On the Uniqueness of the Bounded Solution for the Fractional Nonlinear Partial Integro-Differential Equation with Approximations

This paper studies the uniqueness of the bounded solution to a new Cauchy problem of the fractional nonlinear partial integro-differential equation based on the multivariate Mittag–Leffler function as well as Banach’s contractive principle in a complete function space. Applying Babenko’s approach, w...

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Published in:	Mathematics (Basel) Vol. 11; no. 12; p. 2752
Main Authors:	Li, Chenkuan, Saadati, Reza, Beaudin, Joshua, Hrytsenko, Andrii
Format:	Journal Article
Language:	English
Published:	Basel MDPI AG 01.06.2023
Subjects:	adomian’s decomposition method Algorithms analytic and approximate solution babenko’s approach Banach spaces banach’s contractive principle Cauchy problems Decomposition Differential equations Function space Integral equations Mathematics multivariate Mittag–Leffler function Nonlinear equations Partial differential equations Uniqueness
ISSN:	2227-7390, 2227-7390
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Abstract	This paper studies the uniqueness of the bounded solution to a new Cauchy problem of the fractional nonlinear partial integro-differential equation based on the multivariate Mittag–Leffler function as well as Banach’s contractive principle in a complete function space. Applying Babenko’s approach, we convert the fractional nonlinear equation with variable coefficients to an implicit integral equation, which is a powerful method of studying the uniqueness of solutions. Furthermore, we construct algorithms for finding analytic and approximate solutions using Adomian’s decomposition method and recurrence relation with the order convergence analysis. Finally, an illustrative example is presented to demonstrate constructions for the numerical solution using MATHEMATICA.
AbstractList	This paper studies the uniqueness of the bounded solution to a new Cauchy problem of the fractional nonlinear partial integro-differential equation based on the multivariate Mittag–Leffler function as well as Banach’s contractive principle in a complete function space. Applying Babenko’s approach, we convert the fractional nonlinear equation with variable coefficients to an implicit integral equation, which is a powerful method of studying the uniqueness of solutions. Furthermore, we construct algorithms for finding analytic and approximate solutions using Adomian’s decomposition method and recurrence relation with the order convergence analysis. Finally, an illustrative example is presented to demonstrate constructions for the numerical solution using MATHEMATICA.
Audience	Academic
Author	Hrytsenko, Andrii Beaudin, Joshua Saadati, Reza Li, Chenkuan
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Cites_doi	10.1016/j.apnum.2005.02.008 10.1016/S0045-7825(98)00108-X 10.1002/mma.8890 10.1016/j.jtusci.2016.06.002 10.1016/j.physleta.2007.01.046 10.1016/j.chaos.2017.04.027 10.22436/jnsa.016.01.05 10.1016/j.physleta.2007.06.071 10.3390/math10121971
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SubjectTerms	adomian’s decomposition method Algorithms analytic and approximate solution babenko’s approach Banach spaces banach’s contractive principle Cauchy problems Decomposition Differential equations Function space Integral equations Mathematics multivariate Mittag–Leffler function Nonlinear equations Partial differential equations Uniqueness
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Title	On the Uniqueness of the Bounded Solution for the Fractional Nonlinear Partial Integro-Differential Equation with Approximations
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