A high-accuracy algorithm for solving nonlinear PDEs with high-order spatial derivatives in \(1+1\) dimensions (Q1331450)

A high-accuracy algorithm is proposed to solve a system of nonlinear partial differential equations (PDEs) \(u_ t (x,t) = F(x,t,u,u_ x,u_{xx}, u_{xxx}, u_{xxxx})\), \(x \in [x_ L, x_ R]\), \(t \in\mathbb{R}_ +\), \(u(x,0) = f(x)\), \(V_ i (x_ L, t,u,u_ x, u_{xx}, u_{xxx}) = 0\), \(W_ i (x_ R, t,u,u_ x, u_{xx}\), \(u_{xxx}) = 0\), \(i=1,2\) where \(F,f,V_ i,W_ i\) are \(N\)-dimensional vectors. The method is based on the method of lines and piecewise ninth-order Hermite polynomials.