Computational Study of Time-Fractional Kawahara and Modified Kawahara Equations with Caputo Derivatives Using Natural Homotopy Transform Method
Abstract
1. Introduction
2. Basic Definitions of Fractional Calculus and Natural Transform
3. Formulation of Natural Homotopy Transform Scheme
- Step 3. The exact solution of Equation (5) is specified as
- Step 4. Upon conducting an analysis of p on the two sides, the resulting outcome is presented as
- Step 5. As a result, we are able to summarize the findings of this iterative series as follows:
4. Natural Transformation over Existence and Convergence Analysis
4.1. Existence of NT with Sufficient Condition
4.2. Uniqueness Theorem for NHTS
4.3. Convergence Study of NHTS
5. Numerical Problems
5.1. Problem 1
5.2. Problem 2
6. Results and Discussion
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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Nadeem, M.; Iambor, L.F.; Alzahrani, E.; Ajmal, A.H.P. Computational Study of Time-Fractional Kawahara and Modified Kawahara Equations with Caputo Derivatives Using Natural Homotopy Transform Method. Fractal Fract. 2025, 9, 247. https://doi.org/10.3390/fractalfract9040247
Nadeem M, Iambor LF, Alzahrani E, Ajmal AHP. Computational Study of Time-Fractional Kawahara and Modified Kawahara Equations with Caputo Derivatives Using Natural Homotopy Transform Method. Fractal and Fractional. 2025; 9(4):247. https://doi.org/10.3390/fractalfract9040247
Chicago/Turabian StyleNadeem, Muhammad, Loredana Florentina Iambor, Ebraheem Alzahrani, and Azeem Hafiz P. Ajmal. 2025. "Computational Study of Time-Fractional Kawahara and Modified Kawahara Equations with Caputo Derivatives Using Natural Homotopy Transform Method" Fractal and Fractional 9, no. 4: 247. https://doi.org/10.3390/fractalfract9040247
APA StyleNadeem, M., Iambor, L. F., Alzahrani, E., & Ajmal, A. H. P. (2025). Computational Study of Time-Fractional Kawahara and Modified Kawahara Equations with Caputo Derivatives Using Natural Homotopy Transform Method. Fractal and Fractional, 9(4), 247. https://doi.org/10.3390/fractalfract9040247