In this paper, the adomian decomposition method is applied to nonlinear heat equation with exponential nonlinearity. This method is tested for some examples. The results obtained show that the method is efficient and accurate. This study showed also, the speed of the convergent of Adomian decomposition method.

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The main purpose of this paper is to establish the existence, uniqueness and positive solution of a system of second-order boundary value problem with integral conditions. Using Banach's fixed point theorem and the Leray-Schauder nonlinear alternative, we discuss the existence and uniqueness solution of this problem, and we apply Guo-Krasnoselskii's fixed point theorem in cone to study the existence of positive solution. We also give some examples to illustrate our results. Keywords: nonlinear system of second-order boundary value problem; integral conditions; Banach's fixed point theorem; Leray-Schauder nonlinear alternative; Guo-Krasnoselskii's fixed point theorem in cone

Abstract. The main purpose of this paper is to study the problem of the existence, uniqueness and positivity of solutions of a system of higher order fractional differential equations with boundary value problem expressed by fractional and integral conditions. Using fixed point theorems, we discuss the existence and the uniqueness of solutions of this problem, and we apply GuoKrasnoselskii's fixed point theorem in cone to study the existence of positive solutions. We give some examples to illustrate our results.Mathematics subject classification (2010): 26A33, 34B15, 34B18, 34B27.