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This study presents a new algorithm for effectively solving the nonlinear coupled Drinfeld’s–Sokolov–Wilson (DSW) system using a hybrid explicit finite difference technique with the Adomian polynomial (EFD-AP). The suggested approach addresses the problem of accurately solving the DSW system. Numerical results are obtained by comparing the exact solution with absolute and mean square errors using a test problem to assess the EFD-AP accuracy against the exact solution and the conventional explicit finite difference (EFD) method. The results exhibit excellent agreement between the approximate and exact solutions at different time values, and the results showed that the proposed EFD-AP method achieves superior accuracy and efficiency compared to the EFD method, which makes it a promising method for solving nonlinear partial differential systems of higher order.

In this article, we implemented the Elzaki decomposition technique (EDM) to solve Volterra–Fredholm integro-differential equations of higher-order. Illustrations are used to test the technique’s accuracy and validity. Comparison among the acquired consequences by EDM and actual solutions have proven the power and accuracy of this technique. This technique is dependable and able to supply analytic remedies for solving such equations.

The objective of this research paper is to find the approximate solution to the fractional partial differential equations, which are difficult to find the exact solution, and therefore, by relying on the modified Bernstein polynomial, the study suggested generalizing the method to be available for both space and time fractional partial differential equations. In order to investigate the ability of the method to reach the approximate solution, the method was applied to the coupled of space-time-fractional of the equal width wave equations(FCEWE) using Caputo’s definition of the fractional derivative, the results of the application showed the ability of the method to find the approximate solution.

This paper deals with approximation solution for coupled of space-time-fractional of both the equal width wave equation(FCEWE) and the modified equal width wave equation (FCMEWE) using Bernstein polynomials with collocation method and employing the Caputo definition for fractional derivatives. The method reduces the coupled system to a system of algebraic equations which is simple in handling and gives the best results.

In this work, a combined technique the Variation Iteration Method (VIM) with the Trapezoidal Rule (TR) was recommend to solve linear and nonlinear fractional ordinary differential equations (F.O.D.E.), where the results obtained from the Variation Iteration method were improved, and numerical results were obtained by determining the maximum absolute errors (MAE) and mean square error (MSE) for the given examples. As the results It is proved that that the proposed method is better than the default method.

In this paper, we used Bernstein polynomials to modify the Adomian decomposition method which can be used to solve linear and nonlinear equations. This scheme is tested for four examples from ordinary and partial differential equations; furthermore, the obtained results demonstrate reliability and activity of the proposed technique. This strategy gives a precise and productive system in comparison with other traditional techniques and the arrangements methodology is extremely straightforward and few emphasis prompts high exact solution. The numerical outcomes showed that the acquired estimated solutions were in appropriate concurrence with the correct solution.

In this thesis, a novel wavelet method is successfully used to find the numerical solution to the Burger's Fisher problem. This strategy exhibits reasonably rapid convergence when compared to other strategies currently in use. Examples are given as illustrations to show how the new wavelet method works and how strong it is. By comparing the results, finding the precise solution, and using the numerical output from the cas wavelet and the haar wavelet, we were able to develop the general formulas for n different wavelet integrals

In this paper, To solve nonlinear PDE, we employed a new wavelet formula derived from the notion of convolution. Analyzing the operational matrix of integration, we developed the general formulas for n of integrals to new wavelets. To solve the nonlinear Korteweg-de Vries problem, the proposed approach with collocation points is applied (KDV). Even though the step size utilized was huge, the results were good, and the accuracy of the derived answers is pretty high despite the tiny number of calculation points. The procedure is more precise, better, and gets closer to the correct solution

In this paper we study a numerical solution of coupled BBM systems of Boussinesq type, which describes approximately the two ways propagation of surface wave in a uniform horizontal channel of length l filled in its undisturbed depth h. This paper is devoted to drive the matrix algebraic equation for the one-dimensional nonlinear BBM system which is obtaining from using the implicit finite difference method. The convergence analysis of the solution is proved. Numerical experiments are presented with a variety of initial conditions describing the generation and evolution of such waves, and their interactions.

في هذا البحث، تم تطبيق مصفوفة العوامل للمتكاملات التي تعتمد على مويجة Haar لإيجاد الحل العددي لمعادلة korteweg-de vries-Burger's غير الخطية من الرتبة الثالثة وقد قورنت النتائج مع الحل المضبوط. إن دقة الحلول التي حصلنا عليها عالية حتى إذا كان عدد نقاط الشبكة المحسوبة قليلا وكلما زادت عدد نقاط الشبكة المحسوبة فان الدقة تزداد والخطأ يتناقص وقد تم توضيح ذلك من خلال حل مثال. لقد تم أيضا تخفيض رتبة الشروط الحدودية المطلوبة في الحل العددي وذلك باستخدام طريقة الفروقات المنتهية بالنسبة للزمن وكذلك تم تخفيض رتبة الشروط الحدودية بالنسبة للبعد وذلك باستخدام الشروط الحدودية عند نهاية الفترة x=L بدلا من الشروط الحدودية للمشتقة الثانية.