Inverse problem for an equation of mixed parabolic-hyperbolic type with a characteristic line of change

D. K. Durdiev

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 27 (2023) no. 4, pp. 607-620.

Résumé

This study investigates direct and inverse problems for a model equation of mixed parabolic-hyperbolic type. In the direct problem, an analogue of the Tricomi problem is considered for this equation with a characteristic line of type change. The unknown in the inverse problem is a variable coefficient of the lower-order term in the parabolic equation. To determine it relative to the solution defined in the parabolic part of the domain, an integral overdetermination condition is specified. Local theorems of unique solvability of the posed problems in terms of classical solutions are proven.

Détail

Keywords: parabolic-hyperbolic equation, characteristic, Green's function, inverse problem, principle of compressed mappings

Bibliographie

[1] Babich V. M., Kapilevich M. B., Mikhlin S. G., et al., Lineinye uravneniia matematicheskoi fiziki [Linear Equations of Mathematical Physics], Nauka, Moscow, 1964, 368 pp. (In Russian)

[2] Aziz Kh., Settari A., Matematicheskoe modelirovanie plastovykh sistem [Petroleum Reservoir Simulation], Regul. Khaotich. Dinam., Moscow, Izhevsk, 2004, 416 pp. (In Russian)

[3] Uflyand Ya. S., Integral'nye preobrazovaniia v zadachakh teorii uprugosti [Integral Transforms in the Problems of Elasticity Theory], Nauka, Moscow, 1968, 402 pp. (In Russian)

[4] Shashkov A. G., Sistemno-strukturnyi analiz protsessov teploobmena i ego primenenie [System-Structural Analysis of the Heat Transfer Process and its Application], Energoatomizdat, Moscow, 1983, 280 pp. (In Russian)

[5] Nakhushev A. M., Uravneniia matematicheskoi biologii [Equations of Mathematical Biology], Vyssh. shk., Moscow, 1995, 301 pp. (In Russian)

[6] Zolina L. A., “On a boundary value problem for a model equation of hyperbolo-parabolic type”, U.S.S.R. Comput. Math. Math. Phys., 6:6 (1966), 63–78 | DOI | MR

[7] Sabitov K. B., Priamye i obratnye zadachi dlia uravnenii smeshannogo parabolo-giperbolicheskogo tipa [Direct and Inverse Problems for Mixed Parabolic-Hyperbolic Type Equations], Nauka, Moscow, 2016, 271 pp. (In Russian)

[8] Sabitov K. B., Safin E. M., “The inverse problem for a mixed-type parabolic-hyperbolic equation in a rectangular domain”, Russian Math. (Iz. VUZ), 54:4 (2010), 48–54 | DOI | MR

[9] Sabitov K. B., Safin E. M., “The inverse problem for an equation of mixed parabolic-hyperbolic type”, Math. Notes, 87:6 (2010), 880–889 | DOI | DOI | MR

[10] Sabitov K. B., “Initial boundary and inverse problems for the inhomogeneous equation of a mixed parabolic-hyperbolic equation”, Math. Notes, 102:3 (2017), 378–395 | DOI | DOI | MR

[11] Djamalov S. Z., “The nonlocal boundary value problem with constant coefficients for the multidimensional mixed type equation of the first kind”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 21:4 (2017), 597–610 (In Russian) | DOI | Zbl

[12] Sabitov K. B., Martem'yanova N. V., “A nonlocal inverse problem for a mixed-type equation”, Russian Math. (Iz. VUZ), 55:2 (2011), 61–74 | DOI | MR

[13] Yunusova G. R., “Nonlocal problems for the equation of the mixed parabolic-hyperbolic type”, Vestn. Samar. Gos. Univ., Estestvennonauchn. Ser., 2011, no. 8(89), 108–117 (In Russian)

[14] Sabitov K. B., Sidorov S. N., “Inverse problem for degenerate parabolic-hyperbolic equation with nonlocal boundary condition”, Russian Math. (Iz. VUZ), 59:1 (2015), 39–50 | DOI

[15] Sidorov S. N., “Inverse problems for a degenerate mixed parabolic-hyperbolic equation on finding time-depending factors in right hand sides”, Ufa Math. J., 11:1 (2019), 75–89 | DOI

[16] Sabitov K. B., Sidorov S. N., “On a nonlocal problem for a degenerating parabolic-hyperbolic equation”, Differ. Equ., 50:3 (2014), 352–361 | DOI | DOI

[17] Sabitov K. B., Sidorov S. N., “Initial-boundary-value problem for inhomogeneous degenerate equations of mixed parabolic-hyperbolic type”, J. Math. Sci. (N. Y.), 236:6 (2019), 603–640 | DOI | MR | Zbl

[18] Durdiev D. K., “Inverse source problem for an equation of mixed parabolic-hyperbolic type with the time fractional derivative in a cylindrical domain”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 26:2 (2022), 355–367 | DOI

[19] Prilepko A. I., Kostin A. B., Solov'ev V. V., “Inverse source and inverse coefficients problems for elliptic and parabolic equations in Hölder and Sobolev spaces”, Sib. J. Pure and Appl. Math., 17:3 (2017), 67–85 (In Russian) | DOI

[20] Ivanchov N. I., “On the inverse problem of simultaneous determination of thermal conductivity and specific heat capacity”, Sib. Math. J., 35:3 (1994), 547–555 | DOI | MR | Zbl

[21] Durdiev D. K., Durdiev D. D., “The Fourier spectral method for determining a heat capacity coefficient in a parabolic equation”, Turk. J. Math., 46:8 (2022), 3223–3233 | DOI

[22] Denisov A. M., Elements of the Theory of Inverse Problems, Inverse and Ill-Posed Problems Series, 14, VSP, Utrecht, 1999, iv+272 pp. | Zbl

[23] Prilepko A. I., Orlovsky D. G., Vasin I. A., Methods for Solving Inverse Problems in Mathematical Physics, Pure and Applied Mathematics, 231, Marcel Dekker, New York, NY, 2000, xiii+709 pp. | DOI | Zbl

[24] Durdiev D. K., Zhumaev Zh. Zh., “Memory kernel reconstruction problems in the integro-differential equation of rigid heat conductor”, Math. Meth. Appl. Sci., 45:14 (2022), 8374–8388 | DOI

[25] Durdiev D. K., Zhumaev Zh. Zh., “One-dimensional inverse problems of finding the kernel of integrodifferential heat equation in a bounded domain”, Ukr. Math. J., 73:11 (2022), 1723–1740 | DOI

[26] Durdiev D. K., Jumaev J. J., Atoev D. D., “Inverse problem on determining two kernels in integro-differential equation of heat flow”, Ufa Math. J., 15:2 (2023), 119–134 | DOI

[27] Durdiev D. K., Zhumaev Zh. Zh., “Problem of determining the thermal memory of a conducting medium”, Differ. Equ., 56:6 (2020), 785–796 | DOI | DOI

[28] Romanov V. G., Obratnye zadachi matematicheskoi fiziki [Inverse Problems of Mathematical Physics], Nauka, Moscow, 1984, 264 pp. (In Russian)

[29] Kabanikhin S. I., Obratnye i nekorrektnye zadachi [Inverse and Ill-posed Problems], Sibirskoe Nauchnoe Izd., Novosibirsk, 2009, 457 pp. (In Russian)

[30] Hasanoğlu A. H., Romanov V. G., Introduction to Inverse Problems for Differential Equations, Springer, Cham, 2017, xiii+261 pp. | DOI | Zbl

[31] Durdiev D. K., Totieva Z. D., Kernel Determination Problems in Hyperbolic Integro-Differential Equations, Infosys Science Foundation Series, Springer, Singapore, 2023, xxvi+368 pp. | DOI | Zbl

[32] Durdiev D. K., “Determining the coefficient of a mixed parabolic-hyperbolic equation with noncharacteristic type change line”, Differ. Equ., 58:2 (2022), 1618–1629 | DOI | DOI

[33] Tikhonov A. N., Samarskii A. A., Uravneniia matematicheskoi fiziki [Equations of Mathematical Physics], Nauka, Moscow, 1977, 734 pp. (In Russian)

[34] Smirnov V. I., Kurs vysshei matematiki [Course of Higher Mathematics], v. 4, part 2, Nauka, Moscow, 1981, 551 pp. (In Russian)