Non-local boundary value problem for a system of equations with the partial derivatives of fractional order

  • Mamchuev Murat O., mamchuev@rambler.ru Institute of Applied Mathematics and Automation KBSC RAS, 89A Shortanov Street, Nalchik 360000, Russia
Keywords: fractional derivatives, fractional hyperbolic systems, non-local boundary value problem, conditions for unique solvability

Abstract

We study a non-local boundary value problem in a rectangular domain for a linear system of equations with partial fractional Riemann-Liouville derivatives with constant coefficients. The eigenvalues of matrix coefficients in the main part have fixed sign, which is an essential feature of such systems. These systems can be divided into two types which differ in terms of formulation of the correct boundary value problems. The system under investigation relates to the type II, i.e. to systems with the eigenvalues of matrix coefficients in the main part having different signs. We prove the existence and uniqueness theorem for the solution of the investigated boundary value problem. The conditions for the unique solvability of the problem are obtained in terms of the eigenvectors of the matrix coefficients in the main part of the system.

References


[1]
Nakhushev A. M., Fractional calculus and its application [in Russian], Fizmatlit, Moscow (2003).

[2]
Heibig A., “Existence of solutions for a fractional derivative system of equations,” Integral Equation Oper. Theory, 72, 483–508 (2012).

[3]
Kochubei A. N., “Fractional-parabolic systems,” Potential Anal., 37, 1–30 (2012).

[4]
Kochubei A. N., “Fractional-hyperbolic systems,” Fract. Calc. Appl. Anal., 16, No. 4, 860–873 (2013).

[5]
Mamchuev M. O., “Boundary value problem for a system of fractional partial differential equations,” Differ. Equ., 44, No. 12, 1737–1749 (2008).

[6]
Mamchuev M. O., “Boundary value problem for a system of multidimentional differential equations of fractional order [in Russian],” Vestn. Samarsk. Gos. Univ., Esstestvennonauchn. Ser., 8, No. 67, 164–175 (2008).

[7]
Mamchuev M. O., “Boundary value problem for a linear system of equations with a partial derivative of fractional order [in Russian],” Chelyab. Fiz. Mat. Zh., 2, No. 3, 295–311 (2017).

[8]
Mamchuev M. O., “Cauchy problem for a system of equations with a partial derivative of fractional order [in Russian],” Vestn. KRAUNTS, Fiz.-Mat. Nauki, 3, No. 23, 76–82 (2018).

[9]
Mamchuev M. O., “Boundary value problem for a system of differential equations with a partial derivative of a fractional order in unbounded domains [in Russian],” Dokl. Adygsk. (Cherkess.) Mezhdunar. Akad. Nauk, 7, No. 1, 60–63 (2003).

[10]
Mamchuev M. O., “Fundamental solution of a system of fractional partial differential equations,” Differ. Equ., 46, No. 8, 1123–1134 (2010).

[11]
Mamchuev M. O., “Cauchy problem in non-local statement for a system of fractional partial differential equations,” Differ. Equ., 48, No. 3, 354–361 (2012).

[12]
Mamchuev M. O., “Mixed problem for a loaded system of equations with Riemann–Liouville derivatives,” Math. Notes, 97, No. 3, 412–422 (2015).

[13]
Mamchuev M. O., “Mixed problem for a system of fractional partial differential equations,” Differ. Equ., 52, No. 1, 133–138 (2016).

[14]
Mamchuev M. O., Boundary Value Problems for Equations and Systems of Equations with Partial Derivatives of Fractional Order [in Russian], Izdat. KBNTS RAN, Nal’chik (2013).
How to Cite
Mamchuev, M. ( ) “Non-local boundary value problem for a system of equations with the partial derivatives of fractional order”, Mathematical notes of NEFU, 26(1), pp. 23-31. doi: https://doi.org/10.25587/SVFU.2019.101.27244.
Section
Mathematics