This article investigates different nonlinear systems of fractional partial differential equations analytically using an attractive modified method known as the Laplace residual power series technique. Based on a combination of the Laplace transformation and the residual power series technique, we achieve analytic and approximation results in rapid convergent series form by employing the notion of the limit, with less time and effort than the residual power series method. Three challenges are evaluated and simulated to validate the suggested method’s practicability, efficiency, and simplicity. The analysis of the acquired findings demonstrates that the method mentioned above is simple, accurate, and appropriate for investigating the solutions to nonlinear applied sciences models.

This article is dealing with the analytical solution of Fornberg–Whitham equations in fractional view of Caputo operator. The effective method among the analytical techniques, natural transform decomposition method, is implemented to handle the solutions of the proposed problems. The approximate analytical solutions of nonlinear numerical problems are determined to confirm the validity of the suggested technique. The solution of the fractional-order problems are investigated for the suggested mathematical models. The solutions-graphs are then plotted to understand the effectiveness of fractional-order mathematical modeling over integer-order modeling. It is observed that the derived solutions have a closed resemblance with the actual solutions. Moreover, using fractional-order modeling various dynamics can be analyzed which can provide sophisticated information about physical phenomena. The simple and straight-forward procedure of the suggested technique is the preferable point and thus can be used to solve other nonlinear fractional problems.

In this paper, the fractional view analysis of the Keller–Segal equations with sensitivity functions is presented. The Caputo operator has been used to pursue the present research work. The natural transform is combined with the homotopy perturbation method, and a new scheme for implementation is derived. The modified established method is named as the homotopy perturbation transform technique. The derived results are compared with the solution of the Laplace Adomian decomposition technique by using the systems of fractional Keller–Segal equations. The solution graphs and the table have shown that the obtained results coincide with the solution of the Laplace Adomian decomposition method. Fractional-order solutions are determined to confirm the reliability of the current method. It is observed that the solutions at various fractional orders are convergent to an integer-order solution of the problems. The suggested procedure is very attractive and straight forward and therefore can be modified to solve high nonlinear fractional partial differential equations and their systems.

Wavelet transforms or wavelet analysis represent a recently created mathematical tool for assistance in resolving various issues. Wavelets can also be used in numerical analysis. In this study, we solve pantograph delay differential equations using the Modified Laguerre Wavelet method (MLWM), an effective numerical technique. Fractional derivatives are defined using the Caputo operator. The convergence of the suggested strategy is carefully examined. The suggested strategy is straightforward, effective, and simple in comparison with previous approaches. Specific examples are provided to demonstrate the current scenario’s reliability and accuracy. Compared with other methodologies, our results show a higher accuracy level. With the aid of tables and graphs, we demonstrate the effectiveness of the proposed approach by comparing results of the actual and suggested methods and demonstrating their strong agreement. For better understanding of the proposed method, we show the pointwise solution in the tables provided which confirm the accuracy at each point of the proposed method. Additionally, the results of employing the suggested method to various fractional-orders are compared, which demonstrates that when a value shifts from fractional-order to integer-order, the result approaches the exact solution. Owing to its novelty and scientific significance, the suggested technique can also be used to solve additional nonlinear delay differential equations of fractional-order.

The resolution of a system for modeling the competition between opponents in a chemostat when one of these can produce a toxin has been studied. We propose a novel method to overcome the analytical difficulties of standard mathematical methods. The method is based on the variational iteration method and combined with the Gauss-Seidel technique for increasing the convergence rate. Numerical examples are considered to demonstrate the practicality and improve the convergence of the proposed method.