RATIONAL APPROXIMATION OF THE SOLUTION TO A STIFF ORDINARY DIFFERENTIAL EQUATION USING EXTRAPOLATION

While many numerical methods are available to solve ordinary differential equations, these methods continue to have difficulties with stiff equations. Extrapolation methods have been successful in providing a solution on a coarse grid. Lindberg has an algorithm for non-stiff equations that gives accurate values on a finer grid using only results from the extrapolation table. Our thesis provides values throughout the interval of integration for stiff and non-stiff equations. The solution is a rational function of Clenshaw and Lord generated from the Fourier-Chebyshev coefficients. As with Lindberg's algorithm, these coefficients are produced using only values from the extrapolation table.;Our numerical examples display Chebyshev series coefficients generated from the Euler, backward Euler and trapezoidal methods via both linear and rational extrapolation. The coefficients were generated for stiff and non-stiff equations. The approximate solution gained is a truncated Chebyshev series that can produce an even more accurate rational function approximation via the algorithm of Clenshaw and Lord. An ordinary differential equation with a pole in its solution was correctly found. The result of our method is a uniform solution approximation that can be examined for a closed form solution of the differential equation.;The prototype stiff equation, y' = (lamda)y with very negative (lamda), has an exponential solution. We have shown with analysis and numerical results that Clenshaw and Lord rational functions are a feasible approach to approximating this solution. A one-step discrete variable method--such as the Euler, backward Euler or trapezoidal method, can be used with the extrapolation process to yield a very accurate solution of an ordinary differential equation at the coarse grid points. The extrapolation utilizes the existence of an Euler-Maclaurin Sum formula type asymptotic expansion for the discrete variable solution. We prove an extension of the Euler-Maclaurin formula for integrals with an oscillating integrand. Thus, the Fourier-Legendre or Fourier-Chebyshev coefficients can be generated for a known function by a Romberg quadrature type process. We prove a further extension of the Euler-Maclaurin with respect to ordinary differential equations following the ideas of Gragg and Stetter. It provides an expansion for oscillating integrand type sums from our discrete variable values. These sums can be extrapolated upon to yield Legendre or Chebyshev series coefficients.