In this paper, we consider the nonlinear equations with the additive white noise, which are commonly impossible to be solved by an analytical procedure. The Block-Pulse functions as basic functions are proposed to solve these equations. In order to investigate the validity of this method, we used the Adomian decomposition method to Approximate the Solution of the stochastic Du ng equations. The results reveal that the proposed method is very e ective.

In this paper, we introduce an efficient method for solving the quadratic Riccati differential equation. In this technique, combination of Laplace trans-form and new homotopy perturbation methods (LTNHPM) are considered as an algorithm to the exact Solution of the nonlinear Riccati equation. Un-like the previous approach for this problem, so-called NHPM, the present method, does not need the initial approximation to be defined as a power series. Four examples in different cases are given to demonstrate simplicity and efficiency of the proposed method.

In this paper, sinc-collocation method is discussed to solve Volterra functional integral equations with delay function q (t). Also the existence and uniqueness of numerical Solutions for these equations are provided. This method improves conventional results and achieves exponential convergence. Numerical results are included to confirm the efficiency and accuracy of the method.

In this work, we conduct a comparative study among the combine Laplace trans-form and modified Adomian decomposition method (LMADM) and two traditional methods for an analytic and Approximate treatment of special type of nonlinear Volterra integro-differential equations of the second kind. The nonlinear part of integro-differential is approx-imated by Adomian polynomials, and the equation is reduced to a simple equations. The proper implementation of combine Laplace transform and modified Adomian decomposition method can extremely minimize the size of work if compared to existing traditional tech-niques. Moreover, three particular examples are discussed to show the reliability and the performance of method.

Generalized Solution on Neumann problem of the fourth order ordinary differential equation in space W2a(0, b) has been discussed, we obtain the condition on B.V.P when the Solution is in classical form. Formulation of Quintic Spline Function has been derived and the consistency relations are given. Numerical method, based on Quintic spline approximation has been developed. Spline Solution of the given problem has been considered for a certain value of a: Error analysis of the spline method is given and it has been tested by an example.

The Riesz fractional advection-diffusion is a result of the mechanics of chaotic dynamics. It’, s of preponderant importance to solve this equation numerically. Moreover, the utilization of Chebyshev polynomials as a base in several mathematical equations shows the exponential rate of convergence. To this approach, we transform the interval of state space into the interval [−, 1, 1] × [−, 1, 1]. Then, we use the operational matrix to discretize fractional operators. Applying the resulting discretization, we obtain a linear system of equations, which leads to the numerical Solution. Examples show the effectiveness of the method.