In this paper a modified approach have been used to find the approximate solution of ordinary delay differential equations with constant delay using the collocation method based on Bernstien polynomials.

In this paper Volterra Runge-Kutta methods which include: method of order two and four will be applied to general nonlinear Volterra integral equations of the second kind. Moreover we study the convergent of the algorithms of Volterra Runge-Kutta methods. Finally, programs for each method are written in MATLAB language and a comparison between the two types has been made depending on the least square errors.

This paper presents a new transform method to solve partial differential equations, for finding suitable accurate solutions in a wider domain. It can be used to solve the problems without resorting to the frequency domain. The new transform is combined with the homotopy perturbation method in order to solve three dimensional second order partial differential equations with initial condition, and the convergence of the solution to the exact form is proved. The implementation of the suggested method demonstrates the usefulness in finding exact solutions. The practical implications show the effectiveness of approach and it is easily implemented in finding exact solutions.

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The main work of this paper is devoted to a new technique of constructing approximated solutions for linear delay differential equations using the basis functions power series functions with the aid of Weighted residual methods (collocations method, Galerkin’s method and least square method).

The author obtain results on the asymptotic behavior of the nonoscillatory solutions of first order nonlinear neutral differential equations. Keywords. Neutral differential equations, Oscillatory and Nonoscillatory solutions.

The paper is devoted to solve nth order linear delay integro-differential equations of convolution type (DIDE's-CT) using collocation method with the aid of B-spline functions. A new algorithm with the aid of Matlab language is derived to treat numerically three types (retarded, neutral and mixed) of nth order linear DIDE's-CT using B-spline functions and Weddle rule for calculating the required integrals for these equations. Comparison between approximated and exact results has been given in test examples with suitable graphing for every example for solving three types of linear DIDE's-CT of different orders for conciliated the accuracy of the results of the proposed method.

In this study, a brand-new double transform known as the double INEM transform is introduced. Combined with the definition and essential features of the proposed double transform, new findings on partial derivatives, Heaviside function, are also presented. Additionally, we solve several symmetric applications to show how effective the provided transform is at resolving partial differential equation.