The deterministic differential equations of the form constitute the basic form of so-called diffusion or transport problems which appear in relevant models such as: the growth population geometric (or Malthusian) model in biology, where represents the per capita growth rate; the neutron and gamma ray transport model in physics, where coefficient involves the geometry of the cross-sections of the medium; the continuous composed interest rate models for studying the evolution of an investment under time-variable interest rate which can be taken as, etc. Despite the usefulness of these basic models, they do not often cover all possible situations observed from a practical point of view. In fact, as a simple but illustrative example, if, the Malthus model predicts unlimited growth of a species despite the fact that resources are always limited. Then, the logistic (or Verhulst) model introduces a nonlinear term in order to overcome this drawback by considering the differential equation, where the nonlinearity intensity is given by parameter b. In many practical situations it is appropriate to assume that the nonlinear term affecting the phenomena under study is small enough; then its intensity is controlled by means of a frank small parameter, say. Stochastic differential equations based on the white noise process provide a powerful tool for dynamically modeling these complex and uncertain aspects. Over the last few years, new and relevant methods for finding the exact solutions of such Equations have been developed. They include the homotopy perturbation (HPM) method [1,2], Wiener-Hermite expansion with perturbation method (WHEP Cortes [2011]) [3] and the exp-function method [4,5].

HAM is an analytical technique for solving non linear differential equations. Proposed by Liao in 1992, [6], the technique is superior to the traditional perturbation methods, in which it leads to convergent series solutions of strongly nonlinear problems, independent of any small or large physical parameter associated with the problem, [7]. The HAM provides a more viable alternative to non perturbation techniques such as the Adomian decomposition method (ADM) [8] and other techniques that cannot guarantee the convergence of the solution series and may be only valid for weakly nonlinear problems, [7]. We note here that He’s HPM method, [9] is only a special case of the HAM. In recent years, this method has been successfully employed to solve many problems in science and engineering such as the viscous flows of non-Newtonian fluids [10,11], the KdV-type equations [12], Glauert-jet problem [13], Burgers-Huxley equation [14], time-dependent Emden-Fowler type equations [15], differential-difference equation [16], two-point nonlinear boundary value problems [17]. The HAM provides the solution in the form of a rapidly convergent series with easily computable components using symbolic computation software such as Mathematica.

This paper deals with the solution of 1D stochastic differential models of the form

where the diffusion coefficient and initial condition are deterministic, is a small parameter and is the white noise process, whose intensity is given by parameter, which has the following important properties:

where E denotes the ensemble average operator, is the Dirac delta function. And is a random outcome for a triple probability space, where is a sample space, A is a -algebra associated with and P is a probability measure. The current work also deals with the solution of 2D stochastic quadratic nonlinear equation with as non-homogeneity.

where is the diffusion process, is a deterministic scale for the nonlinear term. The non-homogeneity term is spatial white noise scaled by.

The paper is organized as follows. Section 2 summarizes the basic idea of the HAM method. In Section 3, the HAM is applied in order to obtain fourth order approximation of the solution of 1D diffusion model. In Section 4, the HAM is applied up to the third order approximation for the solution of 2D diffusion model. In addition, we compute approximations for the main statistical moments such as the mean and variance. A comparison is done with the results obtained with the (WHEP Cortes [2011], WHEP El-Beltagy [2013]) technique [4,5]. The results are shown in Section 5 along with comments on the results.

A presentation of the standard HAM for deterministic problems can be found in [9]. The following subsection is a brief description of HAM. Consider the following differential equation:

where N is a nonlinear operator and is the unknown function. By means of generalizing the traditional HPM method, Liao [6] constructs the so-called zero-order deformation equation

where denotes the so-called embedding parameter, is an auxiliary parameter and L is an auxiliary linear operator.

The HAM is based on a kind of continuous mapping, where is an unknown function, is an initial guess for, and denotes a non-zero auxiliary function. It is obvious that when the embedding parameter and, Equation (3) becomes

respectively. Thus as q increases from 0 to 1, the solution varies from the initial guess to the solution. In topology, this kind of variation is called deformation; Equation (3) constructs the homotopy.

Having the freedom to choose the auxiliary parameter, the auxiliary function, the initial approximation, and the auxiliary linear operator L, we can assume that all of them are properly chosen so that the solution of the zero-order deformation Equation (4) exists for.

Expanding in Taylor series with respect to q, one has,

where

Assume that the auxiliary parameter, the auxiliary function, the initial approximation and the auxiliary linear operator L are so properly chosen that the series (6) converges at and

which must be one of the solutions of the original nonlinear Equation, as proved by Liao [9]. As and Equation (4) becomes

This is mostly used in the HPM method. According to definition (8), the governing equation and the corresponding initial condition of can be deduced from the zero-order deformation Equation (4). Define the vector

Differentiating Equation (4) m times with respect to the embedding parameter and then setting and finally dividing them by, we have the so-called m^th-order deformation equation:

where

and

The solution is computed as:

It should be emphasized that for is governed by the linear Equation (10) with linear boundary conditions that come from the deterministic problem, which can be solved by any symbolic computation software such as Mathematica, Maple, or Matlab.

To demonstrate the above presented method it will be used to find the mean and variance of 1D stochastic diffusion problem as follows.

The auxiliary linear operator will be chosen as

Furthermore, we define the nonlinear operator as

We construct the zero-order deformation equation,

The m^th-order deformation equation for and is

Subject to the initial condition

where

Now the solution of the m^th-order deformation Equation (12) for becomes

The first order approximation is obtained by setting in (12) as follows

where

Then

The ensemble average of the first order approximation is

The covariance of the first order solution will be

The variance of the first order solution will be

In this manner, we can have more results of and obtained at

The final expression of the mean of the 4^th order solution will be

Since

Then the final expression of the variance of the 2^nd order solution will be

HAM will be used to find mean and variance of stochastic quadratic nonlinear diffusion problem as follows.

The auxiliary linear operator is chosen as

We have many choices in guessing the initial approximation together with its initial conditions which greatly affects the consequent approximation .The choice is a design problem which can be taken as follows:

One can notice that the selected value function satisfies the initial and boundary conditions and it depends on the parameter which is totally free. One can also notice that selection could control the solution convergence.

Furthermore, we define the nonlinear operator as

We construct the zero-order deformation Equation,

The m^th-order deformation Equation for and is

And subject to the boundary conditions

And the initial condition

where

Now the m^th-order deformation equation for becomes

The first order approximation is obtained by substituting to get

The approximated first order solution of (14) can be obtained using Eigen function expansion as follows,

the ensemble average of the first order approximation is

The covariance of the first order solution can be computed as

The covariance is obtained from the following final expression

The variance of the first order solution will be computed as

To give

In this manner, we can have more results of and obtained at

The final expression of mean of the 3^rd order solution will be

Since

Then the final expression of the variance of the 2^nd order solutionwill be

This paper shows that the HAM technique constitutes a powerful tool for constructing approximate solutions for the stochastic process for random diffusion models with nonlinear perturbations where uncertainty is considered by means of an additive term defined by white noise. The HAM method is employed to give a statistical analytic solution for stochastic 1D and 2D diffusion models. Different from all other analytic methods, the HAM provides us with a simple way to adjust and control the convergence region of the series solution by means of the auxiliary parameter ħ. Thus the auxiliary parameter ħ plays an important role within the frame of the HAM which can be determined by the so called ħ-curves. The solution obtained by means of the HAM is an infinite power series for appropriate initial approximation, which can be, in turn, expressed in a closed form. The accuracy for the method is verified on 1D diffusion model by comparisons with WHEP technique and good agreements are obtained. As shown in Figures 1 and 2, we can see that the valid ħ region in the 1D example is −0.9 < < −1.4 and in 2D example the interval is −0.9 < < −1.1, as shown in Figure 6. The results demonstrate reliability and efficiency of the HAM method. Since HAM was used to solve only deterministic problems, we

can say that this is the first time to apply HAM method on stochastic problems and we found that it’s easier than WHEP and more general than HPM since HPM is a special case of HAM obtained at and its results are accurate.