Practical implementation and error bounds of integer-type general algorithm for higher order differential equations

TL;DR

This paper presents a practical integer-based algorithm for solving higher-order Fuchsian differential equations, providing error bounds and demonstrating high-accuracy solutions through numerical experiments.

Contribution

It introduces a practical implementation of an integer-only algorithm with error bounds for higher-order differential equations, extending previous theoretical work.

Findings

Algorithm successfully produces high-accuracy solutions

Error bounds are established for the numerical solutions

Numerical experiments confirm the effectiveness of the method

Abstract

In our preceding paper, we have proposed an algorithm for obtaining finite-norm solutions of higher-order linear ordinary differential equations of the Fuchsian type [\sum_m p_m (x) (d/dx)^m] f(x) = 0 (where p_m is a polynomial with rational-number-valued coefficients), by using only the four arithmetical operations on integers, and we proved its validity. For any nonnegative integer k, it is guaranteed mathematically that this method can produce all the solutions satisfying \int |f(x)|^2 (x^2+1)^k dx < \infty, under some conditions. We materialize this algorithm in practical procedures. An interger-type quasi-orthogonalization used there can suppress the explosion of calculations. Moreover, we give an upper limit of the errors. We also give some results of numerical experiments and compare them with the corresponding exact analytical solutions, which show that the proposed algorithm is successful in yielding solutions with high accuracy (using only arithmetical operations on integers).