Mixed problem for higher-order equations with fractional derivative and degeneration in both variables

  • B. Yu. Irgashev Namangan engeering institute
Keywords: Differential equation, high order, degeneration, fractional Riemann-Liouville derivative, existence, uniqueness, series, uniform convergence.

Abstract

UDC 517.9

We consider an initial-boundary-value problem for a higher-order equation with fractional Riemann – Liouville derivative in a rectangular domain degenerating in both variables. The solution to the problem is constructed in the explicit form by the method of separation of variables. Uniqueness is proved by the spectral method.

References

A. M. Nakhushev, Drobnoe ischislenie i ego primenenie, Fractional Calculus and its Applications, Fizmatlit, Moscow (2003).

A. V. Pskhu, Uravneniya v chastnykh proizvodnykh drobnogo poryadka, Partial Differential Equations of Fractional Order, Nauka, Moscow (2005).

M. O. Mamchuev, On the well-posedness of boundary value problems for a fractional diffusion-wave equation and one approach to solving them, Different. Equat., 56, 756 – 760 (2020); https://doi.org/10.1134/S0012266120060087. DOI: https://doi.org/10.1134/S0012266120060087

A. V. Pskhu, The fundamental solution of a diffusion-wave equation of fractional order, Izv. Math., 73, № 2, 351 – 392 (2009). DOI: https://doi.org/10.1070/IM2009v073n02ABEH002450

A. V. Pskhu, Solution of a boundary value problem for a fractional partial differential equation, Different. Equat., 39, № 8, 1150 – 1158 (2003). DOI: https://doi.org/10.1023/B:DIEQ.0000011289.79263.02

O. P. Agrawal, Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear Dynam., 291, № 4, 145 – 155 (2002).

S. Kh. Gekkieva, M. A. Kerefov, Dirichlet boundary value problem for Aller – Lykov moisture transfer equation with fractional derivative in time, Ufa Math. J., 11, № 2, 71 – 81 (2019). DOI: https://doi.org/10.13108/2019-11-2-71

F. Mainardi, The fundamental solutions for the fractional diffusion-wave equation, Appl. Math. Lett., 9, № 6, 23 – 28 (1996). DOI: https://doi.org/10.1016/0893-9659(96)00089-4

O. Kh. Masaeva, Uniqueness of solutions to Dirichlet problems for generalized Lavrent'ev – Bitsadze equations with a fractional derivative, Electron. J. Different. Equat., 2017, 1 – 8 (2017).

B. J. Kadirkulov, Boundary problems for mixed parabolic-hyperbolic equations with two lines of changing type and fractional derivative, Electron. J. Different. Equat., 2014, № 57, 1 – 7 (2014).

A. S. Berdyshev, A. Cabada, B. J. Kadirkulov, The Samarskii – Ionkin type problem for the fourth order parabolic equation with fractional differential operator, Comput. and Math. Appl., 62, № 10, 3884 – 3893 (2011). DOI: https://doi.org/10.1016/j.camwa.2011.09.038

A. S. Berdyshev, B. E. Eshmatov, B. J. Kadirkulov, Boundary value problems for fourth-order mixed type equation with fractional derivative, Electron. J. Different. Equat., 36, 1 – 11 (2016).

A. N. Artyushin, Fractional integral inequalities and their applications to degenerate differential equations with the Caputo fractional derivative, Siberian Math. J., 61, № 2, 208 – 221 (2020). DOI: https://doi.org/10.1134/S0037446620020032

M. Saigo, A. A. Kilbas, The solution of a class of linear differential equations via functions of the Mittag-Leffler type, Different. Equat., 36, 193 – 202 (2000); https://doi.org/10.1007/BF02754205. DOI: https://doi.org/10.1007/BF02754205

H. Bateman, A. Erdelyi, Bateman manuscript project, Higher Transcendental Functions, vol. 2, McGraw-Hill, New York (1953).

Yu. P. Apakov, B. Yu. Irgashev, Boundary-value problem for a degenerate high-odd-order equation, Ukr. Math. J., 66, № 10 (2015). DOI: https://doi.org/10.1007/s11253-015-1039-7

A. A. Kilbas, Megumi Saigo, Solution in closed form of a class of linear differential equations of fractional order, Different. Equat., 33, № 2, 194 – 204 (1997).

E. M. Wright, J. London Math. Soc. (Ser. 2), 10, (1935).

R. B. Paris, Some remarks on the theorems of Wright and Braaksma on the Wright function ${}_p{Psi _q}$}; arXiv:1708.04824v1 [math.CA] Aug2017.

Published
27.11.2022
How to Cite
Irgashev, B. Y. “Mixed Problem for Higher-Order Equations With Fractional Derivative and Degeneration in Both Variables”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, no. 10, Nov. 2022, pp. 1328 -1338, doi:10.37863/umzh.v74i10.6298.
Section
Research articles