Abstract
In this paper, the problem of solving the parabolic partial differential equations subject to given initial and nonlocal boundary conditions is considered. We change the problem to a system of Volterra integral equations of convolution type. By using Sinc-collocation method, the resulting integral equations are replaced by a system of linear algebraic equations. The convergence analysis is included, and it is shown that the error in the approximate solution is bounded in the infinity norm by the condition number of the coefficient matrix multiplied by a factor that decays exponentially with the size of the system. Some examples are considered to illustrate the ability of this method.
Similar content being viewed by others
References
Stenger, F.: Numerical methods based on Sinc and analytic functions. Springer, New York (1993)
Amore, P.: A variational Sinc Collocation method for Strong-coupling Problems. J. Phys. A Math. Gen. 39(22), L349–L355 (2006)
Lund, J., Vogel, C.: A Fully–Galerkin method for the solution of an inverse problem in a parabolic partial differential equation. Inverse Probl. 6, 205–217 (1990)
Smith, R., Bowers, K.: A Sinc–Galerkin estimation of diffusivity in parabolic problems. Inverse Probl. 9, 113–135 (1993)
Shidfar, A., Zolfaghari, R., Damirchi, J.: Application of Sinc-collocation method for solving an inverse problem. J. Comput. Appl. Math. 233, 545–554 (2009)
Shidfar, A., Zolfaghari, R.: Determination of an unknown function in a parabolic inverse problem by Sinc-collocation method. Numer. Methods Partial Differ. Equ. 27(6), 1584–1598 (2011)
Lund, J., Bowers, K.: Sinc Methods for Quadrature and Differential Equations. SIAM, Philadelphia (1992)
Borovykh, N.: Stability in the numerical solution of the heat equation with nonlocal boundary conditions. Appl. Numer. Math. 42, 17–27 (2002)
Bouziani, A.: On the quasi static flexure of a thermoelastic rod. Commun. Appl. Anal. 6(4), 549–568 (2002)
Cannon, J.R., Lin Y, Wang, S.: Determination of a control parameter in a parabolic differential equation. Austral. Math. Sco. 33, 149–163 (1991)
Cannon, J.R.: The one-dimensional heat equation. Encyclopedia of Mathematics and Its Applications, vol. 23. Addison-Wesley, Reading, MA (1984)
Day, W.A.: Extension of a property of the heat equation to linear thermoelasticity and other theories. Quart. Appl. Math. 40, 319–330 (1982)
Day, W.A.: A decreasing property of solutions of parabolic equations with applications to thermoelasticity. Quart. Appl. Math. 41, 468–475 (1983)
Friedman, A.: Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions. Quart. Appl. Math. 44, 401–407 (1986)
Hokkanen, V., Morosanu, G.: Functional Methods in Differential Equations. Chapman and Hall/CRC (2002)
Lin, Y.: Numerical solution of the heat equation with nonlocal boundary conditions. J. Comput. Appl. Math. 110, 115–127 (1999)
Dehghan, M.: Numerical solution of a parabolic equation with non-local boundary specifications. Appl. Math. Comput. 145, 185–194 (2003)
Dehghan, M.: Efficient techniques for the second-order parabolic equation subject to nonlocal specifications. Appl. Numer. Math. 52, 39–62 (2005)
Dehghan, M.: A computational study of the one-dimensional parabolic equation subject to nonclassical boundary specifications. Numer. Methods Partial Differ. Equ. 22, 220–257 (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zolfaghari, R., Shidfar, A. Solving a parabolic PDE with nonlocal boundary conditions using the Sinc method. Numer Algor 62, 411–427 (2013). https://doi.org/10.1007/s11075-012-9595-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-012-9595-5