Numerical solutions with a priori error bounds for coupled self-adjoint time dependent partial differential systems

Abstract

This paper is concerned with the construction of accurate continuous numerical solutions for partial self-adjoint differential systems of the type ( P( t) u t ) t = Q( t) u xx , u(0, t) = u( d, t) = 0, u( x, 0) = f( x), u t ( x, 0) = g( x), 0 ≤ x ≤ d, t >- 0, where P( t), Q( t) are positive definite oR r× r -valued functions such that P′( t) and Q′( t) are simultaneously semidefinite (positive or negative) for all t ≥ 0. First, an exact theoretical series solution of the problem is obtained using a separation of variables technique. After appropriate truncation strategy and the numerical solution of certain matrix differential initial value problems the following question is addressed. Given T > 0 and an admissible error ϵ > 0 how to construct a continuous numerical solution whose error with respect to the exact series solution is smaller than ϵ, uniformly in D( T) = {( x, t); 0 ≤ x ≤ d, 0 ≤ t ≤ T}. Uniqueness of solutions is also studied.