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In this paper, we investigate the oscillation properties of solutions of a class of highly oscillatory Volterra integral equations and develop a Hermite collocation method to approximate the solution of these equations. We begin our analysis by obtaining an asymptotic expansion for the solution of these equations using their resolvent representation. We then introduce a new Gaussian radial basis function interpolation to provide a numerical solution for these equations. The convergence analysis of the proposed method is also studied, which shows that increasing the number of collocation points or the number of mesh points controls the impact of the oscillation parameter in the whole error. Some numerical examples are presented to show the accuracy of the proposed scheme. [ABSTRACT FROM AUTHOR]

Abstract Integral-algebraic equations (IAEs) are coupled systems of Volterra integral equations of the first and second kind which naturally arise in many applications in mathematical physics. In this paper, we solve the IAEs of index-1 by Sinc-collocation discretization and prove that the discrete solutions converge to the true solutions of the IAEs exponentially. The discrete solutions are determined by linear systems which can be effectively solved by suitable iteration methods. Numerical examples are given to illustrate the effective performance of our method. [ABSTRACT FROM AUTHOR]

In this work, we present an efficient spectral-collocation method for numerical solution of a class of nonlinear weakly singular Volterra integral equations. This type of equations typically has a singular behavior at the left endpoint of the interval of integration. For overcoming this non-smooth behavior, we apply the Jacobi-collocation method. The convergence analysis of the proposed method is investigated in the L ∞ and the weighted L 2 norms and the results of several numerical experiments are presented which support the theoretical results. The computed results are compared wherever possible with those already available in the literature. [ABSTRACT FROM AUTHOR]