First boundary-value problem for the Aller--Lykov equation with the Caputo fractional derivative

M. A. Kerefov ; S.Kh. Gekkieva ; B. M. Kerefov

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference «Classical and Modern Geometry» dedicated to the 100th anniversary of the birth of Professor Levon Sergeyevich Atanasyan (July 15, 1921—July 5, 1998). Moscow, November 1–4, 2021. Part 2, Tome 221 (2023), pp. 63-70.

Résumé

In this paper, we examine boundary-value problems for the inhomogeneous humidity transport equation with variable coefficients and the Caputo fractional derivative in time. Using the method of energy inequalities, we obtain a priori estimates for solutions of the first and third boundary-value problems, which imply the uniqueness and stability of solutions.

Détail

Keywords: boundary-value problem, a priori estimate, fractional differential equation, regularized fractional derivative, Caputo derivative.
Mots-clés : Aller–Lykov equation

Bibliographie

[1] Alikhanov A. A., “Apriornye otsenki reshenii kraevykh zadach dlya uravnenii drobnogo poryadka”, Differ. uravn., 46:5 (2010), 658–664

[2] Gekkieva S. Kh., “Nelokalnaya kraevaya zadacha dlya obobschennogo uravneniya vlagoperenosa Allera—Lykova”, Vestn. KRAUNTs. Fiz.-mat. nauki., 2018, no. 4 (24), 76–86

[3] Gekkieva S. Kh., Kerefov M. A., “Smeshannye kraevye zadachi dlya nagruzhennogo uravneniya s drobnoi proizvodnoi”, Mat. III Mezhdunar. konf. «Nelokalnye kraevye zadachi i rodstvennye problemy matematicheskoi biologii, informatiki i fiziki» (Nalchik, 2006), IPMA KBNTs RAN, Nalchik, 2006, 80–82

[4] Gekkieva S. Kh., Kerefov M. A., “Pervaya kraevaya zadacha dlya uravneniya vlagoperenosa Allera—Lykova s drobnoi po vremeni proizvodnoi”, Ufim. mat. zh., 11:2 (2019), 72–82

[5] Kerefov M. A., “Ob odnoi kraevoi zadache dlya modifitsirovannogo uravneniya vlagoperenosa s drobnoi po vremeni proizvodnoi”, Dokl. Adyg. (Cherkes.) Mezhdunar. akad. nauk., 4:1 (1999), 12–14

[6] Kerefov M. A., Gekkieva S. Kh., “Kraevye zadachi dlya modifitsirovannogo uravneniya vlagoperenosa s drobnoi po vremeni proizvodnoi v mnogomernoi oblasti”, Nauch. ved. Belgorod. gos. un-ta. Ser. Mat. Fiz., 41:23 (2015), 17–23

[7] Kerefov M. A., Gekkieva S. Kh., “Pervaya kraevaya zadacha dlya neodnorodnogo nelokalnogo volnovogo uravneniya”, Vestn. Buryat. gos. un-ta. Mat. Inform., 2016, no. 1, 76–86

[8] Kerefov M. A., Gekkieva S. Kh., “Nelokalnaya kraevaya zadacha dlya obobschennogo uravneniya vlagoperenosa”, Vestn. Voronezh. gos. un-ta. Ser. Fiz. Mat., 2017, no. 2, 106–112

[9] Kerefov M. A., Gekkieva S. Kh., “Kraevaya zadacha dlya nelokalnogo uravneniya vlagoperenosa Allera—Lykova”, Itogi nauki i tekhn. Sovr. mat. prilozh. Temat. obzory., 167 (2019), 27–33

[10] Kerefov M. A., Gekkieva S. Kh., “Vtoraya kraevaya zadacha dlya obobschennogo uravneniya vlagoperenosa Allera–Lykova”, Vestn. Samar. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki., 23:4 (2019), 607–621

[11] Kerefov M. A., Gekkieva S. Kh., “Chislenno–analiticheskii metod resheniya kraevoi zadachi dlya obobschennykh uravnenii vlagoperenosa”, Vestn. Udmurt. un-ta. Mat. Mekh. Kompyut. nauki., 31:1 (2021), 19–34

[12] Kerefov M. A., Karmokov M. M., Gekkieva S. Kh., “Ob odnoi kraevoi zadache dlya obobschennogo uravneniya Allera”, Vestn. Samar. un-ta. Estestvennonauch. ser., 26:2 (2020), 7–14

[13] Kerefov M. A., Nakhusheva F. M., Gekkieva S. Kh., “Kraevaya zadacha dlya obobschennogo uravneniya vlagoperenosa Allera—Lykova s sosredotochennoi teploemkostyu”, Vestn. Samar. un-ta. Estestvennonauch. ser., 24:3 (2018), 23–29

[14] Kochubei A. N., “Diffuziya drobnogo poryadka”, Differ. uravn., 26:4 (1990), 660–670

[15] Ladyzhenskaya O. A., Kraevye zadachi matematicheskoi fiziki, Nauka, M., 1973

[16] Nakhushev A. M., Uravneniya matematicheskoi biologii, Fizmatlit, M., 1995

[17] Nakhushev A. M., Drobnoe ischislenie i ego primenenie, Fizmatlit, M., 2003

[18] Nakhushev A. M., Uravneniya v chastnykh proizvodnykh drobnogo poryadka, Nauka, M., 2005

[19] Nigmatullin R. R., “Drobnyi integral i ego fizicheskaya interpretatsiya”, Teor. mat. fiz., 90:3 (1992), 354–368

[20] Samarskii A. A., Teoriya raznostnykh skhem, Nauka, M., 1989

[21] Al-Refai M., Luchko Yu., “Maximum principle for the multi-term time-fractional diffusion equations with the Riemann–Liouville fractional derivatives”, Appl. Math. Comput., 257 (2014), 40–51

[22] Daftardar-Gejji V., Bhalekar S., “Boundary-value problems for multi-term fractional differential equations”, J. Math. Anal. Appl., 2008, no. 345, 754–735

[23] Li G., Sun C., Jia X., Du D., “Numerical solution to the multi-term time fractional diffusion equation in a finite domain”, Numer. Math. Theor. Meth. Appl., 9:3 (2016), 337–357

[24] Liu F., Meerschaert M. M., McGough R. J., Zhuang P., Liu Q., “Numerical methods for solving the multi-term time-fractional wave-diffusion equation”, Fract. Calc. Appl. Anal., 16:1 (2013), 9–25

[25] Liu X., Wang J., Wang X., Zhou Y., “Exact solutions of multi-term fractional diffusion-wave equations with Robin type boundary conditions”, Appl. Math. Mech., 35:1 (2014), 49–62

[26] Luchko Yu., “Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation”, J. Math. Anal. Appl., 374:2 (2011), 538–548

[27] Oldham K. B., Spanier J., The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press, New York, 1974