Fractional advection differential equation within Caputo and Caputo–Fabrizio derivatives

Dumitru Baleanu,Bahram Agheli,Maysaa Mohamed Al Qurashi

doi:10.1177/1687814016683305

Abstract

In this research, we applied the variational homotopic perturbation method and q-homotopic analysis method to find a solution of the advection partial differential equation featuring time-fractional Caputo derivative and time-fractional Caputo–Fabrizio derivative. A detailed comparison of the obtained results was reported. All computations were done using Mathematica.

Highlights

We have suggested the q-homotopic analysis method (q:HAM) and the variational homotopic perturbation method (VHPIM) to find a solution for the advection partial differential equations (PDEs) with time-fractional derivatives
With the replacement of the primary status u(x, 0) in the recurrent formula (16), the consequence with q:HAM featuring Caputo derivative is stated as u0(x, t) = 0
1 n(n(15n À 26) + 13)t5x 15 In Tables 1 and 2, we may observe the rough answers for n = 1:0, that is taken for several values of x and t using q:HAM and VHPIM with two fractional derivatives, involving singular differential operator which is named Caputo and involving nonsingular differential operator which is named Caputo–Fabrizio

Summary

Introduction

We have suggested the q-homotopic analysis method (q:HAM) and the variational homotopic perturbation method (VHPIM) to find a solution for the advection partial differential equations (PDEs) with time-fractional derivatives. This work at first has been conducted in order to use the homotopic analysis method (HAM) by Liao[3] and further to use it in order to solve PDEs featuring time-fractional derivative. In section ‘‘Application and consequences,’’ the application of q:HAM and VHPIM to the advection differential equation featuring time-fractional derivative is

Results

Conclusion