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This work is to analyze a Legendre collocation approximation for third-kind Volterra integro-differential equations. The rigorous error analysis in the L ∞ and L ω 0 , 0 2 -norms is provided for the proposed method. In fact when converting the original equation to an equivalent second kind one, the integral operator of the obtained equation contains two singularities and may become non-compact under certain conditions. In addition, in order to avoid the low-order accuracy caused by the singularity of the solution at the initial point, we adopted the idea of smooth transformation at the beginning to convert the original equation into a new equation with a more regular solution. Finally, the validity and applicability of the method are verified by several numerical experiments. [ABSTRACT FROM AUTHOR]

This paper is concerned with the stability properties of Runge-Kutta methods for Volterra integro-differential equations. Both the basic test equation and a convolution test equation are considered. Some fixed order recurrence relations and the corresponding stability conditions are derived for general methods. The concept of V 0 -stability is introduced for the convolution test equation and some V 0 -stable one-stage methods are found. The A 0 -stability and V 0 -stability of the fully implicit discretized collocation methods with one or two stages are investigated in details. Finally, some numerical experiments are given to illustrate the obtained theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate multistep Runge–Kutta methods for Volterra integro-differential equations. First, we derive order conditions for methods of order p and stage order q = p − 1 for p ⩽ 4. Then, stability properties of these methods with respect to the basic and convolution test equations are analyzed. V 0 -stable methods of order 2 and 3 are obtained. Finally, some numerical experiments are given to verify the obtained theoretical results. [ABSTRACT FROM AUTHOR]

This paper aims to provide a rigorous analysis of exponential convergence of the Chebyshev collocation method for third kind linear Volterra integral equations. Different from Volterra integral equations of the second kind, the integral operator in third kind equations is noncompact under certain conditions, which brings special challenges to numerical analysis. The key idea of the proposed method is to adopt a smoothing transformation for the Chebyshev collocation method to circumvent the curse of singularity at the beginning of time. Therefore, the solution of the resulting equation will possess better regularity and then the numerical method can achieve the spectral accuracy. Moreover, in order to show the applicability and efficiency of the method, several examples with non-smooth solutions are illustrated. [ABSTRACT FROM AUTHOR]