Abstract
The numerical stability of the polynomial spline collocation method for general Volterra integro‐differential equation is being considered. The convergence and stability of the new method are given and the efficiency of the new method is illustrated by examples. We also proved the conjecture suggested by Danciu in 1997 on the stability of the polynomial spline collocation method for the higher‐order integro‐differential equations.
Highlights
In this paper, we analyze the stability properties of the polynomial spline collocation method for the approximate solution of general Volterra integro-differential equation
To describe the polynomial spline spaces, let N : 0 = t0 < t1 < · · · < tN = T be the mesh for the interval I, and set σn := tn, tn+1, hn := tn+1 − tn, n = 0, 1, . . . , N − 1, h = max{hn : 0 ≤ n ≤ N − 1}, ZN := tn : n = 1, 2, . . . , N − 1, ZN = ZN ∪ {T}
In order to discuss numerical stability, we study the behavior of the method as applied to the pth-order test Volterra integro-differential equation p−1 t y(p)(t) = q(t) + αj y(j)(t) + ν y(s)ds, ν = 0, t ∈ I = [0, T], j=0 y(i)(0) = y0(i), i = 0, 1, . . . , p − 1, (2.1)
Summary
We analyze the stability properties of the polynomial spline collocation method for the approximate solution of general Volterra integro-differential equation. Volterra integro-differential equation (1.1) will be solved numerically using polynomial spline spaces. To describe the polynomial spline spaces, let N : 0 = t0 < t1 < · · · < tN = T be the mesh for the interval I, and set σn := tn, tn+1 , hn := tn+1 − tn, n = 0, 1, . The solution (y) to the initial-value problem (1.1) is approximated by an element u in the polynomial spline space. It is a polynomial spline function of degree m + d, which possesses the knots ZN , and is d times continuously differentiable on I.
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