Numerical Schemes For Partial Differential Equations Of Fractional Order

Abstract

In this report, we explore the 1st-order stiff-cut scheme developed by professor Sun for solving linear ordinary differential equations (ODEs), where the linear part is stiff. Professor Sun demonstrated that the scheme is unconditionally stable and convergence when the stiff-cutter S satisfies two conditions. First, for a symmetric positive definite (SPD) matrix A that exhibits strong stiffness, the largest eigenvalue of S−1A is less than 2. Secondly, there exists a constant ¯c > 0 that is independent of N (where N is a positive integer) such that the smallest eigenvalue of S−1A remains bounded below by ¯c. The objective of this report is to use Toeplitzplus- Hankel matrices and three different circulant matrices (Strang, T.Chan, and R.Chan) as stiff-cutter to approximate linear stiff system of ODEs based on the stiff-cut scheme. To validate the results, we utilize Riesz fractional diffusion equations (RFDEs) as test cases of our study.