On mixed problems for a class of fractional order equations with involution
107 104
Keywords:
mixed task,fractional derivative,involution, nonlocal equation, Hadamard operator, heat conduction equation, Dirichlet condition, the Neumann condition.Abstract
In this paper, we consider new classes of differential equations of fractional order
related to Hadamard derivatives. These equations generalize the well-known heat conduction
equation for the fractional exponent of the time derivative. For the equations under consideration,
mixed problems with Dirichlet and Neumann boundary conditions are studied. The Fourier method
is used to solve these problems. Two auxiliary problems are obtained for ordinary differential
equations of fractional order and ordinary differential equations with involution. The spectral
properties of ordinary differential operators with involution are studied. For the main problems,
theorems on the existence and uniqueness of solutions are proved.
References
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Kilbas A.A., Srivastava H.M, Trujillo J.J.Theory and applications of fractional differential equations.Elsevier, Amsterdam 2006.
Jarad F., Abdeljawad T., Baleanu D. Caputo-type modification of the Hadamard fractional derivatives. Advances in Difference Equations. – 2012. – No.142. – P.1 – 8.
Al-Salti N, Kirane M., Torebek B.T. On a class of inverse problems for a heat equation with involution perturbation // Hacettepe Journal of Mathematics and Statistics. – 2019. – Vol. 48, No.3. – P. 669 – 681.
Berdyshev A.S. , Kadirkulov B.J. On a nonlocal problem for a fourth-order parabolic equation with the fractional Dzhrbashyan–Nersesyan operator//Differential Equations. – 2016.– V. 52,No.1. – P. 121–127.
Kubica A., Yamamoto M. Initial-boundary value problems for fractional diffusion equations with time-dependent coefficients. Fractional Calculus and Applied Analysis. – 2018.– V. 21. – P. 276–311.
Luchko Y., Yamamoto M. General time-fractional diffusion equation: some uniqueness and existence results for the initial-boundary-value problems // Fractional Calculus and Applied Analysis. .– 2016.– V. 19, № 3, – P. 676 – 695.
Naimark M.A. Linear Differential Operators Part II, Ungar, New York, 1968 .
Turmetov B. Kh., Kadirkulov B. J. On the solvability of an initial-boundary value problem for a fractional heat equation with involution // Lobachevskii Journal of Mathematics. – 2022. – Vol. 43, No.1. – P. 249 – 262.
Boudabsa L., Simon Th. Some properties of the Kilbas–Saigo function // Mathematics. – 2021. – Vol.9,No.3. – P.1 – 24.
Le Roy E. Valeurs asymptotiques de certaines series procedant suivant les puissances enteres et positives d’une variable reelle// Darboux Bull. – 1899. – Vol.24. – P.245–268.
