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About convergence of difference schemes for a third-order pseudo-parabolic equation with nonlocal boundary value condition. (English) Zbl 1473.65266

Summary: A nonlocal boundary value problem for a third-order pseudoparabolic equation with variable coefficients is considered. For solving this problem, a priori estimates in the differential and difference forms are obtained. The obtained a priori estimates imply the uniqueness and stability of the solution on a layer with respect to the initial data and the right-hand side and the convergence of the solution of the difference problem to the solution of the differential problem.

MSC:

65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N06 Finite difference methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
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[1] E.S. Dzekcer, Equation of motion of underground water with a free surface in multilayer media, Sov. Phys., Dokl., 20 (1975), 2426. Zbl 0331.76056 · Zbl 0331.76056
[2] L.I. Rubinshtein, On heat propagation in heterogeneous media, Izv. Akad. Nauk SSSR, Ser. Geogr., 12:1 (1948), 2745.
[3] T.W. Ting, A cooling process according to two-temperature theory of heat conduction, J. Math. Anal. Appl., 45:1 (1974), 2331. Zbl 0272.35039 · Zbl 0272.35039
[4] M. Hallaire, L’eau et la production vegetable, Inst. National de la Recherche Agronomique, 9 (1964).
[5] A.F. Chudnovskii, Thermal physics of soils, Nauka, Moskow, 1976.
[6] L.I. Kamynin, A boundary value problem in the theory of heat conduction with a nonclassical boundary condition, U.S.S.R. Comput. Math. Math. Phys., 4:6 (1964), 3359. Zbl 0206.39801
[7] A.F. Chudnovskii Nekotorye korrektivy v postanovke i reshenii zadach teplo i vlagoperenosa v pochve, Sb. trudov po agrozike, 1969, 4154.
[8] A.V. Bitsadze, A.A. Samarskii, On some simple generalizations of linear elliptic boundary problems, Sov. Math., Dokl., 10 (1969), 398400. Zbl 0187.35501 · Zbl 0187.35501
[9] N.I. Ionkin, The solution of a certain boundary value problem of the theory of heat conduction with a nonclassical boundary condition, Dier. Uravn., 13:2 (1977), 294304. Zbl 0349.35040 · Zbl 0403.35043
[10] N.I. Ionkin, Uniform convergence of the dierence scheme for one nonstationary nonlocal boundary-value problem, Comput. Math. Model., 2:3 (1991), 223228. Zbl 0799.65095 · Zbl 0799.65095
[11] V.A. Il’in, E.I. Moiseev, A nonlocal boundary value problem for the Sturm-Liouville operator in the dierential and dierence treatments, Sov. Math., Dokl., 34 (1987), 507511. Zbl 0643.34016
[12] N.I. Ionkin, E.I. Moiseev, A problem for a heat equation with two-point boundary conditions, Dier. Uravn., 15:7 (1979), 12841295. Zbl 0415.35032 · Zbl 0431.35046
[13] D.G. Gordeziani, On the methods of solution for one class of non-local boundary value problems, Izdatel’stvo Tbilisskogo Universiteta, Tbilisi, 1981. Zbl 0464.35037 · Zbl 0464.35037
[14] A.M. Nakhushev, A nonlocal problem and the Goursat problem for a loaded equation of hyperbolic type, and their applications to the prediction of ground moisture, Sov. Math., Dokl., 19 (1978), 12431247. Zbl 0433.35043 · Zbl 0433.35043
[15] A.P. Soldatov, M.Kh. Shkhanukov, Boundary value problems with general nonlocal Samarskij condition for pseudoparabolic equations of higher order, Sov. Math., Dokl., 36:3 (1988), 507 511. Zbl 0701.35092 · Zbl 0701.35092
[16] A.K. Bazzaev, D.K. Gutnova, M.Kh. Shkhanukov-Lashev, Locally one-dimensional scheme for parabolic equation with a nonlocal condition, Zh. Vychisl. Mat. Mat. Fiz., 52:6 (2012), 10481057. Zbl 1274.35032 · Zbl 1274.35032
[17] M.Kh. Beshtokov, Dierence method for solving a nonlocal boundary value problem for a degenerating third-order pseudo-parabolic equation with variable coecients, Comput. Math. Math. Phys., 56:10 (2016), 17631777. Zbl 1358.65054 · Zbl 1358.65054
[18] M.Kh. Beshtokov, Dierential and dierence boundary value problem for loaded third-order pseudo-parabolic dierential equations and dierence methods for their numerical solution, Comput. Math. Math. Phys., 57:12 (2017), 19731993. Zbl 1393.65007
[19] M.Kh. Beshtokov, V.Z. Kanchukoyev, F.A. Erzhibova, A boundary value problem for a degenerate moisture transfer equation with a condition of the third kind, Vladikavkaz. Mat. Zh., 19:4 (2017), 13-26. Zbl 1452.65150 · Zbl 1452.65150
[20] M.Kh. Beshtokov, Riemann method for solving non-local boundary value problems for the third order pseudoparabolic equations, Vestn. Samar. Gos. Tekh. Univ., Ser. Fiz.-Mat. Nauki, 2013:4(33) (2013), 1524. Zbl 1413.35270 · Zbl 1282.65099
[21] M.Kh. Beshtokov, Finite dierence method for a nonlocal boundary value problem for a third-order pseudoparabolic equation, Dier. Equ., 49:9 (2013), 11341141. Zbl 1282.65099 · Zbl 1282.65099
[22] M.KH. Beshtokov, A numerical method for solving one nonlocal boundary value problem for a third-order hyperbolic equation, Comput. Math. Math. Phys., 54:9 (2014), 14411458. Zbl 1327.65157 · Zbl 1327.65157
[23] A.K. Bazzaev, M.Kh. Shkhanukov-Lashev, Locally one-dimensional scheme for fractional diusion equations with Robin boundary conditions, Comput. Math. Math. Phys., 50:7 (2010), 11411149. Zbl 1224.65198
[24] A.K. Bazzaev, The third boundary problem for general parabolic dierential equation of fractional order in multidimensional eld, Vestn. VGU, Ser. Fiz. Mat., 2010:2 (2010), 514. Zbl 1325.35256 · Zbl 1325.35256
[25] A.K. Bazzaev, A.V. Mambetova, M.Kh. Shkhanukov-Lashev, Locally one-dimensional scheme for the heat equation of fractional order with concentrated heat capacity, Zh. Vychisl. Mat. Mat. Fiz., 52:9 (2012), 16561665. Zbl 1274.35154 · Zbl 1274.35154
[26] A.K. Bazzaev, M.Kh. Shkhanukov-Lashev, Locally one-dimensional scheme for fractional diusion equations with Robin boundary conditions, Comput. Math. Math. Phys., 50:7 (2010), 1141-1149. Zbl 1224.65198
[27] D.K. Faddeev, V.N. Faddeeva, Numerical methods of linear algebra, Fizmatgiz, Moscow, 1960. Zbl 0094.11005 · Zbl 0094.11005
[28] O.A. Ladyzhenskaya, The boundary value problems of mathematical physics, Springer-Verlag, New York etc., 1985. Zbl 0588.35003 · Zbl 0164.12501
[29] A.A. Samarskii The Theory of dierence schemes, Marcel Dekker, New York, 2001. Zbl 0971.65076 · Zbl 0971.65076
[30] A.A. Samarskii, A.V. Gulin, Stability of nite dierence schemes, Nauka, Moscow, 1973. Zbl 0304
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