High-Degree Polynomial Approximations for Solving Linear Integral, Integro-Differential, and Ordinary Differential Equations

TL;DR

The paper develops a universal numerical scheme that solves linear equations combining differentiation and integration, including integro-differential and initial/boundary value problems, by representing the solution $y(x)$ as a high-degree piecewise-polynomial on $[\alpha,\beta]$ and recasting the problem as a least-squares minimization. It provides a cohesive framework to discretize derivatives and integral terms with matrices ($Y$, $Y'$, $Y''$, $M$, $V$) and to incorporate initial/boundary constraints, giving a reduced system $\tilde{A}\tilde{c}=\tilde{\phi}$ solved by $\min_{\tilde{c}}||\tilde{A}\tilde{c}-\tilde{\phi}||^2$. For ill-posed first-kind problems, regularization via Tikhonov terms $\min\{||M\hat{y}-\hat{\phi}||^2+\lambda||L\hat{y}||^2\}$ (with $\hat{y}_{\lambda}=(M^T M + \lambda L^T L)^{-1} M^T \hat{\phi}$) or its polynomial-coefficient analogue improves stability, with parameter selection guided by generalized cross-validation and related criteria. The framework is demonstrated on well-posed ODEs, IDEs, and second-kind integral equations, and on ill-posed first-kind problems (including Volterra, Fredholm, Abel-type, and Laplace-transform inversion), showing accurate reconstructions under noise and suppression of Runge-type oscillations via stabilizing operators. Overall, the method offers a flexible, generalizable approach for discretizing and regularizing linear equations across a broad spectrum of differential and integral problems.

Abstract

This paper presents a universal numerical scheme tailored for tackling linear integral, integro-differential, and both initial and boundary value problems of ordinary differential equations. The numerical scheme is readily adapted for resolving ill-posed problems. Central to our approach is high-degree piecewise-polynomial approximation to the exact solution. We illustrate the accuracy and stability of our numerical solutions in the presence of noise through illustrative examples. Additionally, we demonstrate that proposed regularization being applied to high-degree interpolation, effectively eliminates Runge's phenomenon.

Figures (14)

• Figure 1: Polynomial approximation of Runge's function
• Figure 2: Piecewise-polynomial approximation of Runge's function.
• Figure 3: Solution of boundary value problem for Bessel ODE in form (\ref{['32']})
• Figure 4: Solution of initial value problem (\ref{['33']}).
• Figure 5: Solution of IDE (\ref{['33']}). $\sigma = 0.01$