Numerical method for solving nonlinear ordinary and partial differential equations for boundary-layer flows

A numerical method capable of solving nonlinear ordinary and partial differential equations has been developed. It is slightly similar to, but may be more advantageous or more widely applicable than, three existing numerical methods. Its essence is the joint use of residual generation, parametric embedding, parametric differentiation, and remainder estimation. Since the method is new, it is tested against several classical physical problems for which the solutions are documented. The results obtained by the proposed method are in good agreement with, and sometimes more accurate than, the literature values. Furthermore, it is nonrigorously proved that the proposed scheme possesses a stronger tendency to converge than other conventional linearization methods. The advantages and shortcomings of this method compared with those of other existing methods are also discussed. Finally, the authors propose that the present technique be called the method of parameterized residuals.