Impulsive End Condition for Diffusion Equation

for 0 ^ x ^ 1, with initial condition y(x) 0) = 0 and end conditions y(0, t) = 1, 2/(1, i) = 0. Here x is position and t is time. The problem as stated is a nondimensional version of a heat-conduction problem in which a fixed temperature is suddenly applied to one end of an initially cold bar. Other diffusion problems involving impulsive boundary conditions may require a more complicated description than that given by Eq. ( 1 ) ; additional terms, nonconstant coefficients, or nonlinearities may well be present. Nevertheless, it may happen that even in such a case, Eq. (1) provides an adequate description of the short-time behavior of the solution, and so is suitable for a discussion of the computation error over the first few time steps. Such is the case, for example, in the viscous fluid problem which motivated the present study [1]. For the most part, we will deal with an implicit version of the finite difference approximation to Eq. ( 1 ) ; in many problems, such an implicit scheme is generally desirable because of computational stability with respect to large time steps or with respect to the presence of additional terms in Eq. (1). Let the interval [0, 1] of the x-axis be divided into K sub-intervals, with Sx = \/K; denote the time step by &t, and the ratio 5t/(ôx) by A. Then the finite difference equation to be considered is