The repetition factor and numerical stability of volterra integral equations

AbstractIn this paper we prove that direct linear multistep methods for Volterra integral equations of the second kind with repetition factor equal to one are always stable. We show trivially that this result is not true for first kind equations. We also demonstrate constructively that direct linear multistep methods for both first and second kind Volterra integral equations can have repetition factors greater than one, and indeed of arbitrary high order, and be numerically stable. Finally we explain why the first form of Simpson's rule for second kind equations is stable while the second form is unstable

A Novel Third Order Numerical Method for Solving Volterra Integro-Differential Equations

In this paper we introduce a numerical method for solving nonlinear Volterra integro-differential equations. In the first step, we apply implicit trapezium rule to discretize the integral in given equation. Further, the Daftardar-Gejji and Jafari technique (DJM) is used to find the unknown term on the right side. We derive existence-uniqueness theorem for such equations by using Lipschitz condition. We further present the error, convergence, stability and bifurcation analysis of the proposed method. We solve various types of equations using this method and compare the error with other numerical methods. It is observed that our method is more efficient than other numerical methods