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This paper deals with the construction of approximate series solutions of diffusion models with stochastic excitation and nonlinear losses using the homotopy analysis method (HAM). The mean, variance and other statistical properties of the stochastic solution are computed. The solution technique was applied successfully to the 1D and 2D diffusion models. The scheme shows importance of choice of convergence-control parameter

The deterministic differential equations of the form

HAM is an analytical technique for solving non linear differential equations. Proposed by Liao in 1992, [

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

where the diffusion coefficient

where E denotes the ensemble average operator,

where

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 [

A presentation of the standard HAM for deterministic problems can be found in [

where N is a nonlinear operator and

where

The HAM is based on a kind of continuous mapping

respectively. Thus as q increases from 0 to 1, the solution

Having the freedom to choose the auxiliary parameter

Expanding

where

Assume that the auxiliary parameter

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

This is mostly used in the HPM method. According to definition (8), the governing equation and the corresponding initial condition of

Differentiating Equation (4) m times with respect to the embedding parameter ^{th}-order deformation equation:

where

and

The solution is computed as:

It should be emphasized that

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

Subject to the initial condition

where

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

The first order approximation is obtained by setting

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

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

One can notice that the selected value function satisfies the initial and boundary conditions and it depends on the parameter

Furthermore, we define the nonlinear operator as

We construct the zero-order deformation Equation,

The m^{th}-order deformation Equation for

And subject to the boundary conditions

And the initial condition

where

Now the m^{th}-order deformation equation for

The first order approximation is obtained by substituting

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

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

Figures 1 and 2 show the plots of the

The mean and variance results of the WHEP technique are obtained from [

The effect of

In the following figures, results of the solution of 2D stochastic quadratic nonlinear diffusion model using HAM technique are shown at

third order approximation of mean for different

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 <

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