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.