Orthogonal polynomials, Gaussian quadratures, and PDEs

Orthogonal polynomials are important in mathematical analysis. They can be used to separate many partial differential equations (PDES) which makes them particularly important in solving physical problems. Also, Gaussian integration provides a highly accurate and efficient algorithm for integrating functions. The value of the methods I describe in this paper depends on the basic assumption that a finite-order polynomial can effectively approximate a function. Therefore, a finite sum of orthogonal polynomials can accurately represent this function. By using the ideas of Gaussian integration, a function can be integrated or expanded in terms of orthogonal polynomials