Adomian decomposition method for a two-component non-local reaction-diffusion model of the Fisher - Kolmogorov - Petrovsky - Piskunov type

We consider an approach to constructing the approximate analytical solutions for a one-dimensional two-component reaction-diffusion model describing the dynamics of a population interacting with the active substance surrounding the population. The system of model equations includes the nonlocal generalized Fisher - Kolmogorov - Petrovsky - Piskunov equation for the population density and the diffusion equation for the density of the active substance. Both equations contain additional terms describing the mutual influence of the population and the active substance. To find approximate solutions of the system of model equations, we first use the perturbation method with respect to the small parameter of interaction between the population and the active substance. Next, we apply the well-known iterative method developed by G. Adomian to solve the equations determining the perturbation series terms. In the method proposed, the equation solution is presented as a series whose terms are determined by the corresponding iterative procedure. In this work the diffusion operator is taken as the operator for which the inverse operator is expressed in terms of the diffusion propagator. This allows one to find approximate solutions in the class of functions decreasing at infinity. As an illustration, we consider an example of solving the Cauchy problem for the initial functions of a Gaussian form.

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