Construction of an Explicit Solution of a Time-Fractional Multidimensional Differential Equation
Abstract
:1. Introduction to the Problem and Its Setting
2. General Theories
3. Explicit Solution of the Problem (1) and (2)
4. The Integro-Differential Diffusion Equation with the Mittag-Leffler Type Kernel
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
- Boltzmann, L. Zur theorie der elastischen Nachwirkung. Ann. Der Phys. 1878, 241, 430–432. [Google Scholar] [CrossRef]
- Vronsky, A.P. The phenomenon of aftereffect in à solid. Acad. Sci. Ussr. Appl. Math. Mech. 1941, 5, 31–56. (In Russian) [Google Scholar]
- Gerasimov, A.N. Generalization of linear deformation laws and their application to problems of internal friction. Acad. Sci. Ussr. Appl. Math. Mech. 1948, 12, 529–539. (In Russian) [Google Scholar]
- Tarasova, V.V.; Tarasov, V.E. Economic Interpretation of Fractional Derivatives. Progr. Fract. Differ. Appl. 2017, 3, 1–6. [Google Scholar] [CrossRef]
- Uchaikin, V.V. Background and Theory. In Fractional Derivatives for Physicists and Engineers; Springer: Berlin/Heidelberg, Germany, 2013; p. 373. [Google Scholar]
- Volterra, V. Sur les equations integro-differentielles et leurs applications. Acta Math. 1912, 35, 295–356. [Google Scholar] [CrossRef]
- Colombo, F.; Guidetti, D. Some results on the identification of memory kernels. In Modern Aspects of the Theory of Partial Differential Equations; Ruzhansky, M., Wirth, J., Eds.; Springer: Basel, Switzerland, 2011; pp. 121–138. [Google Scholar]
- Durdiev, D.K. The inverse problem of determining two coefficients in an integrodifferential wave equation. Sib. Zh. Ind. Mat. 2009, 12, 28–40. (In Russian) [Google Scholar]
- Durdiev, D.K. An identification problem of memory function of à medium and the form of an impulse source. J. Sib. Fed. Univ. Math. Phys. 2009, 2, 127–136. [Google Scholar]
- Janno, J.; Von Welfersdorf, L. Inverse problems for identification of memory kernels in viscoelasticity. Math. Methods Appl. Sci. 1997, 20, 291–314. [Google Scholar] [CrossRef]
- Durdiev, D.Q.; Totieva, Z.D. The problem of determining the multidimensional kernel of viscoelasticity equation. Vladikavkaz. Mat. Zh. 2015, 17, 18–43. (In Russian) [Google Scholar]
- Durdiev, D.K.; Safarov, Z.S. Inverse problem of determining the one-dimensional kernel of the viscoelasticity equation in à bounded domain. Math. Notes 2015, 97, 867–877. [Google Scholar] [CrossRef]
- Durdiev, D.K.; Totieva, Z.D. The problem of determining the one-dimensional kernel of the viscoelasticity equation. Sib. Zh. Ind. Mat. 2013, 16, 72–82. (In Russian) [Google Scholar]
- Durdiev, D.K. Inverse problem for the identification of à memory kernel from Maxwell’s system integrodifferential equations for à homogeneous anisotropic media. Nanosyst. Phys. Chem. Math. 2015, 6, 268–273. [Google Scholar] [CrossRef]
- Durdiev, D.K.; Rahmonov, A.A. A 2D kernel determination problem in à visco-elastic porous medium with à weakly horizontally inhomogeneity. Math. Methods Appl. Sci. 2020, 43, 8776–8796. [Google Scholar] [CrossRef]
- Durdiev, D.K.; Rakhmonov, A.A. Inverse problem for à system of integro-differential equations for SH waves in à visco-elastic porous medium: Global solvability. Theor. Math. Phys. 2018, 195, 923–937. [Google Scholar] [CrossRef]
- Durdiev, D.K.; Rahmonov, A.A. The problem of determining the 2D-kernel in à system of integrodifferential equations of à viscoelastic porous medium. Jur. Appl. Industr. Math. 2020, 14, 281–295. [Google Scholar] [CrossRef]
- Durdiev, U.D. A problem of identification of à special 2D memory kernel in an integro-differential hyperbolic equation. Eurasian J. Math. Comput. Appl. 2019, 7, 4–19. [Google Scholar]
- Durdiev, U.; Totieva, Z. A problem of determining à special spatial part of 3D memory kernel in an integrodifferential hyperbolic equation. Math. Methods Appl. Sci. 2019, 42, 7440–7451. [Google Scholar] [CrossRef]
- Rahmonov, A.A.; Durdiev, U.D.; Bozorov, Z.R. Problem of determining the speed of sound and the memory of an anisotropic medium. Theor. Math. Phys. 2021, 207, 494–513. [Google Scholar] [CrossRef]
- Durdiev, D.K.; Zhumaev, Z.Z. Problem of determining à multidimensional thermal memory in à heat conductivity equation. Methods Funct. Anal. Topol. 2019, 25, 219–226. [Google Scholar]
- Durdiev, D.K.; Zhumaev, Z.Z. Problem of determining the thermal memory of à conducting medium. Differ. Equ. 2020, 56, 785–796. [Google Scholar] [CrossRef]
- Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; Elsevier: Amsterdam, The Netherlands, 2006; p. 523. [Google Scholar]
- Meilanov, R.P.; Yanpolov, M.S. Features of the Phase Trajectory of à Fractal Oscillator. Tech. Phys. Lett. 2002, 28, 30–32. [Google Scholar] [CrossRef]
- Mainardi, F.; Luchko, Y.; Pagnini, G. The fundamental solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 2001, 4, 153–192. [Google Scholar]
- Aghajani, A.; Jalilian, Y.; Trujillo, J.J. On the existence of solutions of fractional integro-differential equations. Fract. Calc. Appl. Anal. 2012, 15, 44–69. [Google Scholar] [CrossRef]
- Schumer, R.; Benson, D.A.; Meerschaert, M.M.; Baeumer, B. Fractal mobile/immobile solute transport. Water Resour. Res. 2003, 39, 1–12. [Google Scholar] [CrossRef]
- Giusti, A.; Colombaro, I.; Garra, R.; Garrappa, R.; Polito, F.; Popolizio, M.; Mainardi, F. A practical guide to Prabhakar fractional calculus. Fract. Calc. Appl. Anal. 2020, 23, 9–54. [Google Scholar] [CrossRef] [Green Version]
- Mathai, A.M.; Saxena, R.K.; Haubold, H.J. The H-Function: Theory and Application; Springer: Berlin/Heidelberg, Germany, 2010; 268p. [Google Scholar]
- Tomovski, Z.̆.; Sandev, T. Effects of à fractional friction with power-law memory kernel on string fibrations. Comput. Math. Appl. 2011, 62, 1554–1561. [Google Scholar] [CrossRef] [Green Version]
- Tomovski, Z.̆.; Hilfer, R.; Srivastava, H.M. Fractional and operational calculus with generalized fractional derivatives operators and Mittag-Leffler type functions. Integral Transform. Spec. Funct. 2010, 21, 797–814. [Google Scholar] [CrossRef]
- Li, C.; Deng, W.H.; Shen, X.Q. Exact solutions and the asymptotic behaviors for the averaged generalized fractional elastic models. Commun. Theor. Phys. 2014, 62, 443–450. [Google Scholar] [CrossRef]
- Zao, Y.; Zao, F. The analytical solution of parabolic Volterra integro-differential equations in the infinite domain. Entropy 2016, 18, 14. [Google Scholar]
- Hille, E.; Tamarkin, J.D. On the theory of linear integral equations. Ann. Math. 1930, 31, 479–528. [Google Scholar] [CrossRef]
- Luchko, Y. Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation. Fract. Calc. Appl. Anal. 2012, 15, 141–160. [Google Scholar] [CrossRef] [Green Version]
- Gorenflo, R.; Mainardi, F. Fractional calculus: Integral and differential equations of fractional order. In Fractals and Fractional Calculus in Continuum Mechanics; Springer: Wien, Austria; New York, NY, USA, 1997; pp. 223–276. [Google Scholar]
- Haubold, H.J.; Mathai, A.M.; Saxena, R.K. Mittag-Leffl functions and their applications. Jur. App. Math. 2011, 2011, 51. [Google Scholar]
- Fox, C. The G and H functions and as symmetrical Fourier kernels. Trans Amer Math Soc. 1961, 98, 395–429. [Google Scholar]
- Zhang, Y.; Benson, D.A.; Reeves, D.M. Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications. Adv. Water Resour. 2009, 32, 561–581. [Google Scholar] [CrossRef]
- Durdiev, D.K.; Shishkina, E.L.; Sitnik, S.M. The explicit formula for solution of anomalous diffusion equation in the multi-dimensional space. Lobachevskii J. Mat. 2021, 42, 1264–1273. [Google Scholar] [CrossRef]
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Sultanov, M.A.; Durdiev, D.K.; Rahmonov, A.A. Construction of an Explicit Solution of a Time-Fractional Multidimensional Differential Equation. Mathematics 2021, 9, 2052. https://doi.org/10.3390/math9172052
Sultanov MA, Durdiev DK, Rahmonov AA. Construction of an Explicit Solution of a Time-Fractional Multidimensional Differential Equation. Mathematics. 2021; 9(17):2052. https://doi.org/10.3390/math9172052
Chicago/Turabian StyleSultanov, Murat A., Durdimurod K. Durdiev, and Askar A. Rahmonov. 2021. "Construction of an Explicit Solution of a Time-Fractional Multidimensional Differential Equation" Mathematics 9, no. 17: 2052. https://doi.org/10.3390/math9172052
APA StyleSultanov, M. A., Durdiev, D. K., & Rahmonov, A. A. (2021). Construction of an Explicit Solution of a Time-Fractional Multidimensional Differential Equation. Mathematics, 9(17), 2052. https://doi.org/10.3390/math9172052