The analytical solution of the foam drainage equation with time- and space-fractional derivatives was derived by means of the homotopy
analysis method (HAM). The fractional derivatives are described in the Caputo sense. Some examples are given and comparisons are made; the comparisons
show that the homotopy analysis method is very effective and convenient. By choosing different values of the parameters

Many phenomena in engineering, physics, chemistry, and other science can be described very successfully by models using the theory of derivatives and integrals of fractional order. Interest in the concept of differentiation and integration to noninteger order has existed since the development of the classical calculus [

Since many fractional differential equations are nonlinear and do not have exact analytical solutions, various numerical and analytic methods have been used to solve these equations. The Adomian decomposition method (ADM) [

In 1992, Liao [

The HAM offers certain advantages over routine numerical methods. Numerical methods use discretization which gives rise to rounding off errors causing loss of accuracy and requires large computer memory and time. This computational method yields analytical solutions and has certain advantages over standard numerical methods. The HAM method is better since it does not involve discretization of the variables and hence is free from rounding off errors and does not require large computer memory or time.

The study of foam drainage equation is very significant for that the equation is a simple model of the flow of liquid through channels (Plateau borders [

has been investigated by the ADM and VIM method in [

In this paper, we extend the application of HAM to obtain analytic solutions to the space- and time-fractional foam drainage equation. Two cases of special interest such as the time-fractional foam drainage equation and the space-fractional foam drainage equation are discussed in details. Further, we give comparative remarks with the results obtained using ADM and VIM method (see [

The paper has been organized as follows. Notations and basic definitions are given in Section

A real function

The Riemann-Liouville fractional integral operator (

for

For the concept of fractional derivative, there exist many mathematical definitions [

The MittagLeffler function

To describe the basic ideas of the HAM, we consider the following differential equation:

where

By means of generalizing the traditional homotopy method, Liao [

where

respectively. Thus, as

where

If the auxiliary linear operator, the initial guess, the auxiliary parameter

which must be one of solutions of original nonlinear equation, as proved by Liao [

which is used mostly in the homotopy perturbation method [

Differentiating (

where

Applying the Riemann-Liouville integral operator

It should be emphasized that

Liao [

The parameters

In this section we apply this method for solving foam drainage equation with time- and space-fractional derivatives.

Consider the following form of the time-fractional equation:

The exact solution of (

HAM solution with

Exact solution.

HAM solution with

HAM solution with

This example has been solved using ADM and VIM in [

Considering the operator form of the space-fractional equation

HAM solution with

HAM solution with

This example has been solved using ADM and VIM in [

In this paper, we have successfully developed HAM for solving space- and time-fractional foam drainage equation. HAM provides us with a convenient way to control the convergence of approximation series by adapting

Matlab has been used for computations in this paper.