The method of operational matrices based on different types of polynomials such as Bernstein, shifted Legendre and Bernoulli polynomials will be presented and implemented to solve the nonlinear Blasius equations approximately. The nonlinear differential equation will be converted into a system of nonlinear algebraic equations that can be solved using Mathematica®12. The efficiency of these methods has been studied by calculating the maximum error remainder ( ), and it was found that their efficiency increases as the polynomial degree (n) increases, since the errors decrease. Moreover, the approximate solutions obtained by the proposed methods are compared with the solution of the 4th order Runge-Kutta method (RK4), which gives very

        In this paper, double Sumudu and double Elzaki transforms methods are used to compute the numerical solutions for some types of fractional order partial differential equations with constant coefficients and explaining the efficiently of the method by illustrating some numerical examples that are computed by using  Mathcad 15.and graphic in Matlab R2015a.

In the present paper, three reliable iterative methods are given and implemented to solve the 1D, 2D and 3D Fisher’s equation. Daftardar-Jafari method (DJM), Temimi-Ansari method (TAM) and Banach contraction method (BCM) are applied to get the exact and numerical solutions for Fisher's equations. The reliable iterative methods are characterized by many advantages, such as being free of derivatives, overcoming the difficulty arising when calculating the Adomian polynomial boundaries to deal with nonlinear terms in the Adomian decomposition method (ADM), does not request to calculate Lagrange multiplier as in the Variational iteration method (VIM) and there is no need to create a homotopy like in the Homotopy perturbation method (H

This paper aims to propose a hybrid approach of two powerful methods, namely the differential transform and finite difference methods, to obtain the solution of the coupled Whitham-Broer-Kaup-Like equations which arises in shallow-water wave theory. The capability of the method to such problems is verified by taking different parameters and initial conditions. The numerical simulations are depicted in 2D and 3D graphs. It is shown that the used approach returns accurate solutions for this type of problems in comparison with the analytic ones.

This manuscript presents several applications for solving special kinds of ordinary and partial differential equations using iteration methods such as Adomian decomposition method (ADM), Variation iterative method (VIM) and Taylor series method. These methods can be applied as well as to solve nonperturbed problems and 3rd order parabolic PDEs with variable coefficient. Moreover, we compare the results using ADM, VIM and Taylor series method. These methods are a commination of the two initial conditions.

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.

     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.

     In this article, a new efficient approach is presented to solve a type of partial differential equations, such (2+1)-dimensional differential equations non-linear, and nonhomogeneous. The procedure of the new approach is suggested to solve important types of differential equations and get accurate analytic solutions i.e., exact solutions. The effectiveness of the suggested approach based on its properties compared with other approaches has been used to solve this type of differential equations such as the Adomain decomposition method, homotopy perturbation method, homotopy analysis method, and variation iteration method. The advantage of the present method has been illustrated by some examples.

       In the present paper, by making use of the new generalized operator, some results of third order differential subordination and differential superordination consequence for analytic functions are obtained. Also, some sandwich-type theorems are presented.

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.