The analytical solution of the partial differential equation with time and spacefractional derivatives was derived by means of the homotopy decomposition method (HDM). Some examples are given and comparisons are made. The evaluations show that the homotopy decomposition method is extremely successful and suitable. The achieved results make the steadfastness of the HDM and its wider applicability to fractional differential equation obvious. Additionally, the adding up implicated in HDM is exceptionally undemanding and uncomplicated. It is confirmed that HDM is an influential and professional apparatus for FPDEs. It was also established that HDM is supplementary well organized than the ADM, VIM, HAM, and HPM.
Many observable fact in natural science, physics, chemistry, and other knowledge preserve to be illustrated incredibly fruitfully by representations using the supposition of derivatives and integrals with fractional order. Attention in the notion of differentiation and integration to noninteger order has existed since the progress of the conventional calculus [
In this paper, two cases of special interest such as the timefractional foam drainage equation and the spacefractional foam drainage equation are discussed in detail. The paper has been arranged as follows. History of the fractional derivative order is presented in Section
There is in the literature numereous definitions about fractional derivatives [
For the case of RiemannLiouville derivative we have the following definition:
Lately, Jumarie (see [
To demonstrate the fundamental design of this technique, we reflect on a universal nonlinear nonhomogeneous fractional partial differential equation with initial conditions of the following structure:
In the case of Caputo fractional derivative
In the homotopy perturbation method, the fundamental statement is that the solutions can be written as a power series in
The homotopy decomposition method is obtained by the refined combination of homotopy technique with He’s polynomials [
Putting side by side expressions of identical powers of
It is very important to test the computational complexity of a method or algorithm [
Set
Calculate the recursive relation after the comparison of the terms of the same power is done.
If
Print out
If the exact solution of the fractional partial differential equation (
Let
The last inequality follows from [
The complexity of the homotopy decomposition method is of order
The number of computations including product, addition, subtraction, and division are the following.
In Step
Now in Step
In this section we apply this method for solving partial differential equation with time and spacefractional derivatives.
Let us consider the fractional Riccati differential equation
Following the discussion presented in Section
Comparing the terms of the same power of
And the following solutions are obtained:
Using the package Mathematica, in the same manner, one can obtain the rest of the components. But, here, 10 terms were computed and the asymptotic solution is given by
Now to access the accuracy of HDM, we compare the approximated solution (
Notice that when
Therefore, for any
Example
An Eton proverb says “an image is equivalent to one thousand words.” The following figure shows the comparison of the approximated solution and the exact solution.
Figure
Comparison of approximated and exact solutions for
Consider the following form of the timefractional equation:
Following the steps of HDM, we obtain
Using the package Mathematica, in the same manner, one can obtain the rest of the components. But in this case, 5 terms were computed and the asymptotic solution is given by
Now notice that if we set
This is the exact solution of Example
In this section, to access the accuracy of the HDM, we compare the approximation (
Exact solution for
Approximated solution for
From the Figure
Consider the fractional predatorprey equation
Comparing the terms of the same power of
The following solutions are obtained:
Using the package Mathematica, in the same manner, one can obtain the rest of the components. But in this case, 3 terms were computed and the asymptotic solution is given by
Figures
Prey density for
Predator density for
To test the accuracy of the method used in this paper, we present the numerical result of the approximate solution of (
Comparison of the numerical results of approximate solutions obtained via HDM, VIM and HPM with exact solution of (
Variable 
HDM  VIM  HPM  Exact 



1.05  0.78162  0.781619  0.78162  0.781806  0.000186617  0.000187358 
1.06  0.785437  0.785426  0.785437  0.785664  0.000226381  0.000237859 
1.07  0.789187  0.789179  0.789187  0.789461  0.000274099  0.000282221 
1.08  0.792868  0.792789  0.792868  0.793199  0.000331261  0.000410097 
1.09  0.796479  0.796467  0.796479  0.796878  0.000399615  0.000411144 
1.10  0.800018  0.810017  0.800018  0.800499  0.000481214  0.00951798 
1.11 
0.803484  0.8100484  0.804062  0.804062  0.000578458  0.00598601 
The numerical results show that the HPM and HDM are more accurate than the VIM in this case.
The numerical results show that the HPM and HDM are more accurate than the VIM in this case.
In this paper, we have productively developed HDM for solving partial differential equation of space and timefractional derivatives. the dependability of the HDM and its wider applicability to fractional differential equation. It is consequently, that the HDM makes available more practical series solutions that converge very speedily in real physical problems. Also, the amount of the computational effort has been abridged. In the bargain, the computations concerned in HDM are very straightforward. It is established that HDM is a prevailing and resourceful instrument for FPDEs.
The authors declare that there is no conflict of interests for this paper.
The first draft of this paper was portrayed by Abdon Atangana; the revised version was carefully corrected by Samir Brahim Belhaouari, and both authors read and submitted the final version.
The authors will like to thank the reviewer for his valuable spare time to read this paper and for his valuable comments and suggestions toward the enhancement of this paper.