Chebyshev Spectral Collocation Solution of Non-Linear Time Dependent Partial Differential Equation With Time Derivative Boundary Conditions

Abstract

The present paper demonstrate the implementation of time derivative boundary conditions in the Chebyshev differentiation matrices. The non-linear time dependent partial differential equations with time derivative boundary conditions(mixed boundary conditions) is considered and the spectral collocation algorithm is developed and solutions are presented. Quasi-linearization technique is used to convert the non-linear partial differential equation into linear form by using Taylor series approximation about the initial guess. Lagrange interpolating polynomials are used as basis of the solution at the Gauss-Labatto grid points. Also the time derivative boundary conditions are incorporated within the Chebyshev differentiation matrices. MATLAB software is used to implement this algorithm and numerical results are depicted graphically. The case study problem is solved using this approach and the solution found by the this method is more accurate compared to the finite difference method with uniform grid points.