Numerical solution of Hunter-Saxton type equation via exponential cubic B-spline collocation method

The Hunter-Saxton (HS) nonlinear partial differential equation, which is used to model the nematic liquid crystals and describes certain aspects of the orientation wave. In this study, we proposed the exponential cubic B-spline collocation method to solve the HS equation. The equation in space and time is discretized using the exponential cubic B-spline method and the Crank-Nicolson finite difference approach, respectively. The nonlinear components are linearized by quasi-linearization technique. Discretization techniques in both time and space are used over uniform meshes, which leads to a tridiagonal system. The Von Neumann method has been employed to analyze stability and to determine stability conditions. Two numerical examples have been examined at different values of t and mesh points to demonstrate the effectiveness of the proposed method. Error tables and graphs illustrate the results which are compared with the existing literature. The results demonstrate strong agreement with the analytical solution and outperform the trigonometric cubic B-spline collocation method.