A High-Accuracy Algorithm for Solving Nonlinear PDEs with High-Order Spatial Derivatives in 1 + 1 Dimensions

We propose an algorithm to solve a system of partial differential equations of the type u1(x, t) = F(x, t, u, ux, uxx, uxxx, uxxxx) in 1 + 1 dimensions using the method of lines with piecewise ninth-order Hermite polynomials, where u and F are N-dimensional vectors. Nonlinear boundary conditions are easily incorporated with this method. We demonstrate the accuracy of this method through comparisons of numerically determined solutions to the analytical ones. Then, we apply this algorithm to a complicated physical system involving nonlinear and nonlocal strain forces coupled to a thermal field.