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A New Scheme of the ARA Transform for Solving Fractional-Order Waves-Like Equations Involving Variable Coefficients

Yu-Ming Chu1, Sobia Sultana2, Shazia Karim3, Saima Rashid4,*, Mohammed Shaaf Alharthi5

1 Department of Mathematics, Huzhou University, Huzhou, 313000, China
2 Department of Mathematics, Imam Mohammad Ibn Saud Islamic University, Riyadh, 11461, Saudi Arabia
3 Department of Basic Sciences and Humanities, UET Lahore, Faisalabad Campus, 54800, Pakistan
4 Department of Mathematics, Government College University, Faisalabad, 38000, Pakistan
5 Department of Mathematics and Statistics, College of Science, Taif University, P. O. Box 11099, Taif, 21944, Saudi Arabia

* Corresponding Author: Saima Rashid. Email:

(This article belongs to this Special Issue: On Innovative Ideas in Pure and Applied Mathematics with Applications)

Computer Modeling in Engineering & Sciences 2024, 138(1), 761-791.


The goal of this research is to develop a new, simplified analytical method known as the ARA-residue power series method for obtaining exact-approximate solutions employing Caputo type fractional partial differential equations (PDEs) with variable coefficient. ARA-transform is a robust and highly flexible generalization that unifies several existing transforms. The key concept behind this method is to create approximate series outcomes by implementing the ARA-transform and Taylor’s expansion. The process of finding approximations for dynamical fractional-order PDEs is challenging, but the ARA-residual power series technique magnifies this challenge by articulating the solution in a series pattern and then determining the series coefficients by employing the residual component and the limit at infinity concepts. This approach is effective and useful for solving a massive class of fractional-order PDEs. Five appealing implementations are taken into consideration to demonstrate the effectiveness of the projected technique in creating solitary series findings for the governing equations with variable coefficients. Additionally, several visualizations are drawn for different fractional-order values. Besides that, the estimated findings by the proposed technique are in close agreement with the exact outcomes. Finally, statistical analyses further validate the efficacy, dependability and steady interconnectivity of the suggested ARA-residue power series approach.


Cite This Article

Chu, Y., Sultana, S., Karim, S., Rashid, S., Alharthi, M. S. (2024). A New Scheme of the ARA Transform for Solving Fractional-Order Waves-Like Equations Involving Variable Coefficients. CMES-Computer Modeling in Engineering & Sciences, 138(1), 761–791.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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