Abstract
We find conditions for the unique solvability of the problem u xy (x, y) = f(x, y, u(x, y), (D r0 u)(x, y)), u(x, 0) = u(0, y) = 0, x ∈ [0, a], y ∈ [0, b], where (D r0 u)(x, y) is the mixed Riemann-Liouville derivative of order r = (r 1, r 2), 0 < r 1, r 2 < 1, in the class of functions that have the continuous derivatives u xy (x, y) and (D r0 u)(x, y). We propose a numerical method for solving this problem and prove the convergence of the method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. G. Samko, A. A. Kilbas, and O. I. Marichev, Integrals and Derivatives of Fractional Order and Their Applications [in Russian], Nauka i Tekhnika, Minsk (1987).
V. I. Smirnov, A Course of Higher Mathematics [in Russian], Vol. 5, Fizmatgiz, Moscow (1959).
S. Walczak, “Absolutely continuous functions of several variables and their application to differential equations,” Bull. Pol. Acad. Sci. Math., 35, No. 11–12, 733–744 (1987).
P. Günther, “Über die Differentialgleichung zxy = ϕ(x, y, z, z xy ),” Math. Nachr., 33, No 1/2, 73–89 (1967).
A. F. Timan, Theory of Approximation of Functions of a Real Variable [in Russian], Fizmatgiz, Moscow (1960).
Author information
Authors and Affiliations
Additional information
__________
Translated from Neliniini Kolyvannya, Vol. 8, No. 4, pp. 456–467, October–December, 2005.
Rights and permissions
About this article
Cite this article
Vityuk, A.N., Golushkov, A.V. Darboux problem for a differential equation with fractional derivative. Nonlinear Oscill 8, 450–462 (2005). https://doi.org/10.1007/s11072-006-0013-6
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11072-006-0013-6