A Hybrid Local Radial Basis Function Method for the Numerical Modeling of Mixed Diffusion and Wave-Diffusion Equations of Fractional Order Using Caputo’s Derivatives

Abstract

This article presents an efficient method for the numerical modeling of time fractional mixed diffusion and wave-diffusion equations with two Caputo derivatives of order 0<α<1, and 1<β<2. The numerical method is based on the Laplace transform technique combined with local radial basis functions. The method consists of three main steps: (i) first, the Laplace transform is used to transform the given time fractional model into an equivalent time-independent inhomogeneous problem in the frequency domain; (ii) in the second step, the local radial basis functions method is utilized to obtain an approximate solution for the reduced problem; (iii) finally, the Stehfest method is employed to convert the obtained solution from the frequency domain back to the time domain. The use of the Laplace transform eliminates the need for classical time-stepping techniques, which often require very small time steps to achieve accuracy. Additionally, the application of local radial basis functions helps overcome issues related to ill-conditioning and sensitivity to shape parameters typically encountered in global radial basis function methods. To validate the efficiency and accuracy of the proposed method, several test problems in regular and irregular domains with uniform and non-uniform nodes are considered.

Numerical Solution of the Linear Fractional Delay Differential Equation Using Gauss–Hermite Quadrature

“…This lack of interest from the research community is possibly due to two factors: (i) the Laplace transform cannot handle nonlinear differential equations; (ii) the numerical inversion of the Laplace transform is very hard to compute. Despite these drawbacks, Laplace transform methods have been utilized recently by researchers, particularly in the area of linear PDEs [25][26][27][28][29]. The solution of many differential equations may be found in terms of the Laplace transform which is then, however, very difficult for inversion via the techniques of complex analysis.…”