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.

We aim through this paper to present an improved variational iteration method (VIM) based on Bernstein polynomials (BP) approximations to be used with transcendental functions. The key benefits gained from this modification are to reach stable and fairly accurate results and, at the same time, to expand the unknown function’s domain in partial differential equations (PDEs). The proposed approach introduces the Bernstein polynomials in the transcendental functions of nonlinear PDEs. A number of examples were included in order to expound the method’s capacity and reliability. From the results, we conclude that the VIM with BP is a powerful mathematical tool that can be applied to solve nonlinear PDEs.

In this paper, the Domain decomposition method (ADM) with modified polynomials is applied for nonlinear (2+1) -dimensional Wu-Zhang system . we compared the solution of the system with MVIM, HPM and RDTM [ 17,13,15]. The numerical results obtained by this polynomial are very effective, convenient and quite accurate to system of partial differential equations. A comparative between the modifications method and the other methods is present from some examples to show the efficiency of each method.

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In this paper, an operational matrix of integrations based on the Haar wavelet method is applied for finding the numerical solution of non-linear third-order boussinesq system and the numerical results were compared with the exact solution. The accuracy of the obtained solutions is quite high even if the number of calculation points is small, by increasing the number of collocation points the error of the solution rapidly decreases as shown by solving an example. We have been reduced the boundary conditions in the solution by using the finite differences method with respect to time. Also we have reduced the order of boundary conditions used in the numerical solution by using the boundary condition at x=L instead of the derivatives of order two with respect to space.