scholarly journals A unified numerical scheme for the multi-term time fractional diffusion and diffusion-wave equations with variable coefficients

2018 ◽  
Vol 330 ◽  
pp. 380-397 ◽  
Author(s):  
Hu Chen ◽  
Shujuan Lü ◽  
Wenping Chen
Keyword(s):  
Numerical Scheme ◽  
Wave Equations ◽  
Fractional Diffusion ◽  
Variable Coefficients ◽  
Diffusion Wave ◽  
And Diffusion

On an Explicit Difference Method for Fractional Diffusion and Diffusion-Wave Equations

2009 ◽  
Author(s):  
Joaqui´n Quintana Murillo ◽  
Santos Bravo Yuste
Keyword(s):  
Wave Equations ◽  
Its Region ◽  
Fractional Diffusion ◽  
Diffusion Wave ◽  
Von Neumann ◽  
Stability Bound ◽  
Riemann Zeta ◽  
The Stability ◽  
And Diffusion ◽  
Explicit Difference Schemes

An explicit difference scheme for solving fractional diffusion and fractional diffusion-wave equations, in which the fractional derivative is in the Caputo form, is considered. The two equations are studied separately: for the fractional diffusion equation, the L1 discretization formula is employed, whereas the L2 discretization formula is used for the fractional diffusion-wave equation. Its accuracy is similar to other well-known explicit difference schemes, but its region of stability is larger. The stability analysis is carried out by means of a procedure similar to the standard von Neumann method. The stability bound, which is given in terms of the the Riemann Zeta function, is checked numerically.

scholarly journals Approximate Solutions of Time Fractional Diffusion Wave Models

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 923 ◽  
Author(s):  
Abdul Ghafoor ◽  
Sirajul Haq ◽  
Manzoor Hussain ◽  
Poom Kumam ◽  
Muhammad Asif Jan
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Quadrature Rule ◽  
Approximate Solutions ◽  
Fractional Diffusion ◽  
Haar Wavelets ◽  
Test Problems ◽  
Algebraic Equations ◽  
Diffusion Wave ◽  
Wave Models

In this paper, a wavelet based collocation method is formulated for an approximate solution of (1 + 1)- and (1 + 2)-dimensional time fractional diffusion wave equations. The main objective of this study is to combine the finite difference method with Haar wavelets. One and two dimensional Haar wavelets are used for the discretization of a spatial operator while time fractional derivative is approximated using second order finite difference and quadrature rule. The scheme has an excellent feature that converts a time fractional partial differential equation to a system of algebraic equations which can be solved easily. The suggested technique is applied to solve some test problems. The obtained results have been compared with existing results in the literature. Also, the accuracy of the scheme has been checked by computing L 2 and L ∞ error norms. Computations validate that the proposed method produces good results, which are comparable with exact solutions and those presented before.

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