A Spectral-Collocation and Spline-Based Numerical Framework for Solving Nonlinear Boundary Layer Flow Problems

In this work we present a multi-step algorithm with high accuracy to solve nonlinear second- and third-order ODE problems in fluid dynamics. In this method we use the polynomial spectral collocation method to acquire accurate numerical values at the collocation points, followed by spline interpolation to improve the approximated solution between the collocation points. Also, we derived quintic and septic polynomial spline models and computed the associated error bounds with a convergence analysis. To demonstrate the application of the present method, we considered two cases of fluid dynamics problems. We also looked at the numerical solution's validity and convergence in each example, which are assessed by calculating the infinity norm of the absolute and residual errors at every iteration level. Lastly, we illustrate the effect and influence of the physical parameters over velocity and temperature profiles through graphical results.