Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-22T13:04:35.997Z Has data issue: false hasContentIssue false
  • English
  • Français

The Crank-Nicolson Hermite Cubic Orthogonal Spline Collocation Method for the Heat Equation with Nonlocal Boundary Conditions

Published online by Cambridge University Press:  03 June 2015

B. Bialecki ,
G. Fairweather  and
J.C. López-Marcos
B. Bialecki*
Affiliation:
Department of Applied Mathematics and Statistics, Colorado School of Mines, Golden, CO 80401, USA
G. Fairweather*
Affiliation:
Mathematical Reviews, American Mathematical Society, 416 Fourth Street, Ann Arbor, MI 48103, USA
J.C. López-Marcos*
Affiliation:
Departamento de Matemática Aplicada, Universidad de Valladolid, Valladolid, Spain
*
Corresponding author. Email: [email protected]
Email: [email protected]
Email: [email protected]
Get access

Abstract

We formulate and analyze the Crank-Nicolson Hermite cubic orthogonal spline collocation method for the solution of the heat equation in one space variable with nonlocal boundary conditions involving integrals of the unknown solution over the spatial interval. Using an extension of the analysis of Douglas and Dupont [23] for Dirichlet boundary conditions, we derive optimal order error estimates in the discrete maximum norm in time and the continuous maximum norm in space. We discuss the solution of the linear system arising at each time level via the capacitance matrix technique and the package COLROW for solving almost block diagonal linear systems. We present numerical examples that confirm the theoretical global error estimates and exhibit superconvergence phenomena.

Keywords

65N3565N1265N15Heat equationnonlocal boundary conditionsorthogonal spline collocationHermite cubic splinesconvergence analysissuperconvergence
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 5 , Special Issue 4 , August 2013 , pp. 442 - 460
Copyright
Copyright © Global-Science Press 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Ang, W.T., A method of solution for the one-dimensional heat equation subject to nonlocal conditions, Southeast Asian Bull. Math., 26 (2002), pp. 185–191.Google Scholar
[2]Ang, W. T., Numerical solution of a non-classical parabolic problem: an integro-differential approach, Appl. Math. Comput., 175 (2006), pp. 969–979.Google Scholar
[3]Bahuguna, D., Abbas, S. and Shukla, R. K., Laplace transform method for one-dimensional heat and wave equations with nonlocal conditions, Int. J. Appl. Math. Stat., 16 (2010), pp. 96–100.Google Scholar
[4]Bahuguna, D., Ujlayan, A. and Pandey, D. N., ADM series solution to a nonlocal one-dimensional heat equation, Int. Math. Forum, 4 (2009), pp. 581–585.Google Scholar
[5]Borovykh, N., Stability in the numerical solution of the heat equation with nonlocal boundary conditions, Appl. Numer. Math., 42 (2003), pp. 17–27.CrossRefGoogle Scholar
[6]Boor, C.De and Swartz, B., Collocation at Gaussian points, SIAM J. Numer. Anal., 10 (1973), pp. 582–606.CrossRefGoogle Scholar
[7]Boutayeb, A. and Chetouani, A., Global extrapolations of numerical methods for solving a parabolic problem with non local boundary conditions, Int. J. Comput. Math., 80 (2003), pp. 789–797.CrossRefGoogle Scholar
[8]Boutayeb, A. and Chetouani, A., Galerkin approximation for a semi linear parabolic problem with nonlocal boundary conditions, Proyecciones, 23 (2004), pp. 31–49.Google Scholar
[9]Boutayeb, A. and Chetouani, A., A numerical comparison of different methods applied to the solution of problems with non local boundary conditions, Appl. Math. Sci. (Ruse), 1 (2007), pp. 2173–2185.Google Scholar
[10]Buzbee, B. L., Dorr, F. W., George, J. A. and Golub, G. H., The direct solution of the discrete Poisson equation on irregular regions, SIAM J. Numer. Anal., 8 (1971), pp. 722–736.CrossRefGoogle Scholar
[11]Çiegis, R. and Tumanova, N., Numerical solution of parabolic problems with nonlocal boundary conditions, Numer. Funct. Anal. Optim., 31 (2010), pp. 1318–1329.Google Scholar
[12]Conte, S. D. and De Boor, C., Elementary Numerical Analysis, An Algorithmic Approach: Third Edition, McGraw-Hill Book Company, New York, 1980.Google Scholar
[13]Crandall, S. H., An optimum implicit recurrence formula for the heat conduction problem, Quart. Appl. Math., 13 (1955), pp. 318–320.Google Scholar
[14]Day, W. A., Extensions of a property of the heat equation to linear thermoelasticity and other theories, Quart. Appl. Math., 40 (1982), pp. 319–330.Google Scholar
[15]Day, W. A., A decreasing property of solutions of parabolic equations with applications to thermoelasticity, Quart. Appl. Math., 41 (1983), pp. 468–475.Google Scholar
[16]Dehghan, M., The use of the Adomian decomposition method for solving one-dimensional parabolic equation with non-local boundary specifications, Int. J. Comput. Math., 81 (2004), pp. 25–34.CrossRefGoogle Scholar
[17]Dehghan, M., Efficient techniques for the second-order parabolic equation subject to nonlocal specifications, Appl. Numer. Math., 52 (2005), pp. 39–62.Google Scholar
[18]Dehghan, M. and Tatari, M., Use of radial basis functions for solving the second-order parabolic equation with nonlocal boundary conditions, Numer. Methods Partial Differential Equations, 24 (2008), pp. 924–938.CrossRefGoogle Scholar
[19]Diaz, J. C., Fairweather, G. and Keast, P., FORTRAN packages for solving certain almost block diagonal linear systems by modified alternate row and column elimination, ACM Trans. Math. Software, 9 (1983), pp. 358–375.Google Scholar
[20]Diaz, J. C., Fairweather, G. and Keast, P., Algorithm 603 COLROW and ARCECO: FORTRAN packages for solving certain almost block diagonal linear systems by modified alternate row and column elimination, ACM Trans. Math. Software, 9 (1983), pp. 376–380.Google Scholar
[21]Douglas, J. Jr., The solution of the diffusion equation by a high order correct difference equation, J. Math. Phys., 35 (1956), pp. 145–151.CrossRefGoogle Scholar
[22]Douglas, J. Jr., and Dupont, T., A finite element collocation method for quasilinear parabolic equations, Math. Comput., 27(1973), pp. 17–28.Google Scholar
[23]Douglas, J. Jr., and Dupont, T., Collocation methods for parabolic equations in a single space variable, Lecture Notes in Mathematics, 385, Springer-Verlag, New York, 1974.CrossRefGoogle Scholar
[24]Ekolin, G., Finite difference methods for a nonlocal boundary value problem for the heat equation, BIT, 31 (1991), pp. 245–261.Google Scholar
[25]Fairweather, G. and López-marcos, J.C., Galerkin methods for a semilinear parabolic problem with nonlocal boundary conditions, Adv. Comput. Math., 6 (1996), pp. 243–262.Google Scholar
[26]Fairweather, G. and Gladwell, I., Algorithms for almost block diagonal linear systems, SIAM Rev., 46 (2004), pp. 49–58.Google Scholar
[27]Fairweather, G. and Saylor, R. D., The formulation and numerical solution of certain non-classical initial-boundary value problems, SIAM J. Stat. Comput., 12 (1991), pp. 127–144.Google Scholar
[28]Friedman, A., Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions, Quart. Appl. Math., 44 (1986), pp. 401–407.Google Scholar
[29]Golbabai, A. and Javidi, M., A numerical solution for non-classical parabolic problem based on Chebyshev spectral collocation method, Appl. Math. Comput., 190 (2007), pp. 179–185.Google Scholar
[30]ˇjeseviŽiute, and Sapagovas, M., On the stability of finite-difference schemes for parabolic equations subject to integral conditions with applications to thermoelasticity, Comput. Methods Appl. Math., 8 (2008), pp. 360–373.Google Scholar
[31]Kawohl, B., Remarks on a paper by W. A. Day on a maximum principle under nonlocal boundary conditions, Quart. Appl. Math., 44 (1987), pp. 751–752.Google Scholar
[32]Keller, H. B., A new difference scheme for parabolic problems, Numerical Solution of Partial Differential Equations, II (SYNSPADE 1970), Proc. Sympos., Univ. of Maryland, College Park, MD, 1970, Hubbard, B., ed., Academic Press, New York, 1971, pp. 327–350.Google Scholar
[33]Lakestani, M., Solution of a parabolic differential equation with nonlocal boundary conditions using B-spline functions, Int. J. Math. Comput., 11 (2011), pp. 92–98.Google Scholar
[34]Lin, Y., Xu, S. and Yin, H.-C., Finite difference approximation for a class of non-local parabolic equations, Internat. J. Math. Math. Sci., 20 (1997), pp. 147–163.Google Scholar
[35]Lin, Y. and Zhou, Y., Solving the reaction-diffusion equations with nonlocal boundary conditions based on reproducing kernel space, Numer. Methods Partial Differential Equations, 25 (2009), pp. 1468–1481.Google Scholar
[36]Liu, Y., Numerical solution of the heat equation with nonlocal boundary conditions, J. Comput. Appl. Math., 110 (1999), pp. 115–127.Google Scholar
[37]Liu, J. and Sun, Z., Finite difference method for reaction-diffusion equation with nonlocal boundary conditions, Numer. Math. J. Chinese Univ., 16 (2007), pp. 97–111.Google Scholar
[38]Martìn-Vaquero, J., Two-level fourth-order explicit schemes for diffusion equations subject to boundary integral specifications, Chaos Solitons Fractals, 42 (2009), pp. 2364–2372.Google Scholar
[39]Martìn-Vaquero, J., Queiruga-Dios, A. and Encinas, A. H., Numerical algorithms for diffusion-reaction problems with non-classical conditions, Appl. Math. Comput., 218 (2012), pp. 5487–5495.Google Scholar
[40]Martìn-Vaquero, J. and Vigo-Aguiar, J., A note on efficient techniques for the second-order parabolic equation subject to non-local conditions, Appl. Numer. Math., 59 (2009), pp. 1258–1264.Google Scholar
[41]Martìn-Vaquero, J. and Vigo-Aguiar, J., On the numerical solution of the heat conduction equations subject to nonlocal conditions, Appl. Numer. Math., 59 (2009), pp. 2507–2514.Google Scholar
[42]Mu, L. and Du, H., The solution of a parabolic differential equation with non-local boundary conditions in the reproducing kernel space, Appl. Math. Comput., 202 (2008), pp. 708–714.Google Scholar
[43]Pan, Z., On the forward-Euler and backward-Euler difference scheme for a nonlocal boundary value problem for heat equation, Nanjing Univ. J. Math. Biquarterly, 21 (2004), pp. 67–76.Google Scholar
[44]Pao, C. V., Numerical solution of reaction-diffusion equations with nonlocal boundary conditions, J. Comput. Appl. Math., 136 (2001), pp. 227–243.Google Scholar
[45]Sapagovas, M., On the stability of a finite difference scheme for nonlocal parabolic boundary-value problems, Lith. Math. J., 48 (2008), pp. 339–356.Google Scholar
[46]Sajaviĉius, S., Stability of the weighted splitting finite-difference scheme for two-dimensional parabolic equation with two nonlocal integral conditions, Comput. Math. Appl., 64 (2012), pp. 3485–3499.Google Scholar
[47]Sun, Z.-Z., A high-order difference scheme for a nonlocal boundary-value problem for the heat equation, Comput. Methods Appl. Math., 4 (2001), pp. 398–414.Google Scholar
[48]Wang, S. and Lin, Y., A numerical method for the diffusion equation with nonlocal boundary specification, Int. J. Engng. Sci., 28 (1990), pp. 543–546.Google Scholar
[49]Yousefi, S.A., Behroozifar, M. and Dehghan, M., The operational matrices of Bernstein polynomials for solving the parabolic equation subject to the specification of the mass, J. Comput. Appl. Math., 235 (2011), pp. 5272–5283.Google Scholar