Abstract
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.
Highlights
It is common knowledge that a large number of phenomena are in essence nonlinear and can be modeled using nonlinear differential equations [1]
The following nonlinear differential equation elucidates the basic concept of the variational iteration method (VIM), as follows: L (y) + N (y) = g (x) where L and N are the linear and nonlinear operators, respectively, while g(x) is an inhomogeneous term
VIM is more reliable with Bernstein polynomial approximation when compared to dependent variable transcendental functions in differential equations
Summary
It is common knowledge that a large number of phenomena are in essence nonlinear and can be modeled using nonlinear differential equations [1]. The approach to tackle this shortcoming was to apply different numerical methods suffice to reach approximate solutions including the methods of Adomian decomposition [2, 3], homotopy perturbation [4,5,6], differential transform [7,8,9], and variational iteration [10, 11]. The VIM is a modified Lagrange multiplier method with promising results in solving numerous classes of nonlinear problems with approximations that converge to the correct solution within a short number of iterations. We applied the Bernstein polynomial approximations to nonlinear functions in the correction functions before iterating the numerical solution of PDEs. The main objective is to achieve reliable accurate PDEs solutions if using VIM leads to unstable solutions.
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