In this article, a new homotopy technique is presented for the mathematical analysis of finding the solution of a first-order inhomogeneous partial differential equationThe homotopy perturbation method (HPM) and the decomposition of a source function are used together to develop this new technique. The homotopy constructed in this technique is based on the decomposition of a source function. Various decompositions of source functions lead to various homotopies. Using the fact that the decomposition of a source function affects the convergence of a solution leads us to development of a new method for the decomposition of a source function to accelerate the convergence of a solution. The purpose of this study is to show that constructing the proper homotopy by decomposing f (x, y) in a correct way determines the solution with less computational work than using the existing approach while supplying quantitatively reliable results. Moreover, this method can be generalized to all inhomogeneous PDE problems.

The study in this paper mainly concerns the inverse problem of determining an unknown source function in the linear fractional differential equation with variable coefficient using Adomian decomposition method (ADM). We apply ADM to determine the continuous right hand side functionsfxandftin the heat-like diffusion equationsDtαux,t=hxuxxx,t+fxandDtαux,t=hxuxxx,t+ft, respectively. The results reveal that ADM is very effective and simple for the inverse problem of determining the source function.

We develop a formulation for the analytic or approximate solution of fractional differential equations (FDEs) by using respectively the analytic or approximate solution of the differential equation, obtained by making fractional order of the original problem integer order. It is shown that this method works for FDEs very well. The results reveal that it is very effective and simple in determination of solutions of FDEs.

In this paper we consider the neural eld model for two neural populations. We investigate the properties of D-curves and we give some conditions for asymptotic stability. The asymptotic stability region is determined by using the Stépán's formula. Taking various delay terms into account, the eect of delay on the stability is investigated. Moreover we study on the stability cases by considering the real roots of the characteristic equation.

In this paper, a special case for a delayed neural field model is considered. After constructing its characteristic equation a stability analysis is made. Using Routh-Hurwitz criterion, some conditions for characteristic equation are given for the stability of the system.