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On the stability of the one-step exact collocation methods for the numerical solution of the second kind Volterra integral equation. (English) Zbl 0675.65140
The paper concerns the stability analysis of the exact collocation method with \(C_ 1=0\), \(C_ m=1\), for the numerical solution of a second kind Volterra integral equation. Two types of test equations are used - basic test equation, for which the exact collocation method is equivalent to the discretized collocation method using “m” points interpolatory quadrature formula based on \(C_ 1,C_ 2,...,C_ m\) and convolution test equations for which the numerical solution has the same behaviour as the true solution, which is a linear combination of exponential functions with negative argument.
In the first case it is proved that if the collocation parameters are symmetric, the method is \(A_ 0\)-stable and I-stable, which holds to the stability regions in the plane \((h\lambda,h^ 2\mu)\), infinite of \(\lambda\)-axis and bounded along the \(\mu\)-axis. In some special cases a more specified boundary is found. Numerical experiments are carried out to test the sharpness of this bound in the third quadrant. For the convolution test equation a recurrence relationship is derived for the vector containing the values of the numerical solution and the stability conditions are given. Finally, the stability region is estimated from the theoretical and numerical results.
Reviewer: S.Spassov

MSC:
65R20 Numerical methods for integral equations
45D05 Volterra integral equations
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