Abstract
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 to guarantee the convergence of the solutions of nonlinear differential Equations. The results are compared with the Wiener-Hermite expansion with perturbation (WHEP) technique and good agreements are obtained.
Highlights
The deterministic differential equations of the form x t a t x t constitute the basic form of so-called diffusion or transport problems which appear in relevant models such as: the growth population geometric model in biology, where a t represents the per capita growth rate; the neutron and gamma ray transport model in physics, where coefficient a t involves the geometry of the cross-sections of the medium; homotopy analysis method (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 prob
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]
Summary
The deterministic differential equations of the form x t a t x t constitute the basic form of so-called diffusion or transport problems which appear in relevant models such as: the growth population geometric 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].
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