On Analytical Solution of Time-Fractional Type Model of the Fisher’s Equation

In this paper, the time-fractional Fisher’s equation (TFFE) is considered to exam the analytical solution using the Laplace q-Homotopy analysis method (Lq-HAM)”. The Lq-HAM is a combined form of q-homotopy analysis method (q-HAM) and Laplace transform. The aim of utilizing the Laplace transform is to outdo the shortage that is mainly caused by unfulfilled conditions in the other analytical methods. The results show that the analytical solution converges very rapidly to the exact solution.


ISSN: 0067-2904
Huseen Iraqi Journal of Science, 2020, Vol. 61, No. 6, pp: 1419-1425 1420 In this paper, we consider the TFFE ( ) ( ) ( )( ( )) ( ) , -, -(1) subject to the initial condition: , is a real parameter, represents the Caputo fractional derivative in time [21], ( ) is the given function and is the linear differential operator. This problem is considered a mathematical model for a wide scope of significant physical phenomena. It has become one of the most important classes of nonlinear equations due to its occurrence in many chemical and biological processes. The time-fractional Fisher's equation was solved by homotopy perturbation technique (HPM) [8] and homotopy analysis technique (HAM) [9]. The purpose of this paper is to apply the Lq-HAM, which is a combination of q-HAM and Laplace transform to provide an approximate solution for the TFFE. In section two, we state some necessary concepts of fractional calculus. In section three, we introduce the basic idea of Lq-HAM for TFFE. Finally, in section four, we solve two numerical examples.

Preliminaries
In this section, we state some necessary concepts of fractional calculus that will help us to achieve the aim of this paper [21,22].
and it is said to be in the space if and only if ( ) .
( ) ( ). Some of the basic properties of the operator , which are required here, are introduced.
for . Moreover, some of the most important properties are needed here.
for . Lemma 2.1 [23]: If then the Laplace transform of the fractional derivative The Lq-HAM for TFFE Consider the time-fractional Fisher's equation (1) subject to the initial condition (2). Taking the Laplace transform of both sides of equation (1) Thus, as increases from 0 to , the solution ( ) varies from the initial ( ) to the solution ( ). By expanding ( ) in Taylor series with respect to , we get Assume that ( ) are chosen such that the series (6) (4) times with respect to then setting and finally dividing them by yields the so-called order deformation equation and
By using the analysis in the previous section equation (9), we obtain , and is defined as in equation (11). By using Mathematica software, the following results are obtained , we obtain the same solution given by HPM [8]. Figure-1 shows the 4 th -order approximate solution for different values of with with the exact solution of problem (12)- (13). Figure-2 shows the absolute error obtained by the 4 th -order approximation of problem (12)(13) for α = 1.

Then the order series solution of Lq-HAM is as follows
When the problem (16)(17) gives the exact solution ( ) ( ) .
Using the analysis in the previous section, we obtain , / ( ) , and as defined in equation (11). Using Mathematica software, the following result are obtained Then the order series solution of Lq-HAM is as follows Figure-3 shows the 5 th -order approximate solution for different values of with when with the exact solution of problem (16)(17). Figure-4 shows the absolute error obtained by the 5 th -order approximation of problem (16)(17) for α = 1, when . Table-1

Conclusions
The major concern of this paper is the demonstration of the successful use of Lq-HAM to obtain analytical solutions of TFFE. The Lq-HAM was used in a direct way that is the restrictive assumptions are avoided in Lq-HAM. These considerations give Lq-HAM a significant advantage in many problems. Our results confirm that the appropriate choices of the convergence control parameters and lead to the accuracy of Lq-HAM in the sense that , unlike other methods, just few terms are needed in our approximations to get close to the exact solution for .