Double-exponential Whittaker cardinal function approximation scheme for Bratu-type and Troesch’s problems
Abstract
The authors attempt here to exercise an efficient approximation scheme for obtaining highly accurate approximate solutions to Bratu-type and Troesch’s problems. The underlying mathematical ingredients of the scheme are the double exponential transformation followed by the finite Whittaker cardinal function approximation of functions in the basis generating Shannon–Kotelnikov multiresolution analysis of $L^{2}(\varOmega ) (\varOmega =[a,b]\subset \mathbb {R})$. We provide a formula relating the exponent $n$ in the desired order ($O(10^{-n})$) of accuracy and the resolution $J$ of the approximation space (Paley–Wiener space of bandwidth $[-2^{J}\pi ,2^{J}\pi ]$) of multiresolution analysis of $L^{2}(\mathbb {R})$, the lower and upper limits in the finite sum in the approximation of the solution, and a formula for the a posteriori error. A comparison of the accuracy of the approximate solutions obtained with that of other results in the literature confirms the better efficiency of the present scheme.
