Peridynamic differential operator-based nonlocal numerical paradigm for a class of nonlinear differential equations

Abstract

This paper presents a novel nonlocal numerical paradigm for a class of general nonlinear ordinary differential equations using the peridynamic differential operator. Differential governing equations and initial/boundary conditions are reformulated from the local differential form to the nonlocal integral form using a meshless orthogonal technique. The solution domain is partitioned into a finite number of points, of which the properties are obtained through weighted summation over the corresponding properties of neighboring points. Using the Lagrange multiplier method and the variational principle, nonlinear ordinary differential equations with initial/boundary conditions can be solved through the Newton–Raphson iteration method. Moreover, the differences between the proposed method and other methods are illustrated by comparing several impact factors. Furthermore, three benchmarks, including the Riccati equation, the Poisson equation, and the fluid flow equation, have been solved to show the applicability and accuracy of the proposed numerical method, and the results are consistent with the numerical results in the previous literature. Finally, the proposed method is applied to the galloping vibration problem to reveal the galloping mechanism.