Local RBF‐FD‐Based Mesh‐free Scheme for Singularly Perturbed Convection‐Diffusion‐Reaction Models with Variable Coefficients

This study is devoted to designing two hybrid computational algorithms to find approximate solutions for a class of singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters. In our approaches, the time discretization is first performed by the well-known Rothe method and Taylor series procedures, which reduce the underlying model problem into a sequence of boundary value problems (BVPs). Hence, a matrix collocation technique based on novel shifted Delannoy functions (SDFs) is employed to solve each BVP at each time step. We show that our proposed hybrid approximate techniques are uniformly convergent in order O ( Δ τ s + M − 1 2 ) {\mathcal{O}}\left(\Delta {\tau }^{s}+{M}^{-\tfrac{1}{2}}) for s = 1 , 2 s=1,2 , where Δ τ \Delta \tau is the time step and M M is the number of SDFs used in the approximation. Numerical simulations are performed to clarify the good alignment between numerical and theoretical findings. The computational results are more accurate as compared with those of existing numerical values in the literature.

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RBF-Based Local Meshless Method for Fractional Diffusion Equations

Kamran

Irfan

Alotaibi

et al. 2023

Fractal Fract

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The fractional diffusion equation is one of the important recent models that can efficiently characterize various complex diffusion processes, such as in inhomogeneous or heterogeneous media or in porous media. This article provides a method for the numerical simulation of time-fractional diffusion equations. The proposed scheme combines the local meshless method based on a radial basis function (RBF) with Laplace transform. This scheme first implements the Laplace transform to reduce the given problem to a time-independent inhomogeneous problem in the Laplace domain, and then the RBF-based local meshless method is utilized to obtain the solution of the reduced problem in the Laplace domain. Finally, Stehfest’s method is utilized to convert the solution from the Laplace domain into the real domain. The proposed method uses Laplace transform to handle the fractional order derivative, which avoids the computation of a convolution integral in a fractional order derivative and overcomes the effect of time-stepping on stability and accuracy. The method is tested using four numerical examples. All the results demonstrate that the proposed method is easy to implement, accurate, efficient and has low computational costs.

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Local RBF‐FD‐Based Mesh‐free Scheme for Singularly Perturbed Convection‐Diffusion‐Reaction Models with Variable Coefficients

Cited by 3 publications

References 39 publications

Fitted cubic spline scheme for two-parameter singularly perturbed time-delay parabolic problems

Fitted cubic spline scheme for two-parameter singularly perturbed time-delay parabolic problems

Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters

RBF-Based Local Meshless Method for Fractional Diffusion Equations

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