# A nonlocal problem with fractional derivatives for third-order equations

### Abstract

The unique solvability of a nonlocal boundary value problem for the equation of mixed hyperbolic-parabolic type of the third order is investigated. The boundary condition of the problem contains a linear combination of fractional, in the sense of Riemann-Liouville, operators of integro-differentiation with hypergeometric Gauss function on the values of the solution on the characteristics pointwise associated with the values of the solution and its derivative on the degeneration line. Theorems of existence and uniqueness of the solution to the problem in various cases of the exponent in the equation under consideration are formulated and proved. The uniqueness of the solution of the problem, under certain restrictions of the inequality type on the given functions and orders of fractional derivatives in the boundary condition, is proved by the method of energy integrals. Functional relations between the trace of the desired solution and its derivative brought to the degeneration line from the hyperbolic and parabolic parts of the mixed domain are written. Under the conditions of the uniqueness theorem, we prove the existence of a solution to the problem by equivalent reduction to Fredholm integral equations of the second kind with respect to the derivative of the trace of the desired solution. Also, the intervals of changing the orders of fractional integro-differentiation operators are determined, at which the solution of the problem exists and is unique. The effect of the coefficient at the lower derivative in the original equation on the unique solvability of the problem is established.

### References

[1]

Nahushev A. M., Problems with Shifts for Partial Differential Equations [in Russian], Nauka, Moscow (2006).

[2]

Ezaova A. G., “On some nonlocal problem for mixed type third-order equations,” Izv. Kabard.-Balkar. Gos. Univ., 1, No. 4, 26–31 (2011).

[3]

Repin O. A. and Kumykova S. K., “A problem with generalized fractional integro-differentiation operators of arbitrary order,” Russ. Math., 56, No. 12, 50–60 (2012).

[4]

Repin O. A. and Kumykova S. K., “Boundary-value problem with Saigo operators for mixed type equation of the third order with multiple characteristics,” Russ. Math., 59, No. 7, 44–51 (2015).

[5]

Repin O. A. and Kumykova S. K., “Interior-boundary value problem with Saigo operators for the Gellerstedt equation,” Differ. Equ., 49, No. 10, 1307–1316 (2013).

[6]

Islomov B. and Baltaeva U. I., “Boundary-value problems for a third-order loaded parabolic-hyperbolic equation with variable coefficients,” Electron. J. Differ. Equ., No. 221, 1–10 (2015).

[7]

Utkina E. A., “Boundary value problems for a third-order hyperbolic equation on the plane,” Differ. Equ., 53, No. 6, 818–824 (2017).

[8]

Korzyuk V. I. and Mandrik A. A., “Classical solution of the first mixed problem for a third-order hyperbolic equation with the wave operator,” Differ. Equ., 50, No. 4, 489–501 (2014).

[9]

Repin O. A., “On nonlocal boundary-value problem with Saigo operator for generalized Euler–Poisson–Darbu equation [in Russian],” in: Sb. Nauch. Tr. Integral Transformations and Boundary-Value Problems, No. 13, pp. 175–181, Inst. Mat. Ukr., Chernovtsy (1996).

[10]

Kumykova S. K., “On some boundary-value problem with nonlocal boundary conditions on characrestics for mixed-type equations [in Russian],” Differ. Uravn., 10, No. 1, 78–88 (1974).

[11]

Sopuev A. and Kozhabekov K. G., “Boundary value problems for mixed-type parabolic-hyperbolic third-order equations with lower-order terms with characteristic line of type change [in Russian],” Tr. Mezhdunar. Konf. Partial Differential Equations and Related Problems of Analysis and Informatics, 1, pp. 14–16, Tashkent (2004).

[12]

Dzhuraev T. D., Sopuev A., and Mamadzhanov M., Boundary Value Problems for Parabolic-Hyperbolic Equations [in Russian], Fan, Tashkent (1986).

[13]

Samko S. G., Kilbas A. A., and Marichev O. I., Fractional Integrals and Derivatives and Their Applications [in Russian], Nauka i Tekhnika, Minsk (1987).

[14]

Bitsadze A. V., Some Classes of Partial Differential Equations [in Russian], Nauka, Moscow (1981).

[15]

Smirnov M. M., Degenerate Elliptic and Hyperbolic Equations [in Russian], Nauka, Moscow (1966).

[16]

Smirnov M. M., Mixed-Type Equations [in Russian], Vysshaya Shkola, Moscow (1985).

[17]

Lebedev N. N., Special Functions and Their Applications [in Russian], Fizmatgiz, Moscow; Leningrad (1963).

[18]

Marichev O. I., Kilbas A. A., and Repin O. A., Boundary Value Problems for Partial Differential Equations with Discontinuous Coefficients [in Russian], Izdat. Samar. Gos. Ekonom. Univ., Samara (2008).

*Mathematical notes of NEFU*, 26(1), pp. 14-23. doi: https://doi.org/10.25587/SVFU.2019.101.27243.

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