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 condition...

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Bibliographic Details
Published in:Fractal and fractional 2023-11, Vol.7 (11), p.796
Main Authors: Shammaky, Amnah E., Youssef, Eslam M., Abdou, Mohamed A., ElBorai, Mahmoud M., ElSayed, Wagdy G., Taha, Mai
Format: Article
Language:English
Subjects:
Analysis
Banach spaces
Calculus
complex plane
Control theory
Differential equations
fixed point theorem
Fixed points (mathematics)
fractional integro-differential equations
Integral equations
Methods
Numerical methods
Polynomials
rationalized Haar wavelet
Riemann–Liouville fractional integral operator
Theorems
Wavelet analysis
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Summary: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.
ISSN:2504-3110
2504-3110
DOI:10.3390/fractalfract7110796