Numerical indefinite integration using the sinc method
Thesis (MSc (Mathematics))--University of Stellenbosch, 2007.
In this thesis, we study the numerical approximation of indefinite integrals with algebraic or logarithmic end-point singularities. We show the derivation of the two quadrature formulas proposed by Haber based on the sinc method, as well as, on the basis of error analysis, by means of variable transformations (Single and Double Exponential), the derivation of two other formulas: Stenger’s Single Exponential (SE) formula and Tanaka et al.’s Double Exponential (DE) sinc method. Important tools for our work are residue calculus, functional analysis and Fourier analysis from which we state some standard results, and give the proof of some of them. Next, we introduce the Paley-Wiener class of functions, define the sinc function, cardinal function, when a function decays single and double exponentially, and prove some of their interesting properties. Since the four formulas involve a conformal transformation, we show how to transform from the interval (−¥,¥) to (−1, 1). In addition, we show how to implement the four formulas on two computational examples which are our test problems, and illustrate our numerical results by means of tables and figures. Furthermore, from an application of the four quadrature formulas on two test problems, a plot of the maximum absolute error against the number of function evaluations, reveals a faster convergence to the exact solution by Tanaka et al.’s DE sinc method than by the other three formulas. Next, we convert the indefinite integrals (our test problems) into ordinary differential equations (ODE) with suitable initial values, in the hope that ODE solvers such as Matlabr ode45 or Mathematicar NDSolve will be able to solve the resulting IVPs. But they all failed because of singularities in the initial value. In summary, of the four quadrature formulas, Tanaka et al.’s DE sinc method gives more accurate results than the others and it will be noted that all the formulas are applicable to both singular and non-singular integrals.