# A problem of the Bitsadze-Samarskii type for a loaded hyperbolic-parabolic equation

### Abstract

We investigate a problem of the Bitsadze-Samarskii type with inner boundary conditions for a model characteristicly loaded equation of a mixed hyperbolicparabolic type with degeneration of order in the hyperbolic domain. The equation is considered in a mixed domain with a rectangle as the parabolic part and a semi-infinite strip bounded by the characteristics of the wave equation as the hyperbolic part. In the parabolic domain, the equation is a loaded nonhomogeneous heat equation, while it is a loaded nonhomogeneous McKendrick equation in the hyperbolic domain. The function’s values are given on one side of the boundary of the parabolic domain; on the other side of that boundary we give a condition binding the value of the unknown function with the function’s values at the interior points of the its domain. The existence and uniqueness conditions for the regular solution to the problem are determined and the solution representations are written out.

### References

[1]

Nakhushev A. M., Loaded Equations and Their Applications [in Russian], Nauka, Moscow (2012).

[2]

Dzhenaliyev M. T. and Ramazanov M. I., Loaded Equations As Perturbed Differential Equations [in Russian], Gylym, Almaty (2010).

[3]

Nakhushev A. M., “Nonlocal boundary value problems with shift and their connection with loaded equations,” Differ. Uravn., 21, No. 1, 92–101 (1985).

[4]

Ogorodnikov E. N., “Some characteristic problems for loaded systems of differential equations and their relationship with non-local boundary value problems,” Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki, 19, 22–28 (2003).

[5]

Kozhanov A. I., “A nonlinear loaded parabolic equation and a related inverse problem,” Math. Notes, 76, No. 6, 784–795 (2004).

[6]

Kozhanov A. I. and Pul’kina L. S., “On the solvability of boundary value problems with a nonlocal boundary condition of integral form for multidimensional hyperbolic equations,” Differ. Equ., 42, No. 9, 1233–1246 (2006).

[7]

Khubiev K. U., “Boundary-value problem for a loaded equation of hyperbolic-parabolic type with degeneracy of order in the domain of hyperbolicity,” Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 149, 113–117 (2018).

[8]

Attaev A. Kh., “On some problems for loaded partial differential equation of the first order,” Vestn. KRAUNC. Fiz.-Mat. Nauki, 16, No. 4-1, 9–14 (2016).

[9]

Attaev A. Kh., “The Cauchy problem for for the MC Kendrick-Von Foerster loaded equation,” Int. J. Pure Appl. Math., 113, No. 4, 47–52 (2017).

[10]

Berezgova R. Z., “On a nonlocal boundary-value problem for the Mckendrick von Foerster loaded equation with Caputo operator,” Vestn. KRAUNC, Fiz.-Mat. Nauki, 19, No. 3, 5–9 (2017).

[11]

Borel’ L. V., “On solvability of degenerate loaded systems of equations,” Mat. Zametki SVFU, 22, No. 4, 3–11 (2015).

[12]

Sabitov K. B., “Initial-boundary problem for parabolic-hyperbolic equation with loaded summands,” Russ. Math. (Iz. VUZ), 59, No. 6, 23–33 (2015).

[13]

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

[14]

Sadarangani K. B. and Abdullaev O. Kh., “About a problem for loaded parabolic-hyperbolic type equation with fractional derivatives,” Int. J. Differ. Equ., No. 2016, 1–6 (2016).

[15]

Zikirov O. S. and Kholikov D. K., “On some problem for a loaded pseudoparabolic equation of the third order,” Mat. Zametki SVFU, 23, No. 2, 19–30 (2016).

[16]

Khubiev K. U., “Analogue of Tricomi problem for characteristically loaded hyperbolic-parabolic equation with variable coefficients,” Ufim. Mat. Zh., 9, No. 2, 92–101 (2017).

[17]

Tarasenko A. V., “On solvability of nonlocal problem for loaded parabolic-hyperbolic equation,” Russ. Math. (Iz. VUZ), 62, No. 3, 53–59 (2018).

[18]

Khubiev K. U., “Boundary value problem with shift for loaded hyperbolic-parabolic type equation involving fractional diffusion operator,” Vestn. Udmurt. Univ., Mat. Mekh. Komp. Nauki, 28, No. 1, 82–90 (2018).

[19]

Nakhushev A. M., “Boundary value problems for loaded integro-differential equations of hyperbolic type and some of their applications to the prediction of ground moisture [in Russian],” Differ. Uravn., 15, No. 1, 96–105 (1979).

[20]

Nakhushev A. M., Equations of Mathematical Biology [in Russian], Vysshaia Shkola, Moscow (1995).

[21]

Il’in V. A. and Moiseev E. I., “A nonlocal boundary value problem for the Sturm–Liouville operator in a differential and a difference treatment [in Russian],” Dokl. Akad. Nauk SSSR, 291, No. 3, 534–539 (1986).

[22]

Nakhusheva Z. A., “On a Bitsadze–Samarsky type nonlocal elliptic boundary value problem,” Dokl. Adyg. (Cherkess.) Mezdunar. Akad. Nauk, 15, No. 1, 18–23 (2013).

[23]

Khubiev K. U., “Inner boundary value problems for the loaded equations of mixed type,” Izv. Vyssh. Uchebn. Zaved., Sev.-Kavk. Reg., Estestv. Nauki, No. 6, 23–25 (2008).

*Mathematical notes of NEFU*, 26(2), pp. 31-40. doi: https://doi.org/10.25587/SVFU.2019.102.31510.

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