Amnah E. Shammaky, Eslam M. Youssef, Mohamed A. Abdou, Mahmoud M. ElBorai, Wagdy G. ElSayed, Mai Taha
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A New Technique for Solving a Nonlinear Integro-Differential Equation with Fractional Order in Complex Space
This work aims to explore the solution of a nonlinear fractional integro-differential equation in the complex domain through the utilization of both analytical and numerical approaches. The demonstration of the existence and uniqueness of a solution is established under certain appropriate conditions with the use of Banach fixed point theorems. To date, no research effort has been undertaken to look into the solution of this integro equation, particularly due to its fractional order specification within the complex plane. The validation of the proposed methodology was performed by utilizing a novel strategy that involves implementing the Rationalized Haar wavelet numerical method with the application of the Bernoulli polynomial technique. The primary reason for choosing the proposed technique lies in its ability to transform the solution of the given nonlinear fractional integro-differential equation into a representation that corresponds to a linear system of algebraic equations. Furthermore, we conduct a comparative analysis between the outcomes obtained from the suggested method and those derived from the rationalized Haar wavelet method without employing any shared mathematical methodologies. In order to evaluate the precision and effectiveness of the proposed method, a series of numerical examples have been developed.
期刊介绍:
Fractal and Fractional is an international, scientific, peer-reviewed, open access journal that focuses on the study of fractals and fractional calculus, as well as their applications across various fields of science and engineering. It is published monthly online by MDPI and offers a cutting-edge platform for research papers, reviews, and short notes in this specialized area. The journal, identified by ISSN 2504-3110, encourages scientists to submit their experimental and theoretical findings in great detail, with no limits on the length of manuscripts to ensure reproducibility. A key objective is to facilitate the publication of detailed research, including experimental procedures and calculations. "Fractal and Fractional" also stands out for its unique offerings: it warmly welcomes manuscripts related to research proposals and innovative ideas, and allows for the deposition of electronic files containing detailed calculations and experimental protocols as supplementary material.