Implementation of variational iteration method for various types of linear and nonlinear partial differential equations

Abstract

There are various linear and nonlinear one-dimensional partial differential equations that are the focus of this research. There are a large number of these equations that cannot be solved analytically or precisely. The evaluation of nonlinear partial differential equations, even if analytical solutions exist, may be problematic. Therefore, it may be necessary to use approximate analytical methodologies to solve these issues. As a result, a more effective and accurate approach must be investigated and analyzed. It is shown in this study that the Lagrange multiplier may be used to get an ideal value for parameters in a functional form and then used to construct an iterative series solution. Linear and nonlinear partial differential equations may both be solved using the variational iteration method (VIM) method, thanks to its high computing power and high efficiency. Decoding and analyzing possible Korteweg-De-Vries, Benjamin, and Airy equations demonstrates the method’s ability. With just a few iterations, the produced findings are very effective, precise, and convergent to the exact answer. As a result, solving nonlinear equations using VIM is regarded as a viable option.

A new approximate solution of the fractional trigonometric functions of commensurate order to a regular linear system

“…These formulas are an important improvement and provide accurate and powerful solutions to Riemann-Liouville fractional derivatives. A study by Shihab et al [16] showed the use of the method of variational iteration in solving several types of partial differential equations, whether linear or nonlinear. In this study, it was shown that the Lagrange multiplier can be used to determine an ideal value of parameters in a functional form, and then use these values to construct an iterative series solution.…”