Approximate solutions for the problem of liquid film flow over an unsteady stretching sheet with thermal radiation and magnetic field

The proposed method is based on replacement of the unknown function by a truncated series of the shifted Legendre polynomial expansion.An approximate formula of the integer derivative is introduced.Special attention is given to study the convergence analysis and derive an upper bound of the error for the presented approximate formula.The introduced method converts the proposed equation by means of collocation points to a system of algebraic equations with shifted Legendre coefficients.Thus,after solving this system of equations,the shifted Legendre coefficients are obtained.This efficient numerical method is used to solve the system of ordinary differential equations which describe the thin film flow and heat transfer with the effects of the thermal radiation,magnetic field,and slip velocity.