Abstract
On the basis of product integration techniques a discrete version of a piecewise polynomial collocation method for the numerical solution of initial or boundary value problems of linear Fredholm integro-differential equations with weakly singular kernels is constructed. Using an integral equation reformulation and special graded grids, optimal global convergence estimates are derived. For special values of parameters an improvement of the convergence rate of elaborated numerical schemes is established. Presented numerical examples display that theoretical results are in good accordance with actual convergence rates of proposed algorithms.
Highlights
Let R = (−∞, ∞) and N = {1, 2, . . .}
In the present paper we study the convergence behaviour of a discrete version of a collocation method for the numerical solution of initial or boundary value problems of the form n1
∆ = {(t, s): 0 ≤ t ≤ b, 0 ≤ s ≤ b, t = s}, is denoted the set of q times continuously differentiable functions g : ∆ → R satisfying for all (t, s) ∈ ∆ and all non-negative integers i and j such that i + j ≤ q the condition
Summary
In the present paper, using an integral equation reformulation of problem (1.1), (1.2), we first discretize the corresponding integral equation by quadrature formulas based on product integration (see, for example, [2, 3]) and apply a piecewise polynomial collocation method on special graded grids reflecting the singular behaviour of the exact solution. With this approach we approximate smooth parts of the corresponding integrands by piecewise polynomial interpolation and integrate exactly the remaining (more singular) parts of these integrands (see Section 4).
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