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Finite difference method for multipoint nonlocal elliptic-parabolic problems. (English) Zbl 1205.65230
Summary: A finite difference method for solving the multipoint elliptic-parabolic partial differential equation with a nonlocal boundary condition is considered. Stable difference schemes accurate to first and second orders for this problem are presented. Stability, almost coercive stability and coercive stability for the solution of these difference schemes are obtained. The theoretical statements for the solution of these difference schemes are supported by numerical examples.

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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[1] Agarwal, R.P.; Bohner, M.; Shakhmurov, V.B., Maximal regular boundary value problems in Banach-valued weighted space, Bound. value probl., 1, 9-42, (2005) · Zbl 1081.35129
[2] Ashyralyev, A.; Aggez, N., A note on the difference schemes of the nonlocal boundary value problems for hyperbolic equations, Numer. funct. anal. optim., 25, 439-462, (2004) · Zbl 1065.35021
[3] Ashyralyev, A.; Sirma, A., Nonlocal boundary value problems for the schrodinger equation, Comput. math. appl., 55, 3, 392-407, (2008) · Zbl 1155.65368
[4] Ashyralyev, A., High-accuracy stable difference schemes for well-posed nonlocal boundary value problems, Oper. theory adv. appl., 191, 229-252, (2009) · Zbl 1179.65056
[5] Ashyralyev, A.; Dural, A.; Sozen, Y., Multipoint nonlocal boundary value problems for reverse parabolic equations: well-posedness, Vestnik odessa nat. univ. math. mech., 13, 1-12, (2009)
[6] Ashyralyev, A.; Yildirim, O., On multipoint nonlocal boundary value problems for hyperbolic differential and difference equations, Taiwanese J. math., 14, 1, 165-194, (2010) · Zbl 1201.65128
[7] Bouziani, A., On a class of parabolic equations with a nonlocal boundary condition, Acad. roy. belg. bull. cl. sci., 10, 61-77, (1999) · Zbl 1194.35200
[8] Ewing, R.E.; Lazarov, R.D.; Lin, Y., Finite volume element approximations nonlocal reactive flows in porous media, Numer. methods partial differential equations, 16, 285-311, (2000) · Zbl 0961.76050
[9] Gulin, A.V.; Ionkin, N.I.; Morozova, V.A., Stability of a nonlocal two-dimensional finite-difference problem, Differ. equ., 37, 7, 970-978, (2001) · Zbl 1004.65091
[10] Gordeziani, N.; Natalini, P.; Ricci, P.E., Finite-difference methods for solution of nonlocal boundary value problems, Comput. math. appl., 50, 1333-1344, (2005) · Zbl 1087.65098
[11] Jangveladze, T.A.; Lobjanidze, G.B., On a variational statement of a nonlocal boundary value problem for a fourth-order ordinary differential equation, Differ. equ., 45, 3, 335-343, (2009) · Zbl 1183.34020
[12] Martín-Vaquero, J.; Vigo-Aguiar, J., On the numerical solution of the heat conduction equations subject to nonlocal conditions, Appl. numer. math., 59, 2507-2514, (2009) · Zbl 1167.65423
[13] Ratyni, A.K., On the solvability of the first nonlocal boundary value problem for an elliptic equation, Differ. equ., 45, 6, 862-872, (2009) · Zbl 1182.35097
[14] Samarskii, A.A.; Bitsadze, A.V., Some elementary generalizations of linear elliptic boundary value problems, Dokl. akad. nauk SSSR, 185, 4, 739-740, (1969) · Zbl 0187.35501
[15] Sapagovas, M.P., On the stability of finite difference scheme for nonlocal parabolic boundary value problems, Lith. math. J., 48, 3, 339-356, (2008) · Zbl 1206.65219
[16] Shakhmurov, V.B., Maximal \(B\)-regular boundary value problems with parameters, J. math. anal. appl., 320, 1, 1-19, (2006) · Zbl 1160.35392
[17] A. Ashyralyev, H. Soltanov, On elliptic – parabolic equations in a Hilbert space, in: Proc. of the IMM and CS of Turkmenistan, Ashgabat, Turkmenistan, 1995, pp. 101-104.
[18] Ashyralyev, A., A note on the nonlocal boundary value problem for elliptic – parabolic equations, Nonlinear stud., 13, 4, 327-333, (2006)
[19] Ashyralyev, A.; Ozdemir, Y., On nonlocal boundary value problems for hyperbolic – parabolic equations, Taiwanese J. math., 11, 3, 1077-1091, (2007)
[20] Ashyralyev, A.; Gercek, O., Nonlocal boundary value problems for elliptic – parabolic differential and difference equations, Discrete dyn. nat. soc., 1-16, (2008) · Zbl 1168.35397
[21] Ashyralyev, A.; Gercek, O., On second order of accuracy difference scheme of the approximate solution of nonlocal elliptic – parabolic problems, Abstr. appl. anal., 1-17, (2010) · Zbl 1198.65120
[22] D. Bazarov, H. Soltanov, Some Local and Nonlocal Boundary Value Problems for Equations of Mixed and Mixed-Composite Types, Ylim, Ashgabat, Turkmenistan, 1995.
[23] Diaz, J.; Lerena, M.; Padial, J.; Rakotoson, J., An elliptic – parabolic equation with a nonlocal term for the transient regime of a plasma in a stellarator, J. differential equations, 198, 2, 321-355, (2004) · Zbl 1050.35151
[24] Dzhuraev, T.D., Boundary value problems for equations of mixed and mixed-composite types, (1979), Fan Tashkent, Uzbekistan · Zbl 0487.35068
[25] Hilhorst, D.; Hulshof, J., An elliptic – parabolic problem in combustion theory: convergence to travelling waves, Nonlinear anal., 17, 6, 519-546, (1991) · Zbl 0761.35114
[26] Glazatov, S.N., Nonlocal boundary value problems for linear and nonlinear equations of variable type, Sobolev inst. math. SB RAS, 46, 26, (1998) · Zbl 0917.35080
[27] Kröner, D.; Rodrigues, J.F., Global behaviour for bounded solutions of a porous media equation of elliptic – parabolic type, J. math. pures appl., 64, 105-120, (1985) · Zbl 0549.35064
[28] Salakhitdinov, M.S., Equations of mixed-composite type, (1974), Fan Tashkent, Uzbekistan · Zbl 0235.35059
[29] Vragov, V.N., ()
[30] Sobolevskii, P.E., The coercive solvability of difference equations, Dokl. akad. nauk SSSR, 201, 5, 1063-1066, (1971) · Zbl 0246.39002
[31] P.E. Sobolevskii, On the stability and convergence of the Crank-Nicolson scheme, in: Variational-Difference Methods in Mathematical Physics, Vychisl. Tsentr Sibirsk. Otdel. Akademii Nauk SSSR, Novosibirsk, 1974, pp.146-151.
[32] Sobolevskii, P.E., Difference methods for the approximate solution of differential equations, (1975), Voronezh State University Press Voronezh, Russia · Zbl 0333.47010
[33] Samarskii, A.A.; Nikolaev, E.S., ()
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