Homotopy analysis method for Burgers’ equation: Application of gradient
descent approach
Abstract
The present work provide a synthesis towards understanding the advantage
of the gradient descent method applied over the homotopy analysis method
(HAM). To demonstrate the idea, the nonlinear Burgers’, equation is
taken into consideration. Due to the possibility of strong nonlinearity,
most of the PDEs are solved by using either numerical or approximate
analytical techniques. The HAM is one of the approximate analytical
technique consist of auxiliary parameter h which allows us to adjust and
control the convergence region of the series solution. The chosen value
of h from the convergence region may possibility to produce diverging
solution due to trial and error approach. Hence, the convergence to
correct solution and its accuracy need to be ascertained. Therefore, the
development of efficient iterative optimization approach namely gradient
descent method may establish the correct values of h and gives the
precise solution. In order to establish efficacy of proposed method, the
Burgers’ equation results obtained from gradient descent approach are
validated with existing analytical and numerical results reported in the
literature. The novelty of proposed work would in a way to demonstrate
the potential and effectiveness of gradient descent method for
evaluating the various kinds of nonlinear equation.