Collocation - A Simple, Flexible and Efficient Method for Integral and Differential Equations

We present a collocatIon method based on piecewise cubic polynomials I In C for solving various Integral and differential equatIons. Two new methods for handling curved boundaries are discussed based on collocation and discrete least-squares. Finally, a summary of the properties of the collocation method Is given which IndIcates the simplicity, flexlbl1fty, efficiency, and generality of the method. Introduction. In this report, we present an approximation theoretical method for solving integral and differential equations. The method is the so-called collocation whose idea is analogous to interpolation. Inspite, Its simplicIty, flexibility, and efficiency, the method is not well known, perhaps because It is dIfficult to theoretically analyse. We consider a rather specific instance of the method based on piecewise cubic polynomials I In C and apply It to '/arlous Integral and differentIal equations. We discuss two new methods for handling curved boundaries based on collocation and dlscrete least-squares. Finally, we summarize the properties of the collocation method which make It more efficient for general use. I. COLLOCATION HETHOD We present an abstract formulation of a differential operator problem and an approxImation theoretic method for solving It. The problem can be defined in terms of four attributes.