Numerical and theoretical treatment based on the compact finite difference and spectral collocation algorithms of the space fractional-order Fisher’s equation

This paper presents an accurate numerical algorithm to solve the space fractional-order Fisher’s equation where the derivative operator is described in the Caputo derivative sense. In the presented discretization process, first, we use the compact finite difference (CFD) for a semi-discrete occurrence in time derivative and implement the Chebyshev spectral collocation method (CSCM) of the third-kind to discretize the spatial fractional derivative. The presented method converts the problem understudy to be a system of algebraic equations which can be easily solved. To study the convergence and stability analysis, some theorems are given with their proofs. A numerical simulation is outputted to test the accuracy and applicability of our presented algorithm.

Computation of Hyperspherical Bessel Functions

In this paper we present a fast and accurate numerical algorithm for the computation of hyperspherical Bessel functions of large order and real arguments. For the hyperspherical Bessel functions of closed type, no stable algorithm existed so far due to the lack of a backwards recurrence. We solved this problem by establishing a relation to Gegenbauer polynomials. All our algorithms are written in C and are publicly available at Github [https://github.com/lesgourg/class_public]. A Python wrapper is available upon request.

An Efficient Operational Matrix Technique for Multidimensional Variable-Order Time Fractional Diffusion Equations

This paper derives a new operational matrix of the variable-order (VO) time fractional partial derivative involved in anomalous diffusion for shifted Chebyshev polynomials. We then develop an accurate numerical algorithm to solve the 1 + 1 and 2 + 1 VO and constant-order fractional diffusion equation with Dirichlet conditions. The contraction of the present method is based on shifted Chebyshev collocation procedure in combination with the derived shifted Chebyshev operational matrix. The main advantage of the proposed method is to investigate a global approximation for spatial and temporal discretizations, and it reduces such problems to those of solving a system of algebraic equations, which greatly simplifies the solution process. In addition, we analyze the convergence of the present method graphically. Finally, comparisons between the algorithm derived in this paper and the existing algorithms are given, which show that our numerical schemes exhibit better performances than the existing ones.