Implicit Collocation Technique for Heat Equation with Non-Classic Initial Condition

In this paper we combine the second-order finite difference approximation for the spatial derivative and collocation technique for the time component to numerically solve the one-dimensional heat equation with non-standard initial condition. After discretization in space of the problem, solution is approximated at each spatial grid point by a polynomial depending on time. This discretization produces a linear system of equations and the matrix of this system is nonsingular. Discussion on numerical experiments is given.

[1]  Mehdi Dehghan.  Numerical solution of a parabolic equation with non-local boundary specifications , 2003, Appl. Math. Comput..

[2]  Jan Chabrowski,et al.  On non-local problems for parabolic equations , 1984, Nagoya Mathematical Journal.

[3]  V. Lakshmikantham,et al.  Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space , 1991 .

[4]  Mehdi Dehghan,et al.  On the solution of an initial‐boundary value problem that combines Neumann and integral condition for the wave equation , 2005 .

[5]  Jules Kouatchou,et al.  Parallel implementation of a high-order implicit collocation method for the heat equation , 2001 .

[6]  Ludwik Byszewski.  Strong maximum principles for parabolic nonlinear problems with nonlocal inequalities together with arbitrary functionals , 1991 .

[7]  F. Jézéquel.  A validated parallel across time and space solution of the heat transfer equation , 1999 .

[8]  Mehdi Dehghan,et al.  On the numerical solution of the diffusion equation with a nonlocal boundary condition. , 2003 .

[9]  Mehdi Dehghan.  The solution of a nonclassic problem for one-dimensional hyperbolic equation using the decomposition procedure , 2004, Int. J. Comput. Math..

[10]  Ji-Huan He.  SOME ASYMPTOTIC METHODS FOR STRONGLY NONLINEAR EQUATIONS , 2006 .

[11]  M. Dehghan.  Efficient techniques for the second-order parabolic equation subject to nonlocal specifications , 2005 .

[12]  Yanping Lin,et al.  Analytical and numerical solutions for a class of nonlocal nonlinear parabolic differential equations , 1994 .

[13]  Ji-Huan He.  Variational iteration method – a kind of non-linear analytical technique: some examples , 1999 .

[14]  H. Yin,et al.  Determination of an unknown function in a parabolic equation with an overspecified condition , 1990 .

[15]  J. Kouatchou.  Finite Differences and Collocation Methods for the Solution of the Two Dimensional Heat Equation , 2001 .

[16]  Mehdi Dehghan,et al.  Numerical schemes for one-dimensional parabolic equations with nonstandard initial condition , 2004, Appl. Math. Comput..