Spectral Meshless Radial Point Interpolation Research Articles

Numerical investigation based on a local meshless radial point interpolation for solving coupled nonlinear reaction-diffusion system

In the present paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to the solution of pattern formation in nonlinear reaction-diffusion systems. Firstly, we obtain a time discrete scheme by approximating the time derivative via a finite difference formula, then we use the SMRPI approach to approximate the spatial derivatives. This method is based on a combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which act as basis functions in the frame of SMRPI. In the current work, the thin plate splines (TPS) are used as the basis functions and in order to eliminate the nonlinearity, a simple predictor-corrector (P-C) scheme is performed. The effect of parameters and conditions are studied by considering the well known Brusselator model. Two test problems are solved and numerical simulations are reported which confirm the efficiency of the proposed scheme.

An efficient meshless radial point collocation method for nonlinear p-Laplacian equation

This paper considered the spectral meshless radial point interpolation (SMRPI) method to unravel for the nonlinear p-Laplacian equation with mixed Dirichlet and Neumann boundary conditions. Through this assessment, which includes meshless methods and collocation techniques based on radial point interpolation, we construct the shape functions, with the Kronecker delta function property, as basis functions in the framework of spectral collocation methods. Studies in this regard require one to evaluate the high-order derivatives without any kind of integration locally over the small quadrature domains. Finally, some examples are given to illustrate the low computing costs and high enough accuracy and efficiency of this method to solve a p-Laplacian equation and it would be of great help to fulfill the implementation related to the element-free Galerkin (EFG) method. Both the SMRPI and the EFG methods have been compared by similar numerical examples to show their application in strongly nonlinear problems.

Turing models in the biological pattern formation through spectral meshless radial point interpolation approach

In the present paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to the solution of pattern formation in nonlinear reaction diffusion systems. Firstly, we obtain a time discrete scheme by approximating the time derivative via a finite difference formula, then we use the SMRPI approach to approximate the spatial derivatives. This method is based on a combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which act as basis functions in the frame of SMRPI. In the current work, to eliminate the nonlinearity, a simple predictor–corrector (P–C) scheme is performed. The effect of parameters and conditions are studied by considering the well-known Schnakenberg model.

Two-dimensional capillary formation model in tumor angiogenesis problem through spectral meshless radial point interpolation

In this paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to a mathematical model for two-dimensional capillary formation model in tumor angiogenesis problem. This is a natural continuation of capillary formation in tumor angiogenesis (Shivanian and Jafarabadi in Eng Comput 34:603–619, 2018), where the capillary (1D problem) has been considered. The mathematical model describes the progression of tumor angiogenic factor in a unit square space domain, namely the extracellular matrix. First, we obtain a time discrete scheme by approximating time derivative via a finite difference formula, and then, we use the SMRPI approach to approximate the spatial derivatives. This approach is based on a combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which act as basis functions in the frame of SMRPI. Because of non-availability of the exact solution, we consider two strategies for checking the stability of time difference scheme and for survey the convergence of the fully discrete scheme. The obtained numerical results show that the SMRPI provides high accuracy and efficiency with respect to the other classical methods in the literature.

Numerical simulation of nonlinear coupled Burgers’ equation through meshless radial point interpolation method

In present paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to the solution of nonlinear coupled Burgers’ equation in two dimensions. Firstly, we obtain a time discrete scheme by approximating time derivative via a finite difference formula, then we use the SMRPI approach to approximate the spatial derivatives. This method is based on a combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which act as basis functions in the frame of SMRPI. In the current work, the thin plate splines (TPS) are used as the basis functions and in order to eliminate the nonlinearity, a simple predictor-corrector (P-C) scheme is performed. The aim of this paper is to show that the SMRPI method is suitable for the treatment of nonlinear coupled Burgers’ equation. With regard to test problems that have not exact solutions, we consider two strategies for checking the stability of time difference scheme and for survey the convergence of the fully discrete scheme. The results of numerical experiments confirm the accuracy and efficiency of the presented scheme.

Time fractional modified anomalous sub-diffusion equation with a nonlinear source term through locally applied meshless radial point interpolation

In this paper, an alternative approach of spectral meshless radial point interpolation (SMRPI) is applied to the modified anomalous fractional sub-diffusion equation with a nonlinear source term in one and two dimensions. The time fractional derivative is described in the Riemann–Liouville sense. The applied approach is based on a combination of meshless methods and the spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct the shape functions which act as basis functions in the frame of the SMRPI. It is proved that the scheme is unconditionally stable with respect to the time variable in [Formula: see text] and convergent with the order of convergence [Formula: see text], [Formula: see text]. In this work, the thin plate splines (TPS) are used as the radial basis functions. In order to eliminate the nonlinearity, a simple predictor–corrector (P–C) scheme is used. The results of numerical experiments are compared to the analytical solutions in order to confirm the accuracy and the efficiency of the presented scheme.

The numerical solution for the time-fractional inverse problem of diffusion equation

In this study, the spectral meshless radial point interpolation (SMRPI) technique is applied to the Cauchy problem of two-dimensional fractional diffusion equation. We obtain the unknown data on the inner boundary when overspecified boundary data is imposed on the outer boundary. The SMRPI is based on a combination of meshfree methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which act as basis functions in the frame of SMRPI. Here, similar to other meshless methods, localization in SMRPI can reduce the ill-posedness of the Cauchy problem. However, it does not require to use regularization algorithms and therefore reduces computational time. Two numerical examples, are tested to show that the SMRPI can overcome the ill-posedness of the Cauchy problem and has acceptable accuracy. Also, by adding some large perturbations, the proposed method is still stable.

An improved meshless algorithm for a kind of fractional cable problem with error estimate

The present paper is devoted to the development of a kind of spectral meshless radial point interpolation (SMRPI) technique for solving fractional cable equation in one and two dimensional cases. The time fractional derivative is described in the Riemann–Liouville sense. The applied approach is based on a combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which act as basis functions in the frame of SMRPI. It is proved the scheme is unconditional stable with respect to the time variable in H1 and convergent by the order of convergence O(δtγ), 0 < γ < 1. In the current work, the thin plate splines (TPS) are used as the basis functions. The results of numerical experiments are compared with analytical solution to confirm the accuracy and efficiency of the presented scheme.

The spectral meshless radial point interpolation method for solving an inverse source problem of the time-fractional diffusion equation

In this paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to the inverse source problem of time-fractional diffusion equation in two dimensions. The missing solely time-dependent source is recovered from an additional integral measurement. The applied approach is based on a combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct the shape functions which act as basis functions in the frame of SMRPI. Firstly, we use a difference scheme for the fractional derivative to discretize the governing equation, and we obtain a finite difference scheme with respect to time. Then we use the SMRPI approach to approximate the spatial derivatives. Also, it is proved that the scheme is unconditionally stable with respect to time in space H1. Consequently, when the input data is contaminated with noise, we use the Tikhonov regularization method in order to obtain a stable solution. Numerical results show that the solution is accurate for exact data and stable for noisy data.

An inverse problem of identifying the control function in two and three-dimensional parabolic equations through the spectral meshless radial point interpolation

This study proposes a kind of spectral meshless radial point interpolation (SMRPI) for solving two and three-dimensional parabolic inverse problems on regular and irregular domains. The SMRPI is developed for identifying the control parameter which satisfies the semilinear time-dependent two and three-dimensional diffusion equation with both integral overspecialization and overspecialization at a point in the spatial domain. This method is based on a combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions are used to construct shape functions which act as basis functions in the frame of SMRPI. It is proved that the scheme is stable with respect to the time variable in H1 and convergent by the order of convergence O(δt). The results of numerical experiments are compared to analytical solutions to confirm the accuracy and efficiency of the presented scheme.

Analysis of the spectral meshless radial point interpolation for solving fractional reaction–subdiffusion equation

The present paper is devoted to the development of spectral meshless radial point interpolation (SMRPI) technique for solving fractional reaction–subdiffusion equation in one and two dimensional cases. The time fractional derivative is described in the Riemann–Liouville sense. The applied approach is based on a combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which act as basis functions in the frame of SMRPI. It is proved that the scheme is unconditionally stable with respect to the time variable in H 1 and we show convergence order of the time discrete scheme is O ( δ t α ) , 0 < α < 1 . In the current work, the thin plate splines (TPS) are used to construct the basis functions. The results of numerical experiments are compared with analytical solutions to confirm the accuracy and efficiency of the presented scheme.

An improved meshless method for solving two- and three-dimensional coupled Klein–Gordon–Schrödinger equations on scattered data of general-shaped domains

In present paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to obtain the solution of two- and three-dimensional coupled Klein–Gordon–Schrodinger (KGS) equations. First, we obtain a time discrete scheme by approximating time derivative via a finite difference formula, then we use the SMRPI approach to approximate the spatial derivatives. This approach is based on erudite combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which play as basis functions in the frame of SMRPI. In the current work, the polyharmonic splines are used as the basis functions and to eliminate the nonlinearity, a simple predictor–corrector (P–C) scheme is performed. The aim of this paper is to show that the SMRPI procedure is suitable for the treatment of the KGS equations. The results of numerical experiments are compared with analytical solution and stability analysis is checked to confirm the accuracy and efficiency of the presented scheme.

Capillary formation in tumor angiogenesis through meshless weak and strong local radial point interpolation

This article is devoted to showing the efficiency and accuracy of two numerical procedures for the numerical solution of the capillary formation in tumor angiogenesis. A meshless local radial point interpolation (MLRPI) scheme based on Galerkin weak form is analyzed. The reason for choosing MLRPI approach is that it does not require any background integration cells; instead, integrations are implemented over local quadrature domains, which are further simplified for reducing the complication of computation using regular and simple shape. The spectral meshless radial point interpolation (SMRPI) method does not need any integration. The radial point interpolation method is proposed to construct shape functions for MLRPI and basis functions for SMRPI. A weak formulation with a Heaviside step function transforms the set of governing equations into local integral equations on local subdomains in MLRPI, whereas the operational matrices convert easily the governing equations (even high order) into a linear system of equations in SMRPI. The finite difference technique is employed to approximate the time derivatives. A stability analysis technique based on the eigenvalue of the matrices is employed to check the stability condition for the presented meshless methods. Furthermore, the methods are successfully applied to solve the problem with high values of the cell diffusion constant which many of the available methods did not investigate for solving these cases. Because of non-availability of the exact solution, we consider two strategies for checking the stability of time difference scheme and for surveying the convergence of the fully discrete scheme. From numerical results, it is considerable that the present methods provide the similar results. Moreover, less CPU time and less computational complexity are two advantages of the SMRPI method with respect to the MLRPI method.

An efficient numerical technique for solution of two-dimensional cubic nonlinear Schrödinger equation with error analysis

In this paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to the solution of two-dimensional cubic nonlinear Schrödinger equations. Firstly, we obtain a time discrete scheme by approximating time derivative via a finite difference formula, then we use the SMRPI approach to approximate the spatial derivatives. This method is based on a combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which act as basis functions in frame of SMRPI. In the current work, the thin plate splines (TPS) are used as the basis functions and in order to eliminate the nonlinearity, a simple predictor-corrector (P-C) scheme is performed. We prove that the time discrete scheme is unconditionally stable and convergent in time variable using the energy method. We show that convergence order of the time discrete scheme is O(δt). The aim of this paper is to show that the SMRPI method is suitable for the treatment of the nonlinear Schrödinger equations. Also, the SMRPI has less computational complexity than the other methods that have already solved this problem. The results of numerical experiments are compared with analytical solution to confirm the accuracy and efficiency of the presented scheme.

Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivatives: a stable scheme based on spectral meshless radial point interpolation

In the present paper, a spectral meshless radial point interpolation (SMRPI) technique is applied to solve the Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivative in two dimensional case. The time fractional derivative is described in the Riemann–Liouville sense. The applied approach is based on combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which act as basis functions in the frame of SMRPI. It is proved the scheme is unconditionally stable with respect to the time variable in \(H^1\) and convergent with the order of convergence \(\mathcal {O}(\delta t^{1+\beta })\), \(0<\beta <1\). In the current work, the thin plate splines (TPS) are used as the basis functions. The results of numerical experiments are compared with analytical solutions to confirm the accuracy and efficiency of the presented scheme. Two numerical examples show that the SMRPI has reliable accuracy in general shape domains.

An improved spectral meshless radial point interpolation for a class of time-dependent fractional integral equations: 2D fractional evolution equation

In this paper, a kind of spectral meshless radial point interpolation (SMRPI) technique is applied to the fractional evolution equation in two-dimensional for arbitrary fractional order. The applied approach is based on erudite combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which play as basis functions in the frame of SMRPI. In the current work, the thin plate splines (TPS) are used as the basis functions, and also the discretization technique employed is patterned after an idea of Ch. Lubich. It is proved the scheme is unconditional stable with respect to the time variable in L 2 -norm and convergent by the order of convergence O ( δ t 1 + α ) , 0 < α < 1 . The results of numerical experiments are compared with analytical solution to confirm the accuracy and efficiency of the presented scheme. Three numerical examples, show that the SMRPI has reliable accuracy in irregular domains.

Influence of size effect on flapwise vibration behavior of rotary microbeam and its analysis through spectral meshless radial point interpolation

In this paper, by utilizing the modified couple stress theory, influence of size effect on flapwise vibration behavior of rotary microbeam is analyzed based on Bernoulli–Euler beam model. It should be mentioned that this theory contains the size effect by considering the material length scale parameter unlike the classical continuum theories. Governing equation and boundary conditions are derived by using Hamilton’s principle. Governing equation is solved for cantilever and propped cantilever boundary conditions by applying spectral meshless radial point interpolation (SMRPI) method. SMRPI is based on meshless methods and benefits from spectral collocation ideas. The point interpolation method with the help of radial basis functions is proposed to construct shape functions which have Kronecker delta function property. Evaluation of high order derivatives is possible by constructing and using operational matrices. The computational cost of the method is modest due to using strong form equation and collocation approach.

Numerical solution of two-dimensional inverse force function in the wave equation with nonlocal boundary conditions

In this paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to the inverse time-dependent force function in the wave equation on regular and irregular domains. The SMRPI is developed for identifying the force function which satisfies in the wave equation subject to the integral overspecification over a portion of the spatial domain or to the overspecification at a point in the spatial domain. This method is based on erudite combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which play as basis functions in the frame of SMRPI. Since the problem is known to be ill-posed, Thikhonov regularization strategy is employed to solve effectively the discrete ill-posed resultant linear system. Three numerical examples are tested to show that numerical results are accurate for exact data and stable with noisy data.

Error and stability analysis of numerical solution for the time fractional nonlinear Schrödinger equation on scattered data of general‐shaped domains

In present work, a kind of spectral meshless radial point interpolation (SMRPI) technique is applied to the time fractional nonlinear Schrödinger equation in regular and irregular domains. The applied approach is based on erudite combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which play as basis functions in the frame of SMRPI. It is proved the scheme is unconditionally stable with respect to the time variable in and also convergent by the order of convergence , . In the current work, the thin plate spline are used as the basis functions and to eliminate the nonlinearity, a simple predictor‐corrector (P‐C) scheme is performed. It is shown that the SMRPI solution, as a complex function, is suitable one for the time fractional nonlinear Schrödinger equation. The results of numerical experiments are compared to analytical solutions to confirm the reliable treatment of these stable solutions. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1043–1069, 2017

Stability and convergence of spectral radial point interpolation method locally applied on two‐dimensional pseudoparabolic equation

In this article, we study a spectral meshless radial point interpolation of pseudoparabolic equations in two spatial dimensions. Shape functions, which are constructed through point interpolation method using the radial basis functions, help us to treat problem locally with the aim of high‐order convergence rate. The time derivatives are approximated by the finite difference time‐stepping method. The stability and convergence of this meshless approach are discussed and theoretically proven. Numerical results are presented to illustrate the theoretical findings. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 724–741, 2017