Abstract
In this paper, a compact difference scheme is established for the heat equations with multi-point boundary value conditions. The truncation error of the difference scheme is \(O(\tau ^2+h^4),\) where \(\tau\) and h are the temporal step size and the spatial step size. A prior estimate of the difference solution in a weighted norm is obtained. The unique solvability, stability and convergence of the difference scheme are proved by the energy method. The theoretical statements for the solution of the difference scheme are supported by numerical examples.
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References
Alikhanov, A.A.: On the stability and convergence of nonlocal difference schemes. Differ. Equ. 46(7), 949–961 (2010)
Ashyralyev, A., Aggez, N.: A Note on the difference schemes of the nonlocal boundary value problems for hyperbolic equations. Numerical Functional Analysis and Optimization 25(5–6), 439–462 (2004)
Ashyralyev, A., Gercek, O.: Nonlocal boundary value problems for elliptic-parabolic differential and difference equations. Discrete Dyn. Nat. Soc. 4, 138–144 (2008)
Ashyralyev, A., Gercek, O.: Finite difference method for multipoint nonlocal elliptic-parabolic problems. Comput. Math. Appl. 60(7), 2043–2052 (2010)
Ashyralyev, A., Yurtsever, A.: On a nonlocal boundary value problem for semilinear hyperbolic-parabolic equations. Nonlinear Anal. Theory Methods Appl. 47(5), 3585–3592 (2001)
Gao, G.H., Sun, Z.Z.: Compact difference schemes for heat equation with Neumann boundary conditions (II). Numer. Methods Partial Differ. Equ. 29(5), 1459–1486 (2013)
Gordeziani, D., Avalishvili, G.: Investigation of the nonlocal initial boundary value problems for some hyperbolic equations. Hiroshima Math. J. 31(3), 345–366 (2001)
Gulin, A.V., Morozova, V.A.: On a family of nonlocal difference schemes. Differ. Equ. 45(7), 1020–1033 (2009)
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)
Gushchin, A.K., Mikhailov, V.P.: On solvability of nonlocal problems for a second-order elliptic equation. Russ. Acad. Sci. Sb. Math. 81(1), 101–136 (1995)
Martin-Vaquero, J., Vigo-Aguiar, J.: A note on efficient techniques for the second-order parabolic equation subject to non-local conditions. Appl. Numer. Math. 59(6), 1258–1264 (2009)
Martin-Vaquero, J., Vigo-Aguiar, J.: On the numerical solution of the heat conduction equations subject to nonlocal conditions. Appl. Numer. Math. 59(10), 2507–2514 (2009)
Sun, Z.Z.: A high-order difference scheme for a nonlocal boundary-value problem for the heat equation. Comput. Methods Appl. Math. 1(4), 398–414 (2001)
Sun, Z.Z.: Compact difference schemes for heat equation with Neumann boundary conditions. Numer. Methods Partial Differ. Equ. 29, 1459–1486 (2013)
Wang, Y.: Solutions to nonlinear elliptic equations with a nonlocal boundary condition. Electron. J. Differ. Equ. 05, 227–262 (2002)
Yildirim, O., Uzun, M.: On the numerical solutions of high order stable difference schemes for the hyperbolic multipoint nonlocal boundary value problems. Appl. Math. Comput. 254, 210–218 (2015)
Zikirov, O.S.: On boundary-value problem for hyperbolic-type equation of the third order. Lith. Math. J. 47(4), 484–495 (2007)
Acknowledgements
The research is supported by the National Natural Science Foundation of China (No. 11671081) and the Fundamental Research Funds for the Central Universities (No. 242017K41044).
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Wang, X., Sun, Z. A Compact Difference Scheme for Multi-point Boundary Value Problems of Heat Equations. Commun. Appl. Math. Comput. 1, 545–563 (2019). https://doi.org/10.1007/s42967-019-00025-w
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DOI: https://doi.org/10.1007/s42967-019-00025-w