A quasi-interpolation method for solving stiff ordinary differential equations

Based on the idea of quasi-interpolation and radial basis functions approximation, a numerical method is developed to quasi-interpolate the forcing term of differential equations by using radial basis functions. A highly accurate approximation for the solution can then be obtained by solving the corresponding fundamental equation and a small size system of equations related to the initial or boundary conditions. This overcomes the ill-conditioning problem resulting from using the radial basis functions as a global interpolant. Error estimation is given for a particular second-order stiff differential equation with boundary layer. The result of computations indicates that the method can be applied to solve very stiff problems. With the use of multiquadric, a special class of radial basis functions, it has been shown that a reasonable choice for the optimal shape parameter is obtained by taking the same value of the shape parameter as the perturbed parameter contained in the stiff equation. Copyright (C) 2000 John Wiley & Sons, Ltd.