Numerical solutions of multidimensional nonlinear hyperbolic partial integro-differential equations using a redefined cubic B-spline-based differential quadrature method

The primary objective of this work is to present an efficient numerical method for solving multidimensional nonlinear hyperbolic partial integro-differential equations (HPIDEs). To implement this method, the following steps are sequentially followed. First, by integrating both sides of the HPIDE, we transform it into a new form of a partial integro-differential equation with a time derivative of first order. This new formulation allows the proposed method to ultimately reduce the problem of finding an approximate solution to a system of linear algebraic equations without employing linearization techniques. Next, for the discretization of the newly obtained form in temporal direction, a combination of the Crank–Nicolson finite difference technique and numerical integration methods, including the trapezoidal and rectangle integration rules, is utilized. This process results in a finite difference scheme with second-order convergence, and its stability and convergence are thoroughly examined using energy method. The Richardson extrapolation technique is also utilized to improve the convergence order in the temporal dimension. Furthermore, a differential quadrature method (DQM) based on a redefined structure of cubic B-splines is employed for the spatial discretization of the problem. Finally, some numerical examples in various dimensions are provided to evaluate the accuracy and effectiveness of the proposed method.