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A Physical Phenomenon for the Fractional Nonlinear Mixed Integro-Differential Equation with Local and Nonlocal Conditions Using a Toeplitz Matrix Technique with a Genetic Application
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DOI：10.3390/fractalfract10080549.png)
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Abstract
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Nonlocal circumstances in genetic engineering are crucial as they pertain to the understanding of genetic material. When these conditions are associated with differential integral equations, particularly concerning the time variable, they yield comprehensive insights into the material’s temporal memory, which can be advantageous for understanding all material properties (including chronic conditions or behavioral characteristics), thereby assisting specialists in managing its future evolution. The novelty of this manuscript resides in the exploration of fractional nonlinear mixed integro-differential equations (FrN-MIo-DE) under nonlocal conditions, employing the Toeplitz matrix method with a genetic application. This issue has previously been examined via the Nyström technique and solely under local conditions. A category of mathematical problems prevalent in many domains, including physics, engineering, and biological systems, is fractional calculus. Fractional calculus, which generalizes classical differentiation and integration to non-integer orders, offers a robust foundation for modeling memory and hereditary characteristics in complex systems. We examine the existence and uniqueness of solutions to FrNMIo-DE under nonlocal restrictions, using a discontinuous kernel dependent on location and time-space L 2 [ − 1,1 ] × C [ 0 , T ] , where T < 1, via analytical methods. According to the features of fractional integrals, FrNMIo-DE adheres to the second-kind Volterra–Hammerstein integral equation (V-HIE), characterized by a discontinuous kernel in position for the Hammerstein integral term and a continuous kernel in time for the Volterra integral (VI) term. Subsequently, we use a separation approach technique to produce HIE with time-dependent physical coefficients. Following an analysis of the system’s convergence, a nonlinear algebraic system (NAS) is constructed using the Toeplitz matrix technique (TMT) and related methodologies. The numerical data and associated errors are shown via the Maple 2022 software.
Keywords:
fractional nonlinear integro-differential equation
discontinuous kernel
nonlinear algebraic system
Toeplitz matrix technique
convergence error rate
genetic algorithm
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