Inverse problems for equations of a mixed parabolic-hyperbolic type with power degeneration in finding the right-hand parts that depend on time

References

[1] S. Agmon, L. Nirenberg and M. H. Protter, A maximum principle for a class of hyperbolic equations and applications to equations of mixed elliptic-hyperbolic type, Comm. Pure Appl. Math. 6 (1953), 455–470. 10.1002/cpa.3160060402Suche in Google Scholar

[2] Y. E. Anikonov, B. A. Bubnov and G. N. Erokhin Inverse and Ill-Posed Sources Problems, VSP, Utrecht, 1997. 10.1515/9783110969412Suche in Google Scholar

[3] Y. E. Anikonov and M. V. Neshchadim, Identity for approximate quantum equations and inverse problems, Sib. J. Ind. Mat. 10 (2007), no. 4, 3–9. Suche in Google Scholar

[4] V. I. Arnold, Small denominators, Russian Acad. Sci. Math. Ser. 25 (1961), 21–86. Suche in Google Scholar

[5] G. Bateman and A. Erdeyi, Higher Transcendental Functions. Vol. 2, Nauka, Moscow, 1974. Suche in Google Scholar

[6] Y. Y. Belov, Inverse Problems for Partial Differential Equations, VSP, Utrecht, 2002. 10.1515/9783110944631Suche in Google Scholar

[7] Y. Y. Belov, On the problem of identifying a source function for tne semi-evolutionary system, Z. SFU. Ser. Mat. Nat. 3 (2010), no. 4, 487–499. Suche in Google Scholar

[8] Y. Y. Belov and K. V. Korshun, On an inverse problem for a Burgers-type equation, Sib. J. Ind. Mat. 16 (2013), no. 3, 28–40. Suche in Google Scholar

[9] A. L. Bukhgeim, Introduction to the Theory of Inverse Problems, VSP, Utrecht, 2000. Suche in Google Scholar

[10] J. R. Cannon and P. DuChateau, An inverse problem for an unknown source term in a wave equation, SIAM J. Appl. Math. 43 (1983), no. 3, 553–564. 10.1137/0143036Suche in Google Scholar

[11] A. M. Denisov, Introduction to the Theory of Inverse Problems, MSU, Moscow, 1994. Suche in Google Scholar

[12] T. D. Dzhuraev, A. Sopuev and M. Mamazhanov, Boundary Value Problems for Parabolic-Hyperbolic Equations, “Fan”, Tashkent, 1986. Suche in Google Scholar

[13] I. M. Gel’fand, Some questions of analysis and differential equations, Uspehi Mat. Nauk 14 (1959), no. 3(87), 3–19. Suche in Google Scholar

[14] V. Isakov, Inverse Problems for Partial Differential Equations, Appl. Math. Sci. 127, Springer, New York, 2006. Suche in Google Scholar

[15] M. Ivanchov, Inverse Problems for Equations of Parabolic Type, VNTL, Lviv, 2003. Suche in Google Scholar

[16] V. K. Ivanov, V. V. Vasin and V. P. Tanana, The Theory of Linear Ill-Posed Problems and its Applications, “Nauka”, Moscow, 1978. Suche in Google Scholar

[17] S. I. Kabanikhin, Inverse and Incorrect Tasks, Siberian Scientific, Novosibirsk, 2009. Suche in Google Scholar

[18] S. I. Kabanikhin, A. D. Satybaev and M. A. Shishlenin, Direct Methods of Solving Multidimensional Inverse Hyperbolic Problems, VSP, Utrecht, 2005. 10.1515/9783110960716Suche in Google Scholar

[19] N. Y. Kapustin, The Tricomi problem for a parabolic-hyperbolic equation with a degenerate hyperbolic side. I, Differ. Equ. 23 (1987), no. 1, 72–78. Suche in Google Scholar

[20] N. Y. Kapustin, The Tricomi problem for a parabolic-hyperbolic equation with a degenerate hyperbolic side. II, Differ. Equ. 24 (1988), no. 8, 898–903. Suche in Google Scholar

[21] M. V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, VSP, Utrecht, 2004. 10.1515/9783110915549Suche in Google Scholar

[22] A. I. Kozhanov, Composite Type Equations and Inverse Problems, VSP, Utrecht, 1999. 10.1515/9783110943276Suche in Google Scholar

[23] A. I. Kozhanov, On a nonlinear loaded parabolic equation and a related inverse problem, Mat. Notes 76 (2004), no. 6, 784–795. 10.1023/B:MATN.0000049678.16540.a5Suche in Google Scholar

[24] A. I. Kozhanov and R. R. Safiullova, Linear inverse problems for parabolic and hyperbolic equations, J. Inverse Ill-Posed Probl. 18 (2010), no. 1, 1–24. 10.1515/jiip.2010.001Suche in Google Scholar

[25] O. A. Ladyzenskaja and L. Stupjalis, Equations of mixed type, Vestnik Leningrad. Univ. 20 (1965), no. 19, 38–46. Suche in Google Scholar

[26] M. M. Lavrent’ev, K. G. Reznitskaya and V. G. Yakhno, One-Dimensional Inverse Problems of Mathematical Physics, “Nauka”, Novosibirsk, 1982. Suche in Google Scholar

[27] A. Lorenzi, An Introduction to Mathematical Problems via Functional Analysis, VSP, Utrecht, 2001. 10.1515/9783110940923Suche in Google Scholar

[28] I. T. Lozanovskaya and Y. S. Uflyand, A class of problems in mathematical physics with a mixed eigenvalue spectrum, Dokl. Acad. Sci. USSR 164 (1965), no. 5, 1005–1007. Suche in Google Scholar

[29] A. G. Megrabov, Forward and Inverse Problems for Hyperbolic, Elliptic, and Mixed Type Equations, VSP, Utrecht, 2003. 10.1515/9783110944983Suche in Google Scholar

[30] C. S. Morawetz, A uniqueness theorem for Frankl’s problem, Comm. Pure Appl. Math. 7 (1954), 697–703. 10.1002/cpa.3160070406Suche in Google Scholar

[31] C. S. Morawetz, Note on a maximum principle and a uniqueness theorem for an elliptic-hyperbolic equation, Proc. Roy. Soc. Lond. Ser. A 236 (1956), 141–144. 10.1098/rspa.1956.0119Suche in Google Scholar

[32] C. S. Morawetz, Mathematical problems in transonic flow, Canad. Math. Bull. 29 (1986), no. 2, 129–139. 10.4153/CMB-1986-023-3Suche in Google Scholar

[33] A. I. Prilepko and A. B. Kostin, An estimate for the spectral radius of an operator and the solvability of inverse problems for evolution equations, Mat. Notes 53 (1993), no. 1, 63–66. 10.1007/BF01208524Suche in Google Scholar

[34] A. I. Prilepko, D. G. Orlovsky and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Monogr. Textb. Pure Appl. Math. 231, Marcel Dekker, New York, 1999. Suche in Google Scholar

[35] A. I. Prilepko and V. V. Soloviev, Solvability theorems and the Rothe method in inverse problems for an equation of parabolic type. I, Differ. Equ. 23 (1987), no. 10, 1791–1799. Suche in Google Scholar

[36] A. I. Prilepko and V. V. Soloviev, Solvability theorems and the Rothe method in inverse problems for an equation of parabolic type. II, Differ. Equ. 23 (1987), no. 11, 1971–1980. Suche in Google Scholar

[37] A. I. Prilepko and I. V. Tikhonov, Uniqueness of the solution of an inverse problem for an evolution equation and applications to the transfer equation, Mat. Zametki 51 (1992), no. 2, 77–87. 10.1007/BF02102123Suche in Google Scholar

[38] A. I. Prilepko and I. A. Vasin, A study of problems of the uniqueness of solutions in some nonlinear inverse problems in hydrodynamics, Differ. Equ. 26 (1990), no. 1, 109–120. Suche in Google Scholar

[39] M. H. Protter, Uniqueness theorems for the Tricomi problem. I, J. Rational Mech. Anal. 2 (1953), 107–114. 10.1512/iumj.1953.2.52005Suche in Google Scholar

[40] M. H. Protter, Uniqueness theorems for the Tricomi problem. II, J. Rational Mech. Anal. 4 (1955), no. 5, 721–733. 10.1512/iumj.1955.4.54027Suche in Google Scholar

[41] V. G. Romanov, Inverse Problems of Mathematical Physics, “Nauka”, Moscow, 1984. Suche in Google Scholar

[42] V. G. Romanov, Stability in Inverse Problems, Scientific World, Moscow, 2005. Suche in Google Scholar

[43] V. G. Romanov and S. I. Kabanikhin, Inverse Geo-Electrical Problems, “Nauka”, Moscow, 1991. Suche in Google Scholar

[44] V. G. Romanov and B. G. Mukanova, Inverse source problem for wave equation, Eurasian J. Math. Comput. Appl. 4 (2016), no. 3, 16–28. 10.32523/2306-6172-2016-4-3-15-28Suche in Google Scholar

[45] K. B. Sabitov, Functional, Differential and Integral Equations, Vysshaya Shkola, Moscow, 2005. Suche in Google Scholar

[46] K. B. Sabitov, Direct and Inverse Problems for Equations of Mixed Parabolic-Hyperbolic Type, Nauka, Moskow, 2016. Suche in Google Scholar

[47] K. B. Sabitov, Initial boundary value and inverse problems for the inhomogeneous equation of a mixed parabolic-hyperbolic equation, Math. Notes 102 (2017), no. 3, 378–395. 10.1134/S0001434617090085Suche in Google Scholar

[48] K. B. Sabitov and E. M. Safin, An inverse problem for an equation of mixed parabolic-hyperbolic type in a rectangular domain, Russian Math. 54 (2010), no. 4, 48–54. 10.3103/S1066369X10040067Suche in Google Scholar

[49] K. B. Sabitov and E. M. Safin, The inverse problem for an equation of mixed parabolic-hyperbolic type, Mat. Notes 87 (2010), no. 6, 880–889. 10.1134/S0001434610050287Suche in Google Scholar

[50] K. B. Sabitov and S. N. Sidorov, On a nonlocal problem for a degenerating parabolic-hyperbolic equation, Differ. Equ. 50 (2014), no. 3, 352–361. 10.1134/S0012266114030094Suche in Google Scholar

[51] K. B. Sabitov and S. N. Sidorov, Inverse problem for degenerate parabolic-hyperbolic equation with nonlocal boundary condition, Russian Math. (Iz. VUZ) 59 (2015), no. 1, 39–50. 10.3103/S1066369X15010041Suche in Google Scholar

[52] K. B. Sabitov and S. N. Sidorov, Initial-boundary-value problem for inhomogeneous degenerate equations of mixed parabolic-hyperbolic type, J. Math. Sci. 236 (2019), no. 6, 603–640. 10.1007/s10958-018-4136-ySuche in Google Scholar

[53] L. Stupelis, Initial-boundary value problems for equations of mixed type, Proc. Steklov Mat. Inst. 127 (1975), 135–170. Suche in Google Scholar

[54] A. N. Tikhonov and V. I. Arsenin, Methods for Solving Incorrect Problems, 3rd ed., Nauka, Moscow, 1986. Suche in Google Scholar

[55] Y. S. Uflyand, On the question of the propagation of oscillations in compound electric lines, Eng. Phys. J. 7 (1964), 89–92. Suche in Google Scholar

[56] A. G. Yagola, V. Yanfei, I. E. Stepanova and V. N. Titarenko, Inverse Problems and Methods for Their Solution. Applications to Geophysics, BINOM, Moscow, 2014. Suche in Google Scholar

[57] M. Yamamoto, Stability, reconstruction formula and regularization for an inverse source hyperbolic problem by a control method, Inverse Problems 11 (1995), no. 2, 481–496. 10.1088/0266-5611/11/2/013Suche in Google Scholar