Initial value problems state vector

These methods are efficient for problems with initial-value ODE models without state variable and final time constraints. Here solutions have been reported that require from several dozen to several hundred model (and adjoint equation) evaluations (Jones and Finch, 1984). Moreover, any additional constraints in this problem require a search for their appropriate multiplier values (Bryson and Ho, 1975). Usually, this imposes an additional outer loop in the solution algorithm, which can easily require a prohibitive number of model evaluations, even for small systems. Consequently, control vector iteration methods are effective only when limited to the simplest optimal control problems. 

Program Description The MATLAB funct onparabolic]D.m is developed to solve the parabolic partial differential equation in an unsteady-state one-dimensional problem. The boundary conditions arc passed to the function in the same format as that of Example 6.1, with the exception that they are given in only the. xi-direction. The function also needs the initial condition, u, which is a vector containing the values of the dependent variable for all x at time f = 0. 

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