Solving Ordinary Differential Equations by Simplex Integrals via Linear Equations

In this paper, we show that liner equations {Sigma(nu-1)(k=0)alpha(k) integral(t1)(t0) t(k)dt = phi(nu+1)(t(1)) 1/1 !Sigma(nu-1)(k=0)alpha(k) integral(t1)(t0) (t(1) - t)t(k)dt = phi(nu+2)(t(1)) 1/(nu-1)!Sigma(nu-1)(k=0)alpha(k) integral(t1)(t0) (t(1) - t)(nu-1) t(k)dt = phi(2 nu)(t(1)) determined polynomial Q(nu-1)(t) = alpha(0) + alpha(1)t + ... +alpha(v-1)t(nu-1) can well approximate to simplex integral phi(nu)(t) = 1/(nu-1)! integral(t1)(t0) y(xi)(t-xi)(nu-1) d xi in mall interval [t(0), t(1)]. Altogether with :recursive relations, we can solve ODE of the form P(n)y((n)) + P(n-1)y((n-1)) + ... + P(1)y' + P(0)y = g(t), where P-n(t), Pn-1(t), ..., P-1(t), P-0(t) are arbitrary degree polynomials. Numerical example about Airy equation illustrates the efficiency of this technique.