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A new numerical technique for solving fractional sub-diffusion and reaction sub-diffusion equations with a non-linear source term

Bhrawy Ali H. (King Abdulaziz University, Faculty of Science, Department of Mathematics, Jeddah, Saudi Arabia + Beni-Suef University, Faculty of Science, Department of Mathematics, Beni-Suef, Egypt)
Baleanu Dumitru (Cankaya University, Department of Mathematics and Computer Sciences, Ankara, Turkey + Institute of Space Sciences, Magurele-Bucharest, Romania)
Mallawi Fouad (King Abdulaziz University, Faculty of Science, Department of Mathematics, Jeddah, Saudi Arabia)

In this paper, we are concerned with the fractional sub-diffusion equation with a non-linear source term. The Legendre spectral collocation method is introduced together with the operational matrix of fractional derivatives (described in the Caputo sense) to solve the fractional sub-diffusion equation with a non-linear source term. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem. In addition, the Legendre spectral collocation methods applied also for a solution of the fractional reaction sub-diffusion equation with a non-linear source term. For confirming the validity and accuracy of the numerical scheme proposed, two numerical examples with their approximate solutions are presented with comparisons between our numerical results and those obtained by other methods.

Keywords: local fractional variational iteration method, diffusion equation, non-differentiable solution, local fractional derivative