In this paper, three approximate methods namely the Bernoulli, the Bernstein, and the shifted Legendre polynomials operational matrices are presented to solve two important nonlinear ordinary differential equations that appeared in engineering and applied science. The Riccati and the Darcy-Brinkman-Forchheimer moment equations are solved and the approximate solutions are obtained. The methods are summarized by converting the nonlinear differential equations into a nonlinear system of algebraic equations that is solved using Mathematica®12. The efficiency of these methods was investigated by calculating the root mean square error (RMS) and the maximum error remainder (𝑀𝐸𝑅n) and it was found that the accuracy increases with increasi

This paper studies a novel technique based on the use of two effective methods like modified Laplace- variational method (MLVIM) and a new Variational method (MVIM)to solve PDEs with variable coefficients. The current modification for the (MLVIM) is based on coupling of the Variational method (VIM) and Laplace- method (LT). In our proposal there is no need to calculate Lagrange multiplier. We applied Laplace method to the problem .Furthermore, the nonlinear terms for this problem is solved using homotopy method (HPM). Some examples are taken to compare results between two methods and to verify the reliability of our present methods.

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.

 In this paper, we study the convergence theorems of the Modified Ishikawa iterative sequence with mixed errors for the uniformly continuous mappings and solving nonlinear uniformly continuous mappings equation in arbitrary real Banach space.

In this paper, we present an approximate analytical and numerical solutions for the differential equations with multiple delay using the extend differential transform method (DTM). This method is used to solve many linear and non linear problems.

An approximate solution of the liner system of ntegral cquations fot both fredholm(SFIEs)and Volterra(SIES)types has been derived using taylor series expansion.The solusion is essentailly

This work describes two efficient and useful methods for solving fractional pantograph delay equations (FPDEs) with initial and boundary conditions. These two methods depend mainly on orthogonal polynomials, which are the method of the operational matrix of fractional derivative that depends on Bernstein polynomials and the operational matrix of the fractional derivative with Shifted Legendre polynomials. The basic procedure of this method is to convert the pantograph delay equation to a system of linear equations and by using, the operational matrices we get rid of the integration and differentiation operations, which makes solving the problem easier. The concept of Caputo has been used to describe fractional derivatives. Finally, some

       In this work, the fractional damped Burger's equation (FDBE) formula    = 0,

EDIRKTO, an Implicit Type Runge-Kutta  Method of Diagonally Embedded pairs, is a novel approach presented in the paper that may be used to solve 4th-order ordinary differential equations of the form . There are two pairs of EDIRKTO, with three stages each: EDIRKTO4(3) and EDIRKTO5(4). The derivation techniques of the method indicate that the higher-order pair is more accurate, while the lower-order pair provides superior error estimates. Next, using these pairs as a basis, we developed variable step codes and applied them to a series of -order ODE problems. The numerical outcomes demonstrated how much more effective their approach is in reducing the quantity of function evaluations needed to resolve fourth-order ODE issues.