Approximate solution of the multi-term time fractional diffusion and diffusion-wave equations

• Published in: Computational & applied mathematics Vol. 39; no. 3
• Main Authors: Rashidinia, Jalil, Mohmedi, Elham
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-09-2020; Springer Nature B.V
• Subjects: Applications of Mathematics; Applied physics; Collocation methods; Computational mathematics; Computational Mathematics and Numerical Analysis; Diffusion; Literature reviews; Mathematical Applications in Computer Science; Mathematical Applications in the Physical Sciences; Mathematics; Mathematics and Statistics; Polynomials; Spectral methods; Wave equations; Legendre collocation method; Multi-term time fractional diffusion and diffusion-wave equations; 78A48; Caputo derivative; Convergence analysis; 65M10
• ISSN: 2238-3603; 1807-0302
• DOI: 10.1007/s40314-020-01241-4

We develop a numerical scheme for finding the approximate solution for one- and two-dimensional multi-term time fractional diffusion and diffusion-wave equations considering smooth and nonsmooth solutions. The concept of multi-term time fractional derivatives is conventionally defined in the Caputo view point. In the current research, the convergence analysis of Legendre collocation spectral method was carried out. Spectral collocation method is consequently tested on several benchmark examples, to verify the accuracy and to confirm effectiveness of proposed method. The main advantage of the method is that only a small number of shifted Legendre polynomials are required to obtain accurate and efficient results. The numerical results are provided to demonstrate the reliability of our method and also to compare with other previously reported methods in the literature survey.