Applications of Inverse Operators to a Fractional Partial Integro-Differential Equation and Several Well-Known Differential Equations
Abstract
1. Introduction and Preliminaries
- We study the uniqueness, existence, and stability for the new Equation (1) using several notable fixed-point theorems, an equivalent implicit integral equation from inverse operators, and the equicontinuity concept. Clearly, there are more studies focusing on ordinary fractional differential equations and far fewer on FPDEs.
- We derive a new analytic solution to the generalized multi-term time-fractional convection problem (5) by the multivariate Mittag-Leffler function, an inverse operator, and a subspace space, S, with several illustrative examples showing applications of our main results.
- We obtain a unique series solution in terms of the Laplacian operators, for the first time, to the generalized time-fractional diffusion-wave Equation (6), and further, we establish the uniform solution to the non-homogeneous wave equation in n dimensions for all , which is consistent with all classical consequences but without any complicated integrals in computation.
2. Uniqueness and Stability
3. Existence
4. Applications of Inverse Operators
4.1. A Partial Integro-Differential Equation
4.2. A Multi-Term Time-Fractional Convection Problem
4.3. A Generalized Time-Fractional Diffusion-Wave Equation
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Kilbas, A.-A.; Srivastava, H.-M.; Trujillo, J.-J. Theory and Applications of Fractional Differential Equations; Elsevier: Dutch, The Netherlands, 2006. [Google Scholar]
- Samko, S.G.; Kilbas, A.A.; Marichev, O.I. Fractional Integrals and Derivatives: Theory and Applications; Gordon and Breach: Basel, Switzerland, 1993. [Google Scholar]
- Li, C.; Beaudin, J.; Rahmoune, A.; Remili, W. A matrix Mittag–Leffler function and the fractional nonlinear partial integro-differential equation in Rn. Fractal Fract. 2023, 7, 651. [Google Scholar] [CrossRef]
- Sadek, L.; Baleanu, D.; Abdod, M.S.; Shatanawi, W. Introducing novel Θ-fractional operators: Advances in fractional calculus. J. King Saud Univ. Sci. 2024, 36, 103352. [Google Scholar] [CrossRef]
- Hadid, S.-B.; Luchko, Y.-F. An operational method for solving fractional differential equations of an arbitrary real order. Panamer. Math. J. 1996, 6, 57–73. [Google Scholar]
- Ervin, V.J.; Führer, T.; Heuer, N.; Karkulik, M. DPG method with optimal test functions for a fractional advection diffusion equation. J. Sci. Comput. 2017, 72, 568–585. [Google Scholar]
- Ervin, V.J.; Roop, J.P. Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial. Equations 2005, 22, 558–576. [Google Scholar] [CrossRef]
- Tayebi, A.; Shekari, Y.; Heydari, M.H. A meshless method for solving two-dimensional variable-order time fractional advection-diffusion equation. J. Comput. Phys. 2017, 340, 655–669. [Google Scholar]
- Li, C.P.; Wang, Z. Non-uniform L1/discontinuous Galerkin approximation for the time-fractional convection equation with weak regular solution. Math. Comput. Simult. 2021, 182, 838–857. [Google Scholar]
- Podlubny, I. Fractional Differential Equations; Academie Press: New York, NY, USA, 1999. [Google Scholar]
- Xu, S.; Ling, X.; Zhao, Y.; Jassim, H.-K. A novel schedule for solving the two-dimensional diffusion in fractal heat transfer. Therm. Sci. 2015, 19, 99–103. [Google Scholar]
- Mahor, T.-C.; Mishra, R.; Jain, R. Analytical solutions of linear fractional partial differential equations using fractional Fourier transform. J. Comput. Appl. Math. 2021, 385, 113202. [Google Scholar] [CrossRef]
- Dehghan, M.; Shakeri, F. A semi-numerical technique for solving the multi-point boundary value problems and engineering applications. Int. J. Numer. Methods Heat Fluid Flow 2011, 21, 794–809. [Google Scholar] [CrossRef]
- Singh, J.; Kumar, D.; Swroop, R. Numerical solution of time- and space-fractional coupled Burger’s equations via homotopy algorithm. Alex. Eng. J. 2016, 55, 1753–1763. [Google Scholar]
- Momani, S.; Odibat, Z. Analytical approach to linear fractional partial differential equations arising in fluid mechanics. Phys. Lett. A 2006, 355, 271–279. [Google Scholar]
- Ionescu, C.; Lopes, A.; Copot, D.; Machado, J.T.; Bates, J.H. The role of fractional calculus in modeling biological phenomena: A review. Commun. Nonlinear Sci. Numer. Simul. 2017, 51, 141–159. [Google Scholar] [CrossRef]
- Li, C. Uniqueness and Hyers–Ulam’s stability for a fractional nonlinear partial integro-differential equation with variable coefficients and a mixed boundary condition. Can. J. Math. 2024, 1–21. [Google Scholar] [CrossRef]
- Kumar, S.; Kumar, A.; Singh, J. Numerical simulation of fractional model of tumor-immune interaction. Chaos Solitons Fract. 2020, 132, 109574. [Google Scholar]
- Lu, L.; Meng, X.; Mao, Z.; Karniadakis, G.E. DeepXDE: A deep learning library for solving differential equations. SIAM Rev. 2021, 63, 208–228. [Google Scholar] [CrossRef]
- Li, C. On boundary value problem of the nonlinear fractional partial integro-differential equation via inverse operators. Fract. Calc. Appl. Anal. 2025, 28, 386–410. [Google Scholar] [CrossRef]
- Strauss, W.A. Partial Differential Equations: An Introduction; Wiley: Hoboken, NJ, USA, 2007. [Google Scholar]
- Li, C. An example of the generalized fractional Laplacian. Contemp. Math. 2020, 1, 215–226. [Google Scholar] [CrossRef]
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Li, C.; Liao, W. Applications of Inverse Operators to a Fractional Partial Integro-Differential Equation and Several Well-Known Differential Equations. Fractal Fract. 2025, 9, 200. https://doi.org/10.3390/fractalfract9040200
Li C, Liao W. Applications of Inverse Operators to a Fractional Partial Integro-Differential Equation and Several Well-Known Differential Equations. Fractal and Fractional. 2025; 9(4):200. https://doi.org/10.3390/fractalfract9040200
Chicago/Turabian StyleLi, Chenkuan, and Wenyuan Liao. 2025. "Applications of Inverse Operators to a Fractional Partial Integro-Differential Equation and Several Well-Known Differential Equations" Fractal and Fractional 9, no. 4: 200. https://doi.org/10.3390/fractalfract9040200
APA StyleLi, C., & Liao, W. (2025). Applications of Inverse Operators to a Fractional Partial Integro-Differential Equation and Several Well-Known Differential Equations. Fractal and Fractional, 9(4), 200. https://doi.org/10.3390/fractalfract9040200