Abstract:
The authors used an effective approximation technique to obtain accurate approximate solutions for a class of second-order non-local, non-linear ordinary differential equations with various non-local terms and boundary conditions that often appear in applied sciences. The underlying mathematical ingredients of the proposed scheme is the finite Whittaker Cardinal function approximation of functions in the basis generating Shannon–Kotelnikov multi-resolution analysis of $L^2(\Omega)$ ($\Omega=[a,b]\subset\mathbb{R}$ or $\mathbb{R}^+$). Formulae relating the exponent $n$ in the desired order $(O(10^{-n}))$ of accuracy, the resolution $J$ of the bandwidth of the approximation space, the dependences of the lower and upper limits in the finite sum in the approximation and a formula for a posteriori error in the approximate solution are provided. The efficiency and elegance of the scheme have been examined for various second-order, non-local, non-linear ordinary differential equations of physical interest and found efficient.
