Technical Report CS0901

TR#:CS0901
Class:CS
Title: IMPROVING THE ACCURACY OF QUADRATURE METHOD SOLUTIONS OF FREDHOLM INTEGRAL EQUATIONS THAT ARISE FROM NONLINEAR TWO-POINT BOUNDARY VALUE PROBLEMS
Authors: Avram Sidi
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Abstract: In this paper we are concerned with high-accuracy quadrature method solutions of nonlinear Fredholm integral equations of the form $y(x) = r(x) + \int^1_0 g(x,t) F(t,y(t)) dt,\ 0 \leq x \leq 1$, where the kernel function $g(x,t)$ is continuous, but its partial derivatives have finite jump discontinuities across $x=t$. Such integral equations arise, e.g., when one applies Green's function techniques to nonlinear two-point boundary value problems of the form $y''(x) = f(x,y(x)),\ 0 \leq x \leq 1$, with $y(0)=y_0$ and $y(1)=y_1$, or other linear boundary conditions. A quadrature method that is especially suitable and that has been employed for such equations is one based on the trapezoidal rule that has a low accuracy. By analyzing the corresponding Euler-Maclaurin expansion, we derive suitable correction terms that we add to the trapezoidal rule, thus obtaining new numerical quadrature formulas of arbitrarily high accuracy that we also use in defining quadrature methods for the integral equations above. We prove an existence and uniqueness theorem for the quadrature method solutions, and show that their accuracy is the same as that of the underlying quadrature formula. The solution of the nonlinear systems resulting from the quadrature methods is achieved through successive approximations whose convergence is also proved. The results are demonstrated with numerical examples.
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