Solving Ordinary Differential Equations by Simplex Integrals

Abstract

This paper is devoted to the proper discrete solution for ordinary differential equations, especially to oscillating solution. In contrast to Lipschitz condition, we define a new condition following that

$$\left|\int_{t_{0}}^{t_{1}}f(t)dt\right|\leq R\max\limits_{\xi_{1},\xi_{2}\in [t_{0},t_{1}]} |f(\xi_{1})-f(\xi_{2})|$$

with small R for all t ₀, t ₁ in the correlative intervals. Under the assumption of this new condition, we obtain a new asymptotic formula

$$\phi_{v}(t)-Q_{v-1}(t)=O((Rh)^{v}),$$

where simplex integral φ _v (t) denotes

$$\int_{t_{0}}^{t}\cdots\int_{t_{0}}^{\xi_{v-2}}\int_{t_{0}}^{\xi_{v-1}}f(\xi_{v}) d\xi_{v}d\xi_{v-1}\cdots d\xi_{1}$$

and the v − 1-th polynomials Q _v − 1(t) in which coefficient correspond to simplex integrals $\phi_{n_{k}}(t)$ with n _k  > v, k = 1, 2,..., v. In other words, the accuracy for approximation increasing rapidly as the integrable functions oscillate rapidly or for small step h while it’s difficult for us to pursuit a polynomial to approximate a highly oscillatory function.

Applying this idea of approximation to ODE, this paper surveys the algorithmic issues. If ODE has the form

$$P_{n}y^{(n)}+P_{n-1}y^{(n-1)}+\cdots+P_{1}y'+P_{0}y=g(t),$$

where P _n (t),P _n − 1(t),..., P ₀(t) are arbitrary degree polynomials, then we can solve it by the recursive relation about simplex integrals altogether with approximate relation. Finally, numerical examples about Airy and Bessel equations illustrate the efficiency of this technique.