On a problem for the third order equation with parabolic-hyperbolic operator including a fractional derivative. (English) Zbl 1490.35231

Summary: This work devoted to one valued solvability of a local problem for a parabolic-hyperbolic type differential equation of third order involving Gerasimov-Caputo fractional operator. Boundary value problem, involving third boundary condition on the parabolic domain and discontinuous gluing condition on the line \(x=0\), is considered. The problem replaced by Volterra type nonlinear integral equations. The existence of solution is proved by the method of successive approximations of factorial law.


35M10 PDEs of mixed type
35R11 Fractional partial differential equations
Full Text: DOI


[1] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Vol. 204 of North-Holland Mathematical Studies (Elsevier B. V. Science, Amsterdam, 2006). · Zbl 1092.45003
[2] Nakhushev, A. M., Fractional Calculus and its Applications (2003), Moscow: Fizmatlit, Moscow · Zbl 1066.26005
[3] Podlubniy, I., Fractional Differential Equations (1999), New York: Academic, New York
[4] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integral and Derivatives: Theory and Applications (1993), Longhorne, PA: Gordon and Breach, Longhorne, PA · Zbl 0818.26003
[5] Pskhu, A. V., Solution of a boundary value problem for a fractional partial differential equation, Differ. Equat., 39, 1150-1158 (2003) · Zbl 1065.35096
[6] Pskhu, A. V., Solution of boundary value problems for a diffusion equation of fractional order by the Green’s function method, Differ. Equat., 39, 1509-1513 (2003) · Zbl 1070.45010
[7] Berdyshev, A. S.; Cabada, A.; Karimov, E. T., On a non-local boundary problem for a parabolic-hyperbolic equation involving a Riemann-Liouville fractional differential operator, Nonlin. Anal. Theory Methods Appl., 75, 3268-3273 (2012) · Zbl 1242.35180
[8] Kadirkulov, B. J., Boundary problems for mixed parabolic-hyperbolic equations with two lines of changing type and fractional derivative, El. J. Differ. Equat., 2014, 1-7 (2014) · Zbl 1417.11019
[9] Kilbas, A. A.; Repin, O. A., Analogue of the Bitsadze-Samarskiy problem for an equation of mixed type with a fractional derivative, Differ. Equat., 39, 638-719 (2003) · Zbl 1065.35206
[10] Kilbas, A. A.; Repin, O. A., An analog of the Tricomi problem for a mixed type equation with a partial fractional derivative, Fract. Calc. Appl. Anal., 13, 69-84 (2010) · Zbl 1201.35148
[11] M. S. Salakhitdinov and E. T. Karimov, ‘‘On a nonlocal problem with gluing condition of integral form for parabolic-hyperbolic equation with Caputo operator,’’ Dokl. Akad. Nauk Resp. Uzbekist., No. 4, 6-9 (2014).
[12] T. K. Yuldashev and B. J. Kadirkulov, ‘‘Boundary value problem for weak nonlinear partial differential equations of mixed type with fractional Hilfer operator,’’ Axioms 9, 68-1-19 (2020).
[13] T. K. Yuldashev and E. T. Karimov, ‘‘Inverse problem for a mixed type integro-differential equation with fractional order Caputo operators and spectral parameters,’’ Axioms 9, 121-1-24 (2020).
[14] Mamazhanov, M.; Khalmuratov, D., Boundary-value problems for third-order parabolic-hyperbolic equations with non characteristic type-change boundaries, Differ. Equat., 25, 200-203 (1989) · Zbl 0708.35057
[15] Mamazhonov, M.; Mamadalieva, Kh. B., “Some boundary value problems for a third-order parabolic-hyperbolic equation in a pentagonal domain,” Bull. KRASEC, Phys. Math. Sci., 13, 27-34 (2016) · Zbl 1413.35348
[16] Abdullaev, O. Kh., “Non-local problem for the loaded mixed type equations with integral operator,” Vest. Samar. Tech. Univ., Fiz.-, Mat. Nauki, 20, 220-240 (2016) · Zbl 1413.35320
[17] Melisheva, E. P., “The Dirichlet problem for the loaded Lavrent’ev-Bitsadze equation,” Vestn. SamGU, Estestv.-, Nauch. Ser., 6, 39-47 (2010)
[18] Ramazanov, M. I.; Kosmakova, M. T.; Kasymova, L. Zh., On a problem of heat equation with fractional load, Lobachevskii J. Math., 41, 1873-1885 (2020) · Zbl 1452.35245
[19] Yuldashev, T. K.; Islomov, B. I.; Alikulov, E. K., Boundary-value problems for loaded third-order parabolic-hyperbolic equations in infinite three-dimensional domains, Lobachevskii J. Math., 41, 926-944 (2020) · Zbl 1450.35187
[20] Yuldashev, T. K.; Kadirkulov, B. J., Inverse boundary value problem for a fractional differential equations of mixed type with integral redefinition conditions, Lobachevskii J. Math., 42, 649-662 (2021) · Zbl 1465.35412
[21] Yuldashev, T. K.; Kadirkulov, B. J., Nonlocal problem for a mixed type fourth-order differential equation with Hilfer fractional operator, Ural Math. J., 6, 153-167 (2020) · Zbl 1448.35341
[22] Abdullaev, O. Kh., About a problem for the degenerate mixed type equation involving Caputo and Erdelyi-Kober operators of fractional order, Ukr. Math. J., 71, 723-738 (2019) · Zbl 1436.93064
[23] Abdullaev, O. Kh., Gellerstedt type problem for the loaded parabolic-hyperbolic type equation with Caputo and Erdlyi-Kober operators of fractional order, Russ. Math., 64, 33-46 (2020) · Zbl 1465.35312
[24] Yuldashev, T. K.; Abdullaev, O. Kh., Unique solvability of a boundary value problem for a loaded fractional parabolic-hyperbolic with nonlinear terms, Lobachevskii J. Math., 42, 1113-1123 (2021) · Zbl 1466.35269
[25] Abdullayev, O. Kh.; Matchanova, A. A., Non-local boundary value problems for a loaded parabolic-hyperbolic type equation of third order involving Caputo operator, Bull. Romanovskii Inst. Math., 2018, 36-43 (2018)
[26] P. Agarwal and O. Kh. Abdullaev, ‘‘A non-local problem with integral gluing condition for a third-order loaded equation with parabolic-hyperbolic operator involving fractional derivatives,’’ Math. Meth. Appl. Sci. 248 (2019). · Zbl 1447.35344
[27] Islomov, B.; Baltaeva, U., Boundary value problems for a third-order loaded parabolic-hyperbolic type equation with variable coefficients, El. J. Differ. Equat., 2015, 1-10 (2015) · Zbl 1350.34016
[28] Pskhu, A. V., Partial Differential Equations of Fractional Order (2005), Moscow: Nauka, Moscow · Zbl 1193.35245
[29] Pskhu, A. V., The fundamental solution of a diffusion-wave equation of fractional order, Izv. Math., 73, 351-392 (2009) · Zbl 1172.26001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.