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Numerical methods in the area of nonlinear systems are extensively implemented for computing their approximate solutions because these systems are very difficult to tackle analytically. There are various numerical techniques available in the literature to find the solutions of nonlinear oscillators. Variational iteration method (VIM) is one of these approaches which is convenient to implement for these kinds of problems. In this work, our study aims to identify the numerical solution of nonlinear oscillator by making use of variational iteration method associated with Formable transformation. For the smooth utilization of this approach, we have to formulate the Lagrange multiplier through variational theory. Furthermore, we develop a new unified iterative scheme for the correction functional of VIM, considering the Formable transformation. Several new schemes of correction functional can be deduced from the newly proposed method considering the duality relation of Formable transform. In support of our primary finding, we discuss numerical example as application. A number of Physical applications of nonlinear oscillators are available in the field of vibrations and oscillations but in recent times nonlinear oscillators are used to describe complicated systems or to address mechanical, electrical, and other engineering phenomenon.

This paper aims to investigate a new efficient method for solving time fractional partial differential equations. In this orientation, a reliable formable transform decomposition method has been designed and developed, which is a novel combination of the formable integral transform and the decomposition method. Basically, certain accurate solutions for time-fractional partial differential equations have been presented. The method under concern demands more simple calculations and fewer efforts compared to the existing methods. Besides, the posed formable transform decomposition method has been utilized to yield a series solution for given fractional partial differential equations. Moreover, several interesting formulas relevant to the formable integral transform are applied to fractional operators which are performed as an excellent application to the existing theory. Furthermore, the formable transform decomposition method has been employed for finding a series solution to a time-fractional Klein-Gordon equation. Over and above, some numerical simulations are also provided to ensure reliability and accuracy of the new approach.

The present work introduces a novel approach, the Adomian Decomposition Formable Transform Method (ADFTM), and its application to solve the fractional order Sharma-Tasso-Olver problem. The method’s distinctive outcomes are highlighted through a comparative analysis with established non-local Caputo fractional derivatives and the non-singular Atangana–Baleanu (ABC) fractional derivatives. To provide a comprehensive understanding, the proposed ADFTM’s approximate solution is compared with the homotopy perturbation method (HPM) and residual power series method (RPSM). Further, numerical and graphical results demonstrate the reliability and accuracy of the ADFTM approach. The novel outcomes presented in this work emphasize its capability to address complex engineering problems effectively. By demonstrating its efficacy in solving the fractional order problems, the new ADFTM proves to be a valuable tool in solving scientific problems.

Chemistry, physics, and many other applied fields depend heavily on partial differential equations. As a result, the literature contains a variety of techniques that all have a symmetry goal for solving partial differential equations. This study introduces a new double transform known as the double formable transform. New results on partial derivatives and the double convolution theorem are also presented, together with the definition and fundamental characteristics of the proposed double transform. Moreover, we use a new approach to solve a number of symmetric applications with different characteristics on the heat equation to demonstrate the usefulness of the provided transform in solving partial differential equations.

In this paper, the solutions of the (2+1)-D and (3+1)-D Schrodinger equations are investigated mathematically using two efficient semi-analytical techniques. One proposed technique is based on the combination of the formable transform and the homotopy perturbation method (FTHPM), whereas another technique is the classical variational iteration method (VIM). A comparison study between the formable transform-based homotopy perturbation method (FTHPM) and the variational iteration method (VIM) for solving these equations is discussed. Some theorems are presented to illustrate the convergence of both semi-analytical techniques. To verify the accuracy and efficiency of the proposed schemes, two test examples are discussed.