Legendre wavelet operational matrix of derivative for optimal control in a convective–diffusive fluid problem

We consider the problem of controlling the model of one-dimensional fluid flow through a soil packed tube in which a contaminant is initially distributed. A fluid is pumped through a tube to remove the contaminant. The control problem is to determine the optimal convective velocity due to the fluid being pumped by minimizing a given performance criterion. The performance criterion is chosen to be a combination of the total contaminant at the final time and the cost of the control. The set of orthogonal Fourier trigonometry series is used as a basis function of the Galerkin procedure to lump the distributed parameter system. A Legendre wavelet operational matrix of derivative is used to approximate the control and modal state variables. The main characteristics of this technique is that it reduces these problems to those of solving a system of algebraic equations, thus greatly simplifying the problem. The effectiveness of the proposed approach is illustrated numerically and the results are quite satisfactory.