Lucas collocation method for approximate solutions of HIV enfection models of CD4+T cells

In this thesis, the problem of obtaining Lucas collocation solutions of two models on the HIV infection of T4 cells, which are of central importance in the human immune system, has been considered. The method in question begins by specifying the degree N of the polynomials that represents the approximate solutions and the time interval over which the problem will be considered. N equidistant collocation points on the speciffied interval are then formed. Next, the condition of satisfying the differential equation system at each of the collocation points is imposed to the three approximate solution polynomials. With the addition of the equations corresponding to the initial conditions at t=0 to the thus obtained 3N nonlinear algebraic equations, a nonlinear system of 3N+3 equations with three unknowns is obtained. Upon solving this system the coefficients of the three approximate solution polynomials, hence the approximate solutions themselves, are obtained. Two criteria have been employed in evaluating the accuracy of the obtained approximate solutions. First, the residuals obtained upon substituting the computed approximate solutions in the equations have been considered and it has been observed that these residuals have a general tendency to decrease with increasing N values. Secondly, the obtained solutions have been compared with those resulting from an existing method which is known to be reliable. According to the results of these comparisons, some remarks have been made regarding the conclusions that may be drawn as to effect of the value of the parameter N on the accuracy of Lucas collocation solutions.