In this article we decide to define a modified homotopy perturbation for solving non-linear integral equations. Almost, all of the papers that was presented to solve non-linear problems by the homotopy method, they used from two non-linear and linear operators. But we convert a non-linear problem to two suitable non-linear operators also we use from appropriate bases functions such as Legendre ...

We show the location of so called critical points, i.e., couples diffusion coefficients for which a non-trivial solution linear reaction-diffusion system activator-inhibitor type on an interval with Neumann boundary conditions and additional non-linear unilateral condition at one or two points and/or in interior exists. Simultaneously, we profile such solutions.