English abstract
Adomian Decomposition Method (ADM) has proven to be an efficient and reliable method for solving many types of problems. The ADM has been used for a wide range of equation types (like boundary value problems, integral equations, equations arising in flow of incompressible and compressible fluids etc...). In this thesis we deal with fractional-order models which are more adequate than the integer-order models. Moreover the fractional derivatives and integrals enable the description of memory and hereditary properties inherent in various materials and processes. So our objective in this study is to study the solution of fractional nonlinear Volterra-Fredholm integro differential equation of the form: 1 y ve (= + P&* (x+5(*)+,4,6,8)F,(() dt ŹB,(x,t},v(t)}dt, m-l<asm dt, m-1<asm a /= The fractional derivative y(a)(x) is described in the Caputo sense. In the thesis, we implement the Adomian Decomposition method linked with Laplace transform to solve the above form. This technique is more powerful than ADM because we combine Laplace and ADM to obtain ALDM and it will provide exact and approximate analytical solutions. Furthermore, ALDM, can provide high accuracy of numerical results, reduce the computational time and volume of work. So this work is devoted to an evaluation of the effectiveness of the proposed method. Illustrative examples are given, and numerical results are provided to demonstrate the efficiency of the proposed methods. One more important contribution in the thesis is the demonstration of the solution to the above form when 2 =1, = 0 and 4; (X,1) has the form K(x-t). Moreover, as new and important special case, partial integro-differential equations with fractional order have thoroughly been solved by (ADM) and illustrative examples are provided. The obtained results in this work show that the proposed method is an efficient one in terms of its simplicity, implementation and high accuracy
