Surprising solutions for some challenging problems arising from boundary layer theory by a new technique: the homotopy contraction mapping technique (HCMT)

Abstract

In this work, the homotopy structure $$ (1 - p)L\,[f] = - p\,N\,[f] $$ is considered to solve some newly developed nonlinear problems in fluid dynamics. In summary, deformation in homotopy series solutions occurs from the initial trial function to the actual solution. The main challenging part in many similarity equations associated with the boundary layer theory is to identify a certain quantity of the exact solution such as $$ f^{{\prime \prime }} (0) $$ . This quantity is initially unknown and is automatically guessed through the zeroth function. Therefore, it makes sense that in the previous homotopy series solutions one would need at least another parameter i.e. the controlling parameter ( $$ \hbar $$ ) to handle the convergence through the so-called $$ \hbar $$ -curves analysis and eventually get an approximate value for $$ f^{{\prime \prime }} (0) $$ . Here it is accounted the basic homotopy structure with No controlling parameter ( $$ \hbar $$ ) and it is shown for the 1st time that the zeroth order solution in homotopy series may be potential to contain some certain quantities of the exact solution, here is to be $$ f^{{\prime \prime }} (0) $$ ; i.e. $$ f_{0}^{{\prime \prime }} (0) $$ could be $$ f^{{\prime \prime }} (0) $$ . This hypothesis is checked through a theorem, being totally linked to the Fixed Point Property (FPP), a topological invariant; i.e. being preserved by any homeomorphism. The theorem enables us Not only to check the validity of the hypothesis, but also to extract the value of the quantity as the unique description of the homotopy series solutions truncated at any order of approximation. The technique is initially introduced in a simple and straightforward manner and then it is applied to some fluid mechanical problems to be further compared with traditional homotopy analysis method, homotopy perturbation method and Numerical solutions. The outcomes indicated an excellent improvement by the present approach as only a few series terms were accounted to obtain highly accurate solutions.