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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: bbialeck@mines.edu
Email: gxf@ams.org
Email: lopezmar@mac.uva.es
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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

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