First boundary-value problem on the semiaxis for an isotropic superdiffusion equation of order greater than one
DOI:
https://doi.org/10.3842/umzh.v78i5-6.9437Keywords:
Fractional Laplacian, Riesz fractional derivative operator, random Levy processes, Dirichlet boundary value problem, isotropic superdiffusion equationAbstract
UDC 517.95, 517.983, 519.21
The Dirichlet problem is studied for a one-dimensional isotropic superdiffusion equation of order $\alpha\in(1;2)$ on the semiaxis. The analytic solutions of this problem are obtained by the mapping method in the classes of continuous Hўоlder functions, which may have an integrable singularity with respect to time. The probability interpretation of the solutions is clarified and the uniqueness of the regular solution is proved on an infinite time interval.
References
1. B. Baeumer, M. Kovacs, M. Meerschaert, H. Sankaranarayanan, Reprint of: boundary conditions for fractional diffusion, J. Comput. Appl. Math., 339, 414–430 (2018); DOI: https://doi.org/10.1016/j.cam.2018.03.007. DOI: https://doi.org/10.1016/j.cam.2018.03.007
2. W. Bu, Y. Tang, Y. Wu, J. Yang, Finite difference/finite element method for two-dimensional space and time fractional Bloch–Torrey equations, J. Comput. Phys., 293, 264–279 (2015); DOI: https://doi.org/10.1016/j.jcp.2014.06.031. DOI: https://doi.org/10.1016/j.jcp.2014.06.031
3. C. Bucur, E. Valdinoci, Nonlocal diffusion and applications, Lect. Notes Unione Mat. Ital., 20, Springer, Berlin (2016); DOI: doi:10.1007/978-3-319-28739-3. DOI: https://doi.org/10.1007/978-3-319-28739-3
4. L. Brasco, E. Lindgren, M. Stromqvist, Continuity of solutions to a nonlinear fractional diffusion equation, J. Evol. Equat., 21, 4319–4381 (2021); https://doi.org/10.1007/s00028-021-00721-2. DOI: https://doi.org/10.1007/s00028-021-00721-2
5. N. Cusimano, K. Burrage, I. Turner, D. Kay, On reflecting boundary conditions for space-fractional equations on a finite interval: Proof of the matrix transfer technique, Appl. Math. Model., 42, 554–565 (2017); DOI: https://doi.org/10.1016/j.apm.2016.10.021. DOI: https://doi.org/10.1016/j.apm.2016.10.021
6. S. Eidelman, S. Ivasyshen, A. Kochubei, Analytic methods in the theory of differential and pseudo-differential equations of parabolic type, Oper. Theory Adv. Appl., 152, Birkhäuser Verlag, Basel (2004). DOI: https://doi.org/10.1007/978-3-0348-7844-9
7. L. Feng, I. Turner, T. Moroney, F. Liu, Fractional potential: A new perspective on the fractional Laplacian problem on bounded domains, Commun. Nonlinear Sci. Numer. Simul., 125, 1–28 (2023); DOI: https://doi.org/10.1016/j.cnsns.2023.107368. DOI: https://doi.org/10.1016/j.cnsns.2023.107368
8. O. C. Ibe, Markov processes for stochastic modeling, 2nd ed., Elsevier, Amsterdam (2013); DOI: doi.org/10.1016/C2012-0-06106-6.
9. J. Kelly, H. Sankaranarayanan, M. Meerschaert, Boundary conditions for two-sided fractional diffusion, J. Comput. Phys., 376, 1089–1107 (2019); DOI: https://doi.org/10.1016/j.jcp.2018.10.010. DOI: https://doi.org/10.1016/j.jcp.2018.10.010
10. V. P. Knopova, Some generators of $L_p$-sub-Markovian semigroups in the half-space $R^{n+1}_{0+}$, PhD Thesis, University of Wales Swansea (2003).
11. V. P. Knopova, A. N. Kochubei, A. M. Kulik, Parametrix methods for equations with fractional Laplacians. vol. 2, Fractional Differential Equations, De Gruyter, Boston (2019), pp. 267–298; DOI: doi.org/10.1515/9783110571660-013. DOI: https://doi.org/10.1515/9783110571660-013
12. А. Н. Кочубей, Параболічні псевдодиференціальні рівняння, гіперсингулярні інтеграли та марковські процеси, Вісн. АН СРСР. Сер. мат., 52, № 5, 909–934 (1988) [рос.].
13. A. Krageloh, Feller semigroups generated by fractional derivatives and pseudo-differential operators, PhD Thesis, Friedrich-Alexander Universität Erlangen-Nürnberg (2001).
14. V. A. Litovchenko, The Cauchy problem and distribution of local fluctuations of one Riesz gravitational field, Fract. Calc. Appl. Anal., 25, 668–686 (2022); DOI: https://doi.org/10.1007/s13540-022-00034-2. DOI: https://doi.org/10.1007/s13540-022-00034-2
15. V. A. Litovchenko, Classical solutions of the equation of local fluctuations of Riesz gravitational fields and their properties, Ukr. Math. J., 74, 69–86 (2022); DOI: https://doi.org/10.1007/s11253-022-02048-8. DOI: https://doi.org/10.1007/s11253-022-02048-8
16. V. A. Litovchenko, Local polya fluctuations of Riesz gravitational fields and the Cauchy problem, Carpathian Math. Publ., 15, 222–235 (2023); DOI: https://doi.org/10.15330/cmp.15.1.222-235. DOI: https://doi.org/10.15330/cmp.15.1.222-235
17. D. Monroy, E. Raposo, Solution of the space-fractional diffusion equation on bounded domains of superdiffusive phenomena, Phys. Rev. E, 110, № 5, 1–22 (2024); DOI: https://doi.org/10.1103/PhysRevE.110.054119. DOI: https://doi.org/10.1103/PhysRevE.110.054119
18. M. Osypchuk, M. Portenko, On the third initial-boundary value problem for some class of pseudo-differential equations related to a symmetric $α$-stable process, J. Pseudo-Differ. Oper. Appl., 9, № 3, 811–835 (2018); DOI: 10.1007/s11868-017-0210-3. DOI: https://doi.org/10.1007/s11868-017-0210-3
19. A. Reynolds, Liberating L'evy walk research from the shackles of optimal foraging, Phys. Life Rev., 14, 59–83 (2015). DOI: https://doi.org/10.1016/j.plrev.2015.03.002
20. S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives: theory and applications, Gordon and Breach Science Publ., Yverdon (1993).
21. H. Sun, Y. Zhang, W. Chen, D. Reeves, Use of a variable-index fractional-derivative model to capture transient dispersion in heterogeneous media, J. Contam. Hydrol., 157, 47–58 (2014); DOI: https://doi.org/10.1016/j.jconhyd.2013.11.002. DOI: https://doi.org/10.1016/j.jconhyd.2013.11.002
22. H. Sun, Y. Zhang, D. Baleanu, W. Chen, Y. Chen, A new collection of real world applications of fractional calculus in science and engineering, Comm. Nonlinear Sci. Numer. Simul., 64, 213–231 (2018); DOI: https://doi.org/10.1016/j.cnsns.2018.04.019. DOI: https://doi.org/10.1016/j.cnsns.2018.04.019
23. K. Taira, On the existence of Feller semigroup with boundary conditions, Mem. Amer. Math. Soc., 475, 131–148 (1992). DOI: https://doi.org/10.1090/memo/0475
24. H. Triebel, Interpolation theory, function spaces, differential operators, North Holland Mathematical Library, 18, North Holland Publishing Company, Amsterdam, New York (1978).
25. В. В. Учайкін, Метод дробових похідних, Артишок, Ульяновськ (2008) [рос.].
26. V. Uchaikin, R. Sibatov, Fractional kinetics in space: anomalous transport models, World Scientific, Singapore (2017); DOI: https://doi.org/10.1142/10581. DOI: https://doi.org/10.1142/10581
27. G. M. Viswanathan, V. Afanasyev, S. V. Buldyrev, et al., Lévy flights in random searches, Phys. A, 282, № 1-2, 1–12 (2000); DOI: doi.org/10.1016/S0378-4371(00)00071-6. DOI: https://doi.org/10.1016/S0378-4371(00)00071-6
28. Q. Yang, F. Liu, I. Turner, Numerical methods for fractional partial differential equations with Riesz space fractional derivatives, Appl. Math. Model., 34, № 1, 200–218 (2010); DOI: https://doi.org/10.1016/j.apm.2009.04.006. DOI: https://doi.org/10.1016/j.apm.2009.04.006
29. Y. Zhang, X. Yu, X. Li, et al., Impact of absorbing and reflective boundaries on fractional derivative models: Quantification, evaluation and application, Adv. Water Resour., 128, 129–144 (2019); DOI: https://doi.org/10.1016/j.advwatres.2019.02.011. DOI: https://doi.org/10.1016/j.advwatres.2019.02.011
30. A. Zoia, A. Rosso, M. Kardar, Fractional Laplacian in bounded domains, Phys. Rev. E., 76(2), 11–34 (2007); DOI: 10.1103/PhysRevE.76.021116. DOI: https://doi.org/10.1103/PhysRevE.76.021116
31. В. Н. Золотарьов, Одновимірні стаціонарні розподіли, Наука, Москва (1983) [рос.].
