A finite difference method for a non-local boundary value problem for two-dimensional heat equation
A second-order finite difference scheme is given for the numerical solution of a two-dimensional non-local boundary value problem for heat equation. Using a suitable transformation, the solution of this problem is equivalent to the solution of two other problems. The first problem which is a one-dim...
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| Veröffentlicht in: | Applied mathematics and computation Jg. 112; H. 1; S. 133 - 142 |
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| Sprache: | Englisch |
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01.06.2000
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| Abstract | A second-order finite difference scheme is given for the numerical solution of a two-dimensional non-local boundary value problem for heat equation. Using a suitable transformation, the solution of this problem is equivalent to the solution of two other problems. The first problem which is a one-dimensional non-local boundary value problem giving the value of
μ through using a second-order finite difference scheme. Using this result, the second problem will be changed to a classical two-dimensional problem with Nuemann's boundary condition which will be solved numerically. The stability properties and truncation error of the new method are discussed and the results of numerical experiments are presented. |
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| AbstractList | A second-order finite difference scheme is given for the numerical solution of a two-dimensional non-local boundary value problem for heat equation. Using a suitable transformation, the solution of this problem is equivalent to the solution of two other problems. The first problem which is a one-dimensional non-local boundary value problem giving the value of
μ through using a second-order finite difference scheme. Using this result, the second problem will be changed to a classical two-dimensional problem with Nuemann's boundary condition which will be solved numerically. The stability properties and truncation error of the new method are discussed and the results of numerical experiments are presented. |
| Author | Dehghan, Mehdi |
| Author_xml | – sequence: 1 givenname: Mehdi surname: Dehghan fullname: Dehghan, Mehdi organization: Faculty of Mathematics and Computer Science, Amirkabir University of Technology, Hafez Avenue, No. 424, Tehran 15914, Iran |
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| Cites_doi | 10.1080/00036819308840181 10.1090/qam/160437 10.1090/qam/678203 10.2307/2007712 10.1016/0020-7225(93)90010-R 10.1016/0022-247X(86)90012-0 10.1090/qam/963580 10.1016/0020-7225(90)90056-O 10.1088/0266-5611/5/4/013 |
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| Keywords | The range of stability Partial differential equations Non-local boundary value problems Heat conduction equation Numerical integration procedures Finite difference techniques Nuemann's boundary conditions Modified equivalent analysis Neumann problem Numerical integration Boundary value problem Stability Heat equation Numerical solution Diffusion equation Two dimensional model Partial differential equation Finite difference method |
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| References_xml | – volume: 5 start-page: 631 year: 1989 end-page: 640 ident: BIB6 article-title: A finite difference solution to an inverse problem determining a control function in a parabolic partial differential equations publication-title: Inverse Problems – volume: 46 start-page: 431 year: 1988 end-page: 449 ident: BIB8 article-title: A reaction-diffusion system arising in modeling man-environment diseases publication-title: Q. Appl. Math. – reference: R.F. Warming, B.J. Hyett, The modified equation approach to the stability and accuracy analysis of finite difference methods, Journal of Computational Physics – reference: L. Lapidus, G.F. Pinder, Numerical solution of partial differential equations in science and engineering. Wiley, New York, 1982 – volume: 28 start-page: 543 year: 1990 end-page: 546 ident: BIB5 article-title: A numerical method for the diffusion equation with non-local boundary specifications publication-title: Int. J. Engng. Sci. – volume: 115 start-page: 517 year: 1986 end-page: 529 ident: BIB3 article-title: Diffusion subject to specification of mass publication-title: J. Math. Anal. Appl. – reference: A.R. Mitchell, D.F. Griffiths, The finite difference methods in partial differential equations, Wiley, New York, 1980 – reference: C.F. Gerald, Applied Numerical Analysis, 5th ed., Addison–Wesley, Reading, MA, 1995 – reference: J.R. Cannon, J. van der Hoek, Implicit finite difference scheme for the diffusion of mass in porous media, in: B.J. Noye (Ed.), Numerical Solutions of Partial Differential Equations, North-Holland, Amsterdam, 1982, pp. 527–539 – volume: 21 start-page: 155 year: 1963 end-page: 160 ident: BIB4 article-title: The solution of the heat equation subject to the specification of energy publication-title: Q. Appl. Math. – volume: 50 start-page: 1 year: 1993 end-page: 19 ident: BIB1 article-title: The solution of the diffusion equation in two-space variables subject to the specification of mass publication-title: J. Appl. Anal. – reference: J.R. Cannon, A.L. Matheson, A numerical procedure for diffusion subject to the specification of mass, Int. J. Engng. Sci. 31 (1993) 347 – volume: 130 start-page: 35 year: 1990 end-page: 38 ident: BIB11 article-title: The numerical method for the conduction subject to moving boundary energy specification publication-title: Numer. Heat Transfer – volume: 40 start-page: 319 year: 1982 end-page: 330 ident: BIB7 article-title: Existence of a property of solutions of the heat equation to linear thermoelasticity and other theories publication-title: Q. Appl. Math. – volume: 50 start-page: 1 year: 1993 ident: 10.1016/S0096-3003(99)00055-7_BIB1 article-title: The solution of the diffusion equation in two-space variables subject to the specification of mass publication-title: J. Appl. Anal. doi: 10.1080/00036819308840181 – volume: 21 start-page: 155 year: 1963 ident: 10.1016/S0096-3003(99)00055-7_BIB4 article-title: The solution of the heat equation subject to the specification of energy publication-title: Q. Appl. Math. doi: 10.1090/qam/160437 – volume: 40 start-page: 319 year: 1982 ident: 10.1016/S0096-3003(99)00055-7_BIB7 article-title: Existence of a property of solutions of the heat equation to linear thermoelasticity and other theories publication-title: Q. Appl. Math. doi: 10.1090/qam/678203 – ident: 10.1016/S0096-3003(99)00055-7_BIB12 doi: 10.2307/2007712 – volume: 130 start-page: 35 year: 1990 ident: 10.1016/S0096-3003(99)00055-7_BIB11 article-title: The numerical method for the conduction subject to moving boundary energy specification publication-title: Numer. Heat Transfer – ident: 10.1016/S0096-3003(99)00055-7_BIB13 doi: 10.1016/0020-7225(93)90010-R – ident: 10.1016/S0096-3003(99)00055-7_BIB9 – volume: 115 start-page: 517 year: 1986 ident: 10.1016/S0096-3003(99)00055-7_BIB3 article-title: Diffusion subject to specification of mass publication-title: J. Math. Anal. Appl. doi: 10.1016/0022-247X(86)90012-0 – ident: 10.1016/S0096-3003(99)00055-7_BIB2 – volume: 46 start-page: 431 year: 1988 ident: 10.1016/S0096-3003(99)00055-7_BIB8 article-title: A reaction-diffusion system arising in modeling man-environment diseases publication-title: Q. Appl. Math. doi: 10.1090/qam/963580 – volume: 28 start-page: 543 issue: 6 year: 1990 ident: 10.1016/S0096-3003(99)00055-7_BIB5 article-title: A numerical method for the diffusion equation with non-local boundary specifications publication-title: Int. J. Engng. Sci. doi: 10.1016/0020-7225(90)90056-O – ident: 10.1016/S0096-3003(99)00055-7_BIB14 – volume: 5 start-page: 631 year: 1989 ident: 10.1016/S0096-3003(99)00055-7_BIB6 article-title: A finite difference solution to an inverse problem determining a control function in a parabolic partial differential equations publication-title: Inverse Problems doi: 10.1088/0266-5611/5/4/013 – ident: 10.1016/S0096-3003(99)00055-7_BIB10 |
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| SubjectTerms | Chemistry Exact sciences and technology Finite difference techniques General and physical chemistry Global analysis, analysis on manifolds Heat conduction equation Mathematical analysis Mathematics Modified equivalent analysis Non-local boundary value problems Nuemann's boundary conditions Numerical analysis Numerical analysis in abstract spaces Numerical analysis. Scientific computation Numerical integration procedures Partial differential equations Sciences and techniques of general use The range of stability Theory of reactions, general kinetics Theory of reactions, general kinetics. Catalysis. Nomenclature, chemical documentation, computer chemistry Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds |
| Title | A finite difference method for a non-local boundary value problem for two-dimensional heat equation |
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