An efficient combination of Adomian Decomposition iterative technique coupled with Laplace transformation to solve non-linear Random Integro differential equation (NRIDE) is introduced in a novel way to get an accurate analytical solution. This technique is an elegant combination of theLaplace transform, and the Adomian polynomial. The suggested method will convert differential equations into iterative algebraic equations, thus reducing processing and analytical work. The technique solves the problem of calculating the Adomian polynomials. The method’s efficiency was investigated using some numerical instances, and the findings demonstrate that it is easier to use than many other numerical procedures. It has also been established that (LT

   In this work, a novel technique to obtain an accurate solutions to nonlinear form by multi-step combination with Laplace-variational approach (MSLVIM) is introduced. Compared with the  traditional approach for variational it overcome all difficulties and enable to provide us more an accurate solutions with extended of the convergence region as well as covering to larger intervals which providing us a continuous representation of approximate analytic solution and it give more better information of the solution over the whole time interval. This technique is more easier for obtaining the general Lagrange multiplier with reduces the time and calculations. It converges rapidly to exact formula with simply computable terms wit

     In this paper, the class of meromorphic multivalent functions of the form by using fractional differ-integral operators is introduced. We get Coefficients estimates, radii of convexity and star likeness. Also closure theorems and distortion theorem for the class ,  is calculaed.

    New class A^* (a,c,k,β,α,γ,μ)  is introduced of meromorphic univalent functions with positive coefficient f(z)=â–¡(1/z)+∑_(n=1)^∞▒〖a_n z^n 〗,(a_n≥0,z∈U^*,∀ n∈ N={1,2,3,…}) defined by the integral operator in the punctured unit disc U^*={z∈C∶0<|z|<1}, satisfying |(z^2 (I^k (L^* (a,c)f(z)))^''+2z(I^k (L^* (a,c)f(z)))^')/(βz(I^k (L^* (a,c)f(z)))^''-α(1+γ)z(I^k (L^* (a,c)f(z)))^' )|<μ,(0<μ≤1,0≤α,γ<1,0<β≤1/2 ,k=1,2,3,… ) . Several properties were studied like coefficient estimates, convex set and weighted mean.

In this paper, our aim is to study variational formulation and solutions of 2-dimensional integrodifferential equations of fractional order. We will give a summery of representation to the variational formulation of linear nonhomogenous 2-dimensional Volterra integro-differential equations of the second kind with fractional order. An example will be discussed and solved by using the MathCAD software package when it is needed.

  The aim of this paper is to prove a theorem on the Riesz means of expansions with respect to Riesz bases, which extends the previous results of [1] and [2] on the Schrödinger operator and the ordinary differential operator of 4-th order to the operator of order 2m by using the eigen functions of the ordinary differential operator. Some Symbols that used in the paper:     the uniform norm. <,>   the inner product in L2. G   the set of all boundary elements of G. ˆ u   the dual function of u.

In this article, the solvability of some proposal types of the multi-fractional integro-partial differential system has been discussed in details by using the concept of abstract Cauchy problem and certain semigroup operators and some necessary and sufficient conditions. 

The main aim of this paper is to apply a new technique suggested by Temimi and Ansari namely (TAM) for solving higher order Integro-Differential Equations. These equations are commonly hard to handle analytically so it is request numerical methods to get an efficient approximate solution. Series solutions of the problem under consideration are presented by means of the Iterative Method (IM). The numerical results show that the method is effective, accurate and easy to implement rapidly convergent series to the exact solution with minimum amount of computation. The MATLAB is used as a software for the calculations.           

In this work, we employ a new normalization Bernstein basis for solving linear Freadholm of fractional integro-differential equations  nonhomogeneous  of the second type (LFFIDE_s). We adopt Petrov-Galerkian method (PGM) to approximate solution of the (LFFIDE_s) via normalization Bernstein basis that yields linear system. Some examples are given and their results are shown in tables and figures, the Petrov-Galerkian method (PGM) is very effective and convenient and overcome the difficulty of traditional methods. We solve this problem (LFFIDE_s) by the assistance of Matlab10.