The techniques of fractional calculus are applied successfully in many branches of science and engineering, one of the techniques is the Elzaki Adomian decomposition method (EADM), which researchers did not study with the fractional derivative of Caputo Fabrizio. This work aims to study the Elzaki Adomian decomposition method (EADM) to solve fractional differential equations with the Caputo-Fabrizio derivative. We presented the algorithm of this method with the CF operator and discussed its convergence by using the method of the Cauchy series then, the method has applied to solve Burger, heat-like, and, couped Burger equations with the Caputo -Fabrizio operator. To conclude the method was convergent and effective for solving this type of

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.

This article introduces a novel procedure to detect an approximate solution to Fredholm fractional integro-differential equations with linear type (LFFIDE) defined using Caputo fractional derivative. The new procedure approximates the solution using three types of polynomials: Laguerre polynomials, Hermite polynomials, and Legendre polynomials, thereafter transforming the problem into a linear programming problem. The approximate solutions are compared using testing examples to examine the efficiency of the suggested approach. Also, a comparison with the other methods using the same polynomials illustrates The effectiveness and consistency of the proposed technique. Finally, the error analysis of the proposed technique and convergent are di

The aim of this article is to solve the Volterra-Fredholm integro-differential equations of fractional order numerically by using the shifted Jacobi polynomial collocation method. The Jacobi polynomial and collocation method properties are presented. This technique is used to convert the problem into the solution of linear algebraic equations. The fractional derivatives are considered in the Caputo sense. Numerical examples are given to show the accuracy and reliability of the proposed technique.

In this work, an analytical approximation solution is presented, as well as a comparison of the Variational Iteration Adomian Decomposition Method (VIADM) and the Modified Sumudu Transform Adomian Decomposition Method (M STADM), both of which are capable of solving nonlinear partial differential equations (NPDEs) such as nonhomogeneous Kertewege-de Vries (kdv) problems and the nonlinear Klein-Gordon. The results demonstrate the solution’s dependability and excellent accuracy.

Many numerical approaches have been suggested to solve nonlinear problems. In this paper, we suggest a new two-step iterative method for solving nonlinear equations. This iterative method has cubic convergence. Several numerical examples to illustrate the efficiency of this method by Comparison with other similar methods is given.

     The method of operational matrices is based on the Bernoulli and Shifted Legendre polynomials which is used to solve the Falkner-Skan equation. The nonlinear differential equation converting to a system of nonlinear equations is solved using Mathematica®12, and the approximate solutions are obtained. The efficiency of these methods was studied by calculating the maximum error remainder ( ), and it was found that their efficiency increases as  increases. Moreover, the obtained approximate solutions are compared with the numerical solution obtained by the fourth-order Runge-Kutta method (RK4), which gives  a good agreement.

The Korteweg-de Vries equation plays an important role in fluid physics and applied mathematics. This equation is a fundamental within study of shallow water waves. Since these equations arise in many applications and physical phenomena, it is officially showed that this equation has solitary waves as solutions, The Korteweg-de Vries equation is utilized to characterize a long waves travelling in channels. The goal of this paper is to construct the new effective frequent relation to resolve these problems where the semi analytic iterative technique presents new enforcement to solve Korteweg-de Vries equations. The distinctive feature of this method is, it can be utilized to get approximate solutions for travelling waves of