Galerkin–Legendre spectral method for the velocity and thermal boundary layers over a non-linearly stretching sheet

Analytical solutions for the velocity and temperature fields in a viscous fluid flowing over a nonlinearly stretching sheet are obtained via Galerkin–Legendre spectral method. The application of spectral methods to this problem is novel as well as the concept of the non-uniqueness of the solution and the efficient algorithms developed in this paper. The governing partial differential equations are converted into a nonlinear ordinary differential equation for the velocity field ff and a variable coefficient linear ordinary differential equation for the temperature field θθ by a similarity transformation in the semi-infinite physical domain. A coordinate transformation is introduced to map the physical state space into a bounded computational domain. It is shown that Galerkin–Legendre spectral method is the most efficient among spectral methods that lead to the analytical solution of the governing set of nonlinear differential equations in the computational domain. Computationally efficient algorithms are constructed for the velocity gradient and temperature profile computations and two distinct solutions of the field equations are presented.