Continuous numerical solutions and error bounds for time dependent systems of partial differential equations: Mixed problems - Jódar, L.; Ponsoda, E. (Computers & mathematics with applications (1987), 1995)
The aim of this paper is to construct continuous numerical solutions with a prefixed accuracy in a bounded domain Ω(t 0, t 1) = [0, p] × [t 0, t 1] , for mixed problems of the type u t ( x, t) − D( t) u xx ( x, t) = 0, 0 < x < p, t > 0, subject to u(0, t) = u( p, t) = 0 and u( x, 0) = F( x). Here, u( x, t) and F( x) are r-component vectors and D( t) is a C r × r valued two-times continuously differentiable function, so that D( t 1) D( t 2) = D( t 2) D( t 1) for t 2 ≥ t 1 > 0 and there exists a positive number δ such that every eigenvalue z of (D(t) + D H(t)) 2 with t > 0 is bigger than δ.