Ali, Mustafa Ahmed; Merdan, Mehmet

This study examines approximate analytical solutions of the time-fractional Swift–Hohenberg equation involving the Caputo–Fabrizio fractional derivative. The main objective of this study is to investigate efficient analytical approximation techniques using the Yang transform Adomian decomposition method (YTADM). The proposed approach combines the Yang transform with the Adomian decomposition technique to construct a rapidly convergent series solution for the fractional partial differential equation. The effects of the fractional order and system parameters on the behavior of the solution have been examined via tables and are graphically represented. The results confirm that the proposed method provides accurate and computationally efficient approximate solutions to the fractional Swift–Hohenberg equation. These results show that YTADM is an appropriate and useful analytical tool to solve nonlinear fractional partial differential equations arising in mathematical physics, especially those equations involving nonsingular fractional operators. The principal contribution of this work effort is the development and application of the YTADM framework for solving the fractional Swift–Hohenberg equation; therefore, it serves as a reliable analytic tool for nonlinear fractional partial differential equations arising in mathematical physics.

Ali, Mustafa Ahmed; Merdan, Mehmet

In this paper, the Yang transform Adomian decomposition method (YTADM) is employed in the solution of nonlinear time-fractional coupled Burgers equations. The technique solves the fractional and nonlinear terms successfully via the Adomian decomposition of the Yang transform. Illustrative examples are given, which show the accuracy and ease of implementation of the method. The results show that the YTADM method gives good, reliable results and is very accurate, so that its solutions agree well with the exact solution. Finally, tables and graphs are shown, which illustrate the behavior of the solutions found.

Şişman, Şeyma; Merdan, Mehmet

The differential transformation method (DTM) is one of the best methods easily applied to linear and nonlinear difference equations with random coefficients. In this study, we apply the theorems related to the DTM to the given examples and investigate the behaviour of the approximate analytical solutions. The expected value, variance, coefficient of variation, and confidence intervals of the solutions of random difference equations obtained from discrete probability distributions such as uniform, geometric, Poisson, and binomial distributions will be calculated. Maple and MATLAB software packages are used to plot the solution graphs and also to interpret the solution behaviour.