Stability and Accuracy Analysis of θ Scheme for a Convolutional Integro-Differential Equation

Spatially long-range interactions for linearly elastic media resulting in dispersion relations are modelled by an integro-differential equation of convolution type (IDE) that incorporate non-local effects. This type of IDE is nonstandard (hence, it is almost impossible to obtain exact solutions) and plays an important role in modeling various applied science and engineering problems. In this article, such an IDE describing a linear elastic wave phenomenon has been studied. First, a discrete equivalent of the model IDE in space is proposed and then a class of forward backward average one step θ scheme for the semi-discrete time dependent numerical method has been developed. Further, stability and accuracy of the developed method has been analyzed rigorously.