A New Approach for the Approximate Analytical Solution of Space-Time Fractional Differential Equations by the Homotopy Analysis Method

The motivation of this study is to construct the truncated solution of space-time fractional differential equations by the homotopy analysis method (HAM). The first space-time fractional differential equation is transformed into a space fractional differential equation or a time fractional differential equation before the HAM. Then the power series solution is constructed by the HAM. Finally, taking the illustrative examples into consideration we reach the conclusion that the HAM is applicable and powerful technique to construct the solution of space-time fractional differential equations.


Introduction
The fractional calculus is a very attractive concept for scientists in various fields since it has a wide application in mathematics, physics, biology, and so on [1][2][3][4].Various fractional derivatives are defined to get the solution of fractional differential equations [5,6].In order to get the solution, different methods are introduced and applied [7][8][9][10][11].These methods include the finite difference method [12], the homotopy perturbation method (HPM) [13,14], and the generalized differential transform method [15].
Since the mathematical models include space and time variables in the modelling of mathematical and physical problems, fractional derivatives are required.The HAM provides as simple way to construct truncated solution of various differential equations.The advantage of the HAM is that the accuracy of the HAM does not depend on small parameter in the considered equation [16][17][18].Therefore, compare the other technique HAM has many advantage in the application [19][20][21].
The motivation of this paper is to determine the truncated solution of the following space-time FPDE by implementing the HAM with a new algorithm by which space-time fractional differential equation is reduced to either a space fractional differential equation or a time fractional differential equation making the problem easier to deal with by the HAM: (, 0) =  () , 0 <  ≤ (0, ) =  1 () , 0 <  ≤ (0, ) =  2 () , 0 <  ≤  (4) where 0 <  ⩽ 1, 1 <  ⩽ 2 and the source function F depends on either  or .

Preliminaries
This section is about basic definitions and fundamental properties of the fractional calculus. 2

Basic Ideas of the Homotopy Analysis Method (HAM)
Let us consider the differential equation where  is a nonlinear operator,  and  are independent variables, and (, ) is an unknown function.Liao [17] introduced the so-called zero-order deformation equation by the homotopy method as follows: with the initial conditions where  ∈ [0, 1], ℏ ̸ = 0 are parameters, (, ) is nonzero auxiliary function, and  =    , (−1 <  ≤ ) is an auxiliary linear operator with the following feature: Obviously, for  = 0 and  = 1, we have where  0 (, ) is an initial guess of (, ) and (, ; ) is an unknown function.As  goes to 1, the solution (, ; ) changes from  0 (, ) to (, ).By Taylor series of (, ; ) based on  is written as where If the auxiliary linear operator, the initial guess, the auxiliary operator ℎ, and the auxiliary function are properly chosen, the series (20) converges at  = 1, then we have Define the vector Differentiating ( 16) -times with respect to the embedding parameter  and then setting  = 0 and finally dividing them by !, we obtain the th-order deformation equation, with assumption (, ) = 1 subject to the following initial conditions: where and We can apply the operator   to both sides of (24) to obtain Using the Caputo derivative and the initial conditions, we obtain Finally, we will approximate the HAM solution (22) by the truncated series The th-order deformation (24) is linear and thus can be easily solved, especially by means of symbolic computation software such as Matlab and Maple.

Space-Time FPDE by the HAM
To construct the solution of problem ( 1)-( 4) by the HAM, it is transformed into a space fractional differential problem or time fractional differential problem.The transformation we apply in this paper is motivated by the properties ( 7)- (9).Therefore let us take the problems with and without source functions into consideration: (A) Transforming into Time-Fractional Differential Equation (A ) No Source Function.In order to make the boundary condition (4) equal to zero, the transformation V(, ) = (, ) − (, 0) is applied to (1)-(2): V (, 0) = 0.
In order to make the order of derivative on the right-hand side, integer, the transformation V =  2−   is applied to (31)-(32): Substituting (33) into (26),   ( → −1 ) can be given by The solution of th-order deformation (26) for  ≥ 1 becomes By using (36) with the initial condition given by (34), we now successively obtain  0 (, ) =  (, 0) = 0 (37) and so on.Hence, by using first four terms of the approximate solution (, ) can be obtained in the form of the series below: By the transformations (, ), the solution of problem ( 1)-( 2) is obtained.
(A ) Including the Source Function.In order to make the boundary condition (4) equal to zero, the transformation V(, ) = (, ) − (, 0) is applied to (1)-( 2): In order to make the order of derivative on the right-hand side, integer, the transformation V =  2−   is applied to (42)-(43): Substituting ( 26) into (44),   ( → −1 ) can be given by The solution of th-order deformation (46) for  ≥ 1 becomes By using (47) with the initial condition given by (45), we now successively obtain and so on.Hence, the truncated solutions of (, ) can be constructed in the series form by using the first four terms as follows: By the transformations (, ), the solution of problem ( 1)-( 2) is determined.
(B ) Including Source Function.Let us consider the problem below: Substituting ( 26) into (63),   ( → −1 ) can be given by The solution of th-order deformation (66) for  ≥ 1 becomes By using (64)-(65), we now successively obtain and so on.Hence, the truncated solutions of V(, ) can be constructed in the series form by using the first four terms below: By the transformation (, ) =  1−  V(, ), the solution of problem (1), (3), and (4) is determined.

Examples
Example .Let us take the homogeneous problem.
In Tables 1 and 2, the absolute errors have been worked out for various values of  and .As can be seen from the tables, for the fixed values of  and  as  increases, the absolute error decreases, while as  increases, the absolute error also increases.Moreover as  increases to 1,  increases to 2 and the absolute error decreases.In addition, the absolute error takes the minimum values at the auxiliary values ℏ = −0.94,ℏ = −1.02for case A and case B, respectively.
Example .Let us take the nonhomogeneous problem.
In Tables 3 and 4, the absolute errors have been worked out for various values of  and .As is obvious from the tables, as  increases to 1,  increases to 2 and the absolute error decreases.In addition, the absolute error takes the minimum values at the auxiliary values ℏ = −0.94,ℏ = −0.57for case A and case B, respectively.

Conclusion
In this research, the truncated solution of space-time FPDEs is constructed by the HAM and special transformations which allows us to implement the HAM to deal with harder problems.It is obvious that applying the HAM to the equation including two different fractional derivatives is harder than applying the HAM to the equation including one fractional derivative.Therefore, the transformation in this paper allows us to find the solution of the problem by the HAM much more easier.Examples manifest that when the order of space and time derivatives is fractional, the result of the HAM method gives better results for smaller values of ℏ in some interval unlike the integer order of space and time fractional derivatives.

Table 1 :
The absolute errors for case A in Example 1.

Table 3 :
The absolute errors for case A in Example 2.

Table 4 :
The absolute errors for case B in Example 2.