This study presents a practical method for solving fractional order delay variational problems. The fractional derivative is given in the Caputo sense. The suggested approach is based on the Laplace transform and the shifted Legendre polynomials by approximating the candidate function by the shifted Legendre series with unknown coefficients yet to be determined. The proposed method converts the fractional order delay variational problem into a set of (n ＋ 1) algebraic equations, where the solution to the resultant equation provides us the unknown coefficients of the terminated series that have been utilized to approximate the solution to the considered variational problem. Illustrative examples are given to show that the recommended appro

The aim of this paper is to propose an efficient three steps iterative method for finding the zeros of the nonlinear equation f(x)=0 . Starting with a suitably chosen , the method generates a sequence of iterates converging to the root. The convergence analysis is proved to establish its five order of convergence. Several examples are given to illustrate the efficiency of the proposed new method and its comparison with other methods.

The γ- mixing ratios of γ- transitions from levels of 56Fe populated in reaction are calculated using least square fitting program for the first time in the case of pure and mixed transitions the results obtained have been compound with γ Values determined by other methods .The comparison shows that the agreement is good this confirmed the valilety of this method in calculating of values for such γ- transitions key word: γ- transition ,Multipole mixing ratios ,Least square fitting method.

In this paper, we present new algorithm for the solution of the nonlinear high order multi-point boundary value problem with suitable multi boundary conditions. The algorithm is based on the semi-analytic technique and the solutions are calculated in the form of a rapid convergent series. It is observed that the method gives more realistic series solution that converges very rapidly in physical problems. Illustrative examples are provided to demonstrate the efficiency and simplicity of the proposed method in solving this type of multi- point boundary value problems.

    In this paper, we present new algorithm for the solution of the second order nonlinear three-point boundary value problem with suitable multi boundary conditions. The algorithm is based on the semi-analytic technique and the solutions which are calculated in the form of a rapid convergent series. It is observed that the method gives more realistic series solution that converges very rapidly in physical problems. Illustrative examples are provided to demonstrate the efficiency and simplicity of the proposed method in solving this type of three point boundary value problems.

In this paper, the proposed phase fitted and amplification fitted of the Runge-Kutta-Fehlberg method were derived on the basis of existing method of 4(5) order to solve ordinary differential equations with oscillatory solutions. The recent method has null phase-lag and zero dissipation properties. The phase-lag or dispersion error is the angle between the real solution and the approximate solution. While the dissipation is the distance of the numerical solution from the basic periodic solution. Many of problems are tested over a long interval, and the numerical results have shown that the present method is more precise than the 4(5) Runge-Kutta-Fehlberg 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.

 The aim of this paper is to present a semi - analytic technique for solving singular initial value problems of ordinary differential equations with a singularity of different kinds to construct polynomial solution using two point  osculatory  interpolation.           The efficiency and accuracy of suggested method is assessed by comparisons with exact and other approximate solutions for a wide classes of non–homogeneous, non–linear singular initial value problems.             A new, efficient estimate of the global error is used for adaptive mesh selection. Also, analyze some of the numerical aspects