The Cauchy problem for a linear parabolic partial differential equation

Abstract

Numerical approximation of the solution of the Cauchy problem for the linear parabolic partial differential equation is considered. The problem: ( p( x) u x ) x − q( x) u = p( x) u t , 0 < x < 1,0 < t⩽ T; u(0, t) = ƒ 1(t), 0 < t ⩽ T ; u(1,t) = ƒ 2(t), 0 < t ⩽ T ; p(0) u x (0, t) = g( t), 0 < t 0 ⩽ t ⩽ T, is ill-posed in the sense of Hadamard. Complex variable and Dirichlet series techniques are used to establish Hölder continuous dependence of the solution upon the data under the additional assumption of a known uniform bound for ¦ u(x, t)¦ when 0 ⩽ x ⩽ 1 and 0 ⩽ t ⩽ T. Numerical results are obtained for the problem where the data ƒ 1, ƒ 2 and g are known only approximately.