Remarks on a New Variable-Coefficient Integro-Differential Equation via Inverse Operators
In this paper, we investigate functional inverse operators associated with a class of fractional integro-differential equations. We further explore the existence, uniqueness, and stability of solutions to a new integro-differential equation featuring variable coefficients and a functional boundary c...
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| Language: | English |
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MDPI AG
2025-06-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/9/7/404 |
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| author | Chenkuan Li Nate Fingas Ying Ying Ou |
| author_facet | Chenkuan Li Nate Fingas Ying Ying Ou |
| author_sort | Chenkuan Li |
| collection | DOAJ |
| description | In this paper, we investigate functional inverse operators associated with a class of fractional integro-differential equations. We further explore the existence, uniqueness, and stability of solutions to a new integro-differential equation featuring variable coefficients and a functional boundary condition. To demonstrate the applicability of our main theorems, we provide several examples in which we compute values of the two-parameter Mittag–Leffler functions. The proposed approach is particularly effective for addressing a wide range of integral and fractional nonlinear differential equations with initial or boundary conditions—especially those involving variable coefficients, which are typically challenging to treat using classical integral transform methods. Finally, we demonstrate a significant application of the inverse operator approach by solving a Caputo fractional convection partial differential equation in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></mrow></semantics></math></inline-formula> with an initial condition. |
| format | Article |
| id | doaj-art-e038829a5d804237be06711bd893405a |
| institution | DOAJ |
| issn | 2504-3110 |
| language | English |
| publishDate | 2025-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj-art-e038829a5d804237be06711bd893405a2025-08-20T03:07:58ZengMDPI AGFractal and Fractional2504-31102025-06-019740410.3390/fractalfract9070404Remarks on a New Variable-Coefficient Integro-Differential Equation via Inverse OperatorsChenkuan Li0Nate Fingas1Ying Ying Ou2Department of Mathematics and Computer Science, Brandon University, Brandon, MB R7A 6A9, CanadaDepartment of Mathematics and Computer Science, Brandon University, Brandon, MB R7A 6A9, CanadaDepartment of Mathematics and Computer Science, Brandon University, Brandon, MB R7A 6A9, CanadaIn this paper, we investigate functional inverse operators associated with a class of fractional integro-differential equations. We further explore the existence, uniqueness, and stability of solutions to a new integro-differential equation featuring variable coefficients and a functional boundary condition. To demonstrate the applicability of our main theorems, we provide several examples in which we compute values of the two-parameter Mittag–Leffler functions. The proposed approach is particularly effective for addressing a wide range of integral and fractional nonlinear differential equations with initial or boundary conditions—especially those involving variable coefficients, which are typically challenging to treat using classical integral transform methods. Finally, we demonstrate a significant application of the inverse operator approach by solving a Caputo fractional convection partial differential equation in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></mrow></semantics></math></inline-formula> with an initial condition.https://www.mdpi.com/2504-3110/9/7/404fractional nonlinear integro-differential equationuniqueness and existenceBanach’s contractive principleLeray–Schauder’s fixed-point theoremMittag–Leffler functionfunctional inverse operator |
| spellingShingle | Chenkuan Li Nate Fingas Ying Ying Ou Remarks on a New Variable-Coefficient Integro-Differential Equation via Inverse Operators Fractal and Fractional fractional nonlinear integro-differential equation uniqueness and existence Banach’s contractive principle Leray–Schauder’s fixed-point theorem Mittag–Leffler function functional inverse operator |
| title | Remarks on a New Variable-Coefficient Integro-Differential Equation via Inverse Operators |
| title_full | Remarks on a New Variable-Coefficient Integro-Differential Equation via Inverse Operators |
| title_fullStr | Remarks on a New Variable-Coefficient Integro-Differential Equation via Inverse Operators |
| title_full_unstemmed | Remarks on a New Variable-Coefficient Integro-Differential Equation via Inverse Operators |
| title_short | Remarks on a New Variable-Coefficient Integro-Differential Equation via Inverse Operators |
| title_sort | remarks on a new variable coefficient integro differential equation via inverse operators |
| topic | fractional nonlinear integro-differential equation uniqueness and existence Banach’s contractive principle Leray–Schauder’s fixed-point theorem Mittag–Leffler function functional inverse operator |
| url | https://www.mdpi.com/2504-3110/9/7/404 |
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