The conjugation problem for pseudoparabolic and pseudohyperbolic equations

  • Grigorieva Alexandra I., shadrina_ai@mail.ru M. K. Ammosov North-Eastern Federal University, Institute of Mathematics and Informatics, 48 Kulakovskii Street, Yakutsk 677000, Russia
Keywords: pseudoparabolic equation, pseudohyperbolic equation, discontinuous coefficient, conjugation problem

Abstract

We study solvability of a conjugation problem for pseudoparabolic and pseudohyperbolic equations. The equations are considered as equations with discontinuous coefficients. We prove the existence and uniqueness theorem using natural parameter continuation.

References

[1]Gelfand I. M., “Some questions of analysis and differential equations”, Usp. Mat. Nauk, 14:3 (1959), 3–19  mathnet  zmath
[2]Oleinik O. A., “Boundary value problems for linear equations of elliptic and parabolic type with discontinuous coefficients”, Izv. AN USSR, Ser. Mat., 25:1 (1961), 3–20  mathnet  mathscinet  zmath
[3]Ilyin V. A., “On the solvability of the Dirichlet and Neumann problems for linear elliptic operator with discontinuous coefficients”, Dokl. Akad. Nauk, 137:1 (1961), 28–30  mathnet  zmath
[4]Ilyin V. A., “Fourier method for hyperbolic equations with discontinuous coefficients”, Dokl. Akad. Nauk, 142:1 (1961), 21–24  mathnet  zmath
[5]Ilyin V. A. and Shishmarev I. A., “Potential method for the Dirichlet and Neumann problems in the case of equations with discontinuous coefficients”, Sib. Math. J., 2:1 (1961), 46–58  zmath
[6]Ladyzhenskaya O. A., Solonnikov V. A., and Uraltseva N. N.,, Linear and quasilinear parabolic equations, Nauka, Moscow, 1967  mathscinet
[7]Ilyin V. A. and Luferenko P. V., “Mixed problems describing the longitudinal vibrations of a rod consisting of two segments with different densities, different elastic, but the same impedances”, Dokl. Math., 428:1 (2009), 12–15  mathnet  zmath
[8]Andropova O. A., “Spectral problems interfacing with surface dissipation of energy”, Tr. Inst. Prikl. Mat. Mekh., 19 (2009), 10–22
[9]Rogozhnikov A. M., “Study of the mixed problem, describing the process of shaft vibrations, which consists of several sections, provided match the transit time of waves in each of these areas”, Dokl. Math., 441:4 (2012), 449–451
[10]Kuleshov A. A., “Mixed problem for the equation of longitudinal vibrations of an inhomogeneous bar with a free or fixed right end, consisting of two sections of different density and elasticity”, Dokl. Math., 442:4 (2012), 451–454  zmath
[11]Rogozhnikov A. M., ““Study of the mixed problem, describing the process of shaft vibrations, consisting of several sections with arbitrary lengths”, Dokl. Math., 444:5 (2012), 488–491  mathscinet  zmath
[12]Smirnov I. N., “Oscillations described by the telegraph equation for a system consisting of several sections of different density and elasticity”, Differ. Equ., 49:5 (2012), 643–648
[13]Bitsadze A. V., Equations of Mixed Type, Akad. Nauk SSSR, Moscow, 1959  mathscinet
[14]Ladyzhenskaya O. A. and Stupyalis L., “About mixed type equations”, Vestn. Leningrad.Univ., 1967, no. 18, 38–46
[15]Smirnov M. M.,, Equations of Mixed Type, Nauka, Moscow, 1970  mathscinet
[16]Ladyzhenskaya O. A. and Stupyalis L., “Boundary value problems for mixed type equations”, Tr. Mat. Inst. Steklova, 116 (1971), 101–136  mathnet  mathscinet  zmath
[17]Stupyalis L., “Boundary value problems for for elliptic-hyperbolic equations”, Tr. Mat. Inst. Steklova, 125 (1973), 211–229  mathnet  mathscinet
[18]Tersenov S. A., Introduction to the theory of parabolic equations with a varying direction of time, Inst. Mat., Novosibirsk, 1982  mathscinet
[19]Juraev T. D., Boundary value problems for equations of mixed and mixed-composite types, FAN, Tashkent, 1986  mathscinet
[20]Moiseev E. I., Mixed Type Equations with a Spectral Parameter, MGU, Moscow, 1988  mathscinet
[21]Sabitov K. B. and Martemyanova N. V., “The inverse problem for an equation of elliptichyperbolic type with a nonlocal boundary condition”, Sib. Math. J., 53:3 (2012), 633–647  mathnet  mathscinet  zmath  elib
[22]Kozhanov A. I., “A conjugation problem for a class of composite type equations with changing direction”, Nonclassical Equ. Math. Phys., Inst. Math., Novosibirsk, 2002, 96–109  zmath
[23]Bouziani A., Merazga N., “Solution to a transmission problem for quasilinear pseudoparabolic equations by the Rothe method”, Electronic Journal of Differential Equ., 2007, no. 14, 1–27  mathscinet
[24]Shubin V. V, “Boundary problems for third-order equations with discontinuous coefficients”, Vestn. Novosib. Gos. Univ., 12:1 (2012), 126–138  mathnet  zmath
[25]Potapova S. V., “Boundary value problems for pseudohyperbolic equations with a variable time direction”, TWMS J. Pure Appl. Math., 3:1 (2012), 73–91  mathscinet
[26]Kozhanov A. I. and Sharin E. F., “A conjugation problem for some nonclassical differential equations of higher order”, Ukr. Mat. Vestn., 11:2 (2014), 181–202
[27]Kozhanov A. I. and Potapova S. V., “The Dirichlet problem for a class of composite type equations with a discontinuous coefficient of the highest derivative”, Dalnevost. Math. J., 14:1 (2014), 48–65  mathnet  mathscinet  zmath  elib
[28]Antipin V. I., “The solvability of the boundary value problem for a third-order equation with changing time direction”, Mat. Zamet. YaGU, 18:1 (2011), 8–15  zmath
[29]Pyatkov S. G., Popov S., Antipin V. I., “On solvability of boundary value problem for kinetic operator-differential equations”, Integral Equ. Operator Theory, 80:4 (2014), 557–580  crossref  mathscinet  zmath  elib  scopus
[30]T. von Petersdorff, “Boundary integral equations for mixed Dirichlet, Neumann and transmission problems”, Math. Meth. Appl. Sci., 11 (1989), 185–213  crossref  mathscinet  zmath  scopus
[31]Nikolskiy D. N., “Evolution section of various liquids in non-uniform layers”, Comput. Math. Math. Phys., 50:7 (2010), 1269–1275  mathnet  mathscinet  zmath  elib
[32]Nikolskiy D. N., “The three-dimensional pollution in the border bounded by a piecewise-porous medium”, Comput. Math. Math. Phys., 51:5 (2011), 913–919  mathnet  mathscinet  zmath  elib
[33]Trenogin V. A., Functional analysis, Fizmatlit, Moscow, 2007  mathscinet
[34]Sobolev S. L., “On a boundary problem of mathematical physics”, Izv. Akad. Nauk, 18:2 (1954), 3–50  mathnet  mathscinet  zmath
[35]Demidenko G. V., and Uspenskii S. V., Equations and systems unsolvable with respect to the higher derivative, Nauchn. Kniga, Novosibirsk, 1998  mathscinet
[36]Kozhanov A. I., Composite type equations and inverse problems, VSP, Utrecht, 1999  mathscinet  zmath
[37]Sviridyuk G. A., Fedorov V. E., Linear Sobolev type equations and degenerate semigroup of operators, VSP, Utrecht, 2003  mathscinet
[38]Hayashi N., Kaikina E. I., Naumkin P. I., Shishmarev I. A., Asymptotic for dissipative nonlinear equations, Springer-Verl., Berlin; Heidelberg, 2006  mathscinet
[39]Sveshnikov A. G., Alshin A. B., Korpusov M. O., and Pletner Yu. D., Linear and non-linear equations of Sobolev type, Fizmatlit, Moscow, 2007
[40]Korpusov M. O., The destruction of solutions in the nonclassical non-local equations, Librokom, Moscow, 2011
How to Cite
Grigorieva, A. (2016) “The conjugation problem for pseudoparabolic and pseudohyperbolic equations”, Mathematical notes of NEFU, 23(3), pp. 27-45. Available at: http://mzsvfu.ru/index.php/mz/article/view/the-conjugation-problem-for-pseudoparabolic-and-pseudohyperbolic-equations (Accessed: 22September2020).
Section
Mathematics