion coefficients. We adopt a splitting parameter to split the nonlinear term into two nonlinear parts, which are separately placed on both sides of the nonlinear elliptic equation. Then, a new linearization technique is used to treat the nonlinear part on the left-hand side. In each iteration, the linear system of equations is regularized by the multiple-scale technique. The proposed methods converge very fast to obtain very accurate numerical solutions, which confirm the validity of the presented novel splitting and linearizing technique (NSLT) to solve nonlinear elliptic equations in arbitrary plane domains.

In this paper, our focus is on the multidimensional mathematical physics models. We employ the sub-equation method to obtain new exact solutions to the proposed strongly nonlinear time-fractional differential equations of conformable type. The models considered are generalized Benjamin equation, modified generalized multidimensional Kadomtsev–Petviashvili (KP) equations, modified generalized multidimensional KP–BBM equation and the variant Boussinesq system of equations. We also introduced a new modified generalized multidimensional KP type equation and its exact solutions. As the order of fractional derivative tends to one, the obtain exact solutions by the proposed method reduce to classical solutions. We successfully established varieties of soliton type solutions. The results obtained affirm that sub-equation method is an efficient and powerful technique for analytic solutions of nonlinear fractional partial differential equations.

In this paper we present a general extrapolated elliptic regularity result for second order differential operators in divergence form on fractional Sobolev-type spaces of negative order Xs-1,qD(Ω) for s > 0 small, including mixed boundary conditions and with a fully nonsmooth geometry of Ω and the Dirichlet boundary part D. We expect the result to find applications in the analysis of nonlinear parabolic equations, in particular for quasilinear problems or when treating coupled systems of equations. To demonstrate the usefulness of our result, we give a new proof of local-in-time existence and uniqueness for the van Roosbroeck system for semiconductor devices which is much simpler than already established proofs.

In this paper the problem of mutual interaction of the incompressible turbulent fluid flow with a flexibly supported airfoil is addressed. The problem is mathematically described with the aid of the Reynolds averaged Navier–Stokes equations. The turbulence is modeled with the aid of k - k L - ω transition model. In order to treat the motion of the domain the system of equations are written in the arbitrary Lagrangian–Eulerian. The structure motion is governed by a nonlinear system of ordinary differential equations, where two degrees of freedom for the rotation and the vertical translation are used. The problem is numerically discretized with the aid of the finite volume method using the linear reconstruction technique. The numerical results are shown.

This paper presents a local non-singular knot method (LNKM) to accurately solve the large-scale acoustic problems in complicated geometries. The LNKM is a domain-type meshless collocation method, which relies only on scattered nodes. Firstly, a series of subdomains corresponding to every nodes can be searched based on the Euclidean distance between nodes. To each subdomain, a small linear system can be yielded by using the non-singular general solutions of Helmholtz-type equations. Secondly, the unknown variables at every nodes can be explicitly expressed by the function values at their corresponding supporting nodes. Finally, a large sparse system of linear equations is formed and solved to obtain the numerical solutions of physical quantities at every nodes. The proposed LNKM is mathematically simple, numerically accurate, and more applicable to the large-scale computation. Four numerical examples conform its effectiveness and accuracy for the large-scale computation of Helmholtz-type equations in complicated geometries.

In general, common cause failures (CCFs) have been modeled with the assumption that components within the same group are symmetric. This assumption reduces the number of parameters required for the CCF probability estimation and allows us to use a parametric model, such as the alpha factor model. Although there are various asymmetric conditions in nuclear power plants (NPPs) to be addressed, the traditional CCF models are limited to symmetric conditions. Therefore, this paper proposes the copula-based CCF model to deal with asymmetric as well as symmetric CCFs. Once a joint distribution between the components is constructed using copulas, the proposed model is able to provide the probability of common cause basic events (CCBEs) by formulating a system of equations without symmetry assumptions. In addition, Bayesian inferences for the parameters of the marginal and copula distributions are introduced and Markov Chain Monte Carlo (MCMC) algorithms are employed to sample from the posterior distribution. Three example cases using simulated data, including asymmetry conditions in total failure probabilities and/or dependencies, are illustrated. Consequently, the copula-based CCF model provides appropriate estimates of CCFs for asymmetric conditions. This paper also discusses the limitations and notes on the proposed method.

The Gauss-Seidel and Kaczmarz methods are two classic iteration methods for solving systems of linear equations, which operate in column and row spaces, respectively. Utilizing the connections between these two methods and imitating the exact analysis of the mean-squared error for the randomized Kaczmarz method, we conduct an exact closed-form formula for the mean-squared residual of the iterate generated by the randomized Gauss-Seidel method. Based on this new formula, we further estimate an upper bound for the convergence rate of the randomized Gauss-Seidel method. Theoretical analysis and numerical experiments show that this bound measurably improves the existing ones. Moreover, these theoretical results are also extended to the more general extrapolated randomized Gauss-Seidel method.

The article considers approximation of solution sets to indeterminate systems for non-linear equations. We developed a method to obtain inner and outer approximations of such sets. The method uses interval analysis techniques and relies on the Krawczyk operator and its modification based on the Baumann bicentered interval extension. We developed software that efficiently constructs and visualizes the computed approximations of the solution sets for systems of non-linear equations. The software is implemented in Python programming language and is available for free download and use. Finding the solution set of indeterminate systems of equations finds at least one important application in practice: bounding the workspace of a robotic manipulator. We perform experiments for the 2-RPR robot and evaluate the tightness of the obtained approximations. As expected the Krawczyk bicentered method noticeably improves the quality of approximations as compared with the classical Krawczyk operator.

The subject of the paper is a simply supported circular plate with symmetrically varying mechanical properties in the thickness direction. In extreme cases of this variability, the plate becomes a single-layer or three-layer structure. The main goal of the study is to develop a mathematical description of both a single-layer and three-layer structure using one formula. The mathematical model also includes all intermediate states of the plate in the axisymmetric bending and buckling problems with consideration of the shear effect. To this purpose, two dimensionless functions closely related to the variability of mechanical properties were introduced and the nonlinear hypothesis of deformation of the straight line normal to the plate neutral surface is assumed. Based on the principle of stationary potential energy two differential equations of equilibrium are obtained. The system of equations is analytically solved. The deflections and intensity of critical loads for example plates are derived and presented in Tables.

Uniform exponential stability of a linear delayed integro-differential vector equation x ˙ ( t ) = ∑ k = 1 m A k ( t ) x ( h k ( t ) ) + ∑ k = 1 l ∫ g k ( t ) t P k ( t , s ) x ( s ) d s , t ∈ [ 0 , ∞ ) , where x = ( x 1 , … , x n ) T is an unknown vector-function, is considered. It is assumed that m, l are positive integers, matrices A k , P k and delays h k , g k are Lebesgue measurable. The main result is of an explicit type, depending on all delays, and its proof is based on an a priori estimation of solutions, a Bohl-Perron type result, and utilization of the matrix measure. As particular cases, it includes ( 2 m + l − 1 ) mutually different sufficient conditions. Some of them are formulated separately as corollaries. Advantages of derived explicit results over the existing ones are demonstrated on examples and open problems are proposed as well.

The fluidized bed theory can be complex, especially for heterogeneous flow descriptions with bypasses, generally resulting in large differential and algebraic systems of equations. Consequently, their applicability to model-based process control is limited. In this work, a simplified homogeneous model for pharmaceutical batch drying is derived from the two-phase fluidization theory using physical insights and simplifying assumptions, reducing more than a hundred equations to five. A nonlinear model predictive controller with an internal model structure is designed from these simple equations, showing the simplicity of tuning and implementation. Parameters of the model are calibrated through nonlinear grey-box identification using pilot scale experimental data. The validation demonstrates that the proposed simplifications do not impair the ability to replicate the process dynamics adequately with experimental conditions similar to the ones used for calibration. Closed-loop results in simulation attest the robustness of this control strategy.

Stable multistep schemes based on Caputo fractional derivative approximation are presented for solving 1D and 2D nonlinear fractional model arising from dielectric media. We approximate Caputo fractional derivatives in time with a multistep scheme of order O ( τ 3 − α ) & O ( τ 3 − β ) , 1 β α 2 , spatial Laplacian operator with a central difference scheme, and nonlinear source term g ( B ) by using Taylor series. The discretization of the problem results in a linear system of equations that is tridiagonal and penta-diagonal for 1D and 2D case, respectively. The unique solvability and unconditional stability are derived for both cases. The convergence of schemes is established with the help of optimal error bounds. Further, we establish that the order of convergence for 1D case is O ( τ 3 − α + τ 3 − β + h 2 ) and for 2D case is O ( τ 3 − α + τ 3 − β + h x 2 + h y 2 ) . Moreover, the stability of our schemes are verified numerically by adding some linear and nonlinear noisy inputs. Finally, four test functions are investigated to show the effectiveness and stability of our schemes. The method is simple, easy to implement, and yields very accurate results.

The paper shows how to use the multidimensional interval RDM arithmetic to solve an interval system of linear equations. The main problems that arise when solving this type of equations using the standard interval arithmetic are discussed and the most important flaws of this type of solutions are listed. Next, an example is shown to demonstrate how to correctly solve systems of interval equations by using multidimensional interval RDM arithmetic.

The work aims at the investigation of dynamic stability of an asymmetric sandwich beam varying exponentially and resting on a Pasternak foundation. The stiffness of the Pasternak foundation varies linearly with displacement. The extended Hamilton’s principle is used to derive the governing equations of motion and associated boundary conditions. The system of equations is non-dimensionalized. Using Galerkin’s method, a set of Hill’s equations are obtained. The dynamic stability is analyzed using Saito-otomi conditions. The effects of elastic foundation parameter, thermal gradient, shear parameter, taper parameter, core-loss factor and modulus ratios on dynamic stability of the system is studied for pinned–pinned boundary condition. The results are represented graphically using suitable MATLAB program.

This paper presents applications of boundary element method (BEM) in engineering to numerically analyse the Telegraph equation in two dimensions for two cases. For this purpose, the mathematical model developed is called D-BEM, uses a time independent fundamental solution and the Finite Difference Method is combined to BEM to approximate the derivative time terms and the Gauss Quadrature is used for calculation of domain integrals. The solution of the system of equations is based on the Gaussian elimination method, being verified the stability of the proposed formulation with a correlation level greater than 0.9, estimated from the dispersion coefficient R2 between the analytical solution and the D-BEM solution. The D-BEM formulation was used to solve a variation of the Telegraph equation, called the Reaction-telegraph equation, in which the value of the delay parameter λ was varied, obtaining interesting results for the population density simulating the dynamics of population invasion, similar to the problem of tumour growth or cancer progression in four examples analysed. The numerical results of the Telegraph equation, as well as the entire formulation of the boundary elements are presented below.

We consider the system of equations describing motion of compressible viscoelastic fluids in a whole space. We investigate the large time behavior of solutions around a motionless state, and obtain the L p decay estimates of solutions for 1 p ≤ ∞ , provided that the initial data is sufficiently close to the motionless state. In addition, we clarify the diffusion wave phenomena caused by the sound wave and the elastic shear wave.

We apply a novel computation approach to determine the numerical solution of variable-order fractional optimal control problems. The dynamic constraint of these problems is considered with variable-order (VO) fractional derivatives. The numerical procedure is based on the Genocchi polynomials. To carry out the suggested technique, the Genocchi operational matrix of the VO fractional integral is calculated. By utilizing the mentioned operational matrix and the Gauss–Legendre quadrature rule, a system of equations is made. Then, the produced system is simply solved. Some numerical examples are given. The obtained results show the applicability and usefulness of the numerical technique.

The Parametric High-Fidelity-Generalized-Method-of-Cells (PHFGMC) micromechanical model is extended to include a local cohesive formulation for simulating discontinuities in multiphase composites. The nonlinear PHFGMC governing equations are obtained from a virtual work principle applied to both the local and far-field variables and solved using a new incremental-iterative formulation. The nonlinearity stems from the introduction of the cohesive zones in conjunction with damage evolution. The proposed formulation yields an overall symmetric system of equations and enables the implementation of advanced traction-separation laws available in the literature similar to the displacement-based finite element method (FEM). Two cohesive zone models are integrated with the PHFGMC with a unique nonlinear formulation. These are the non-potential based model proposed by Camanho and Davila (CD) and the potential based Park–Paulino–Roesler (PPR) model. This article presents comparisons with FEM and the applications of this modeling approach. The predictions of damage initiation and evolution, as well as the overall stress–strain response of the cohesive PHFGMC are shown to be in good agreement with those from a finite element analysis for various configurations and loading patterns.

In this paper, a meshless BEM based on the radial integration method is developed to solve transient non-homogeneous convection-diffusion problem with spatially variable velocity and time-dependent source term. The Green function served as the fundamental solution is adopted to derive the boundary domain integral equation about the normalized field quantity. The two-point backward finite difference technique is utilized to discretize the time-dependent terms in the integral equation, which results in that the final integral equation formulation is only related with the normalized field quantity at the current time and has three domain integrals. Both two domain integrals regarding the normalized field quantity at the current and previous times are transformed into boundary integrals by using radial integration method and radial basis function approximation. For domain integral about the source term being known function of time and coordinate, radial basis functions approximation is still adopted to make the transformed boundary integral be evaluated only once, not at each time level. A pure boundary element algorithm with boundary-only discretization and internal points is established and the system of equations is assembled like the corresponding steady problem. Four numerical examples are given to demonstrate the accuracy and effectiveness of the present method.

This work is related to the numerical investigation of two species chemotaxis models in both one and two dimensions. This system in its simpler form is a set of non-linear partial differential equations. First equation represents the dynamics of cell densities and the others are responsible for chemoattractant concentration. The system is known to produce delta-type singularities in finite time. Therefore, a less order accurate numerical schemes are not capable of handling such a complicated spiky solution. A Space Time Conservation Element and Solution Element (CE/SE) method is proposed for solving such systems. This method has distinct attributes which includes treatment of space and time variables in unified fashion. Furthermore, numerical diffusion is reduced on the basis that both conserved quantities and derivatives are anonymous. Due to this feature, diffusion in numerical schemes is inherently reduced. The Nessyahu–Tadmor (NT) central scheme is also implemented for validation and comparison. Several one and two dimensional case studies are carried out. Numerical results obtained through proposed numerical method are analyzed in comparison to Nessyahu–Tadmor (NT) central scheme. Moreover, effects of the different densities and concentration functions are explored to see generic applicability of the proposed method for current system of equations. It is observed that the CE/SE and NT central method has the capability to capture the delta type singularities in the solution profile. However, CE/SE resolves the solution peaks better as compare to NT central scheme.

The present paper investigates the role that the radial component of heat conduction plays in a cone-plate viscometer. The cone and the disk may be taken as stationary or in action; both co-rotating or counter-rotating. Hydrodynamic and thermal fields are resolved by means of computationally simulating the resulting systems of equations. Upon working out the well-documented velocity field in the gap region from the similarity governing equations, the rates of heat transfer at both surfaces are calculated from an amended energy equation by further adding radial diffusive terms, which were missing in the previous data published in the literature. It is shown that addition of such physical streamwise heat conduction terms into the energy equation much influences the well-known results of heat transfer rates, particularly when the conical gap section is not small. The missing heat transfer rates pertaining to the cone wall are also presented here. In particular, it is demonstrated that the best cooling of cone-disk apparatus can be achieved for a rotating disk with a stationary cone, provided that the wall temperatures are kept as uniformly constant. The critical power index of passage from cooling to heating is determined to be 1.54492.

In this paper, we propose a generalized Newton method for solving a class of discrete-time linear complementarity systems consisting of a system of linear equations and a linear complementarity constraints with a Z-matrix. We obtain a complete characterization of the least element solution of a linear complementarity problem with a Z-matrix that a solution is the least element solution if and only if the principal submatrix corresponding to the nonzero components of the solution is an M-matrix. We present a Newton method for solving a linear complementarity problem with a Z-matrix. We propose a generalized Newton method for solving the discrete-time linear complementarity system where the linear complementarity problem constraint is solved by the proposed Newton method. Under suitable conditions, we show that the generalized Newton method converges globally and finds a solution in finitely many iterations. Preliminary numerical results show the efficiency of the proposed method.