An improved localized radial basis-pseudospectral method for solving fractional reaction–subdiffusion problem

AbstractThis paper concerns with the implementation of radial basis function pseudospectral (RBF-PS) method for solving Fisher’s equation. Pseudospectral methods are well known for being highly accurate but are limited in terms of geometric flexibility. Radial basis function (RBF) in combination with the pseudospectral method is capable to overcome this limitation. Using RBF, Fisher’s equation is approximated by transforming it into a system of ordinary differential equations (ODEs). An ODE solver is used to solve the resultant ODEs. In this approach, the optimal value of the shape parameter is discussed with the help of leave-one out cross validation strategy which plays an important role in the accuracy of the result. Several examples are given to demonstrate the accuracy and efficiency of the method. RBF-PS method is applied using different types of basis functions and a comparison is done based upon the numerical results. A two-dimensional problem that generalizes the Fisher’s equation is also solved numerically. The obtained numerical results and comparisons confirm that the use of RBF in pseudospectral mode is in good agreement with already known results in the literature.

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Radial Basis Function-Pseudospectral Method for Solving Non-Linear Whitham-Broer-Kaup Mode

Sohag Journal of Mathematics ◽

10.18576/sjm/040103 ◽

2017 ◽

Vol 4 (1) ◽

pp. 13-18

Author(s):

Rostam K. Saeed ◽

Mohammed I. Sadeeq

Keyword(s):

Radial Basis Function ◽

Basis Function ◽

Pseudospectral Method ◽

Non Linear ◽

Radial Basis

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Radial Basis Function Pseudospectral Method for Solving Standard Fitzhugh-Nagumo Equation

International Journal of Mathematical Engineering and Management Sciences ◽

10.33889/ijmems.2020.5.6.110 ◽

2020 ◽

Vol 5 (6) ◽

pp. 1488-1497

Author(s):

Geeta Arora ◽

Gurpreet Singh Bhatia

Keyword(s):

Radial Basis Function ◽

Basis Function ◽

Shape Parameter ◽

Cross Validation ◽

Pseudospectral Method ◽

Mesh Free ◽

Radial Basis ◽

Nagumo Equation ◽

The Stability ◽

Leave One Out

In this article, a pseudospectral approach based on radial basis functions is considered for the solution of the standard Fitzhugh-Nagumo equation. The proposed radial basis function pseudospectral approach is truly mesh free. The standard Fitzhugh-Nagumo equation is approximated into ordinary differential equations with the help of radial kernels. An ODE solver is applied to solve the resultant ODEs. Shape parameter which decides the shape of the radial basis function plays a significant role in the solution. A cross-validation technique which is the extension of the statistical approach leave-one-out-cross-validation is used to find the shape parameter value. The presented method is demonstrated with the help of numerical results which shows a good understanding with the exact solution. The stability of the proposed method is demonstrated with the help of the eigenvalues method numerically.

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