An optimal and modified homotopy perturbation method for strongly nonlinear differential equations

Homotopy perturbation method (HPM) is one of the most popular semi-analytical methods to solve a nonlinear differential equation. However, in HPM, there is no strict rule for the choice of its linear operator, and its series solution may not always converges. In this study, firstly, we define the linear operator as an auxiliary linear operator (L-a), in the frame of homotopy analysis method (HAM). Then we generalize this La based on the auxiliary roots of L-a = 0. Finally, using the optimization technique (on minimization of the residual error) we determine the best-fitted optimal La for a problem. By doing this we ensure and accelerate the convergence of our semi-analytical homotopy perturbation series solution. Thereby we rename the HPM as Optimal and Modified Homotopy Perturbation Method (OMHPM). We consider three strongly nonlinear differential equations of nonlinear dynamical phenomena associated with the fluid dynamics to certify our technique. The dependencies of the form of the optimal La and the convergence of the solution (obtained by HPM, Optimal HAM and OMHPM) on the values of parameters (involved in the scale transformation), initial/boundary conditions and artificial controlling parameters (involved in optimal HAM) are explored here. It is reported that our OMHPM is highly accurate and efficient than HPM, optimal HAM and Domain decomposition optimal HAM. Moreover, OMHPM is simple and can be applied to directly to any singular/non-singular highly nonlinear ordinary differential equations without any decomposition, special/scale transformation, linearization, artificial controlling parameters and discretization. An attempt is made to apply our optimal auxiliary linear operator onto the optimal HAM for possible fastest convergence.

Automated summary

The study defines the auxiliary linear operator as La and generalizes it based on the zero roots of La = 0. By determining the best-fitted optimal La using the optimization technique, the study ensures and speeds up the convergence of the semi-analytical homotopy perturbation series solution, renaming it as Optimal and Modified Homotopy Perturbation Method (OMHPM). Three strongly nonlinear differential equations associated with fluid dynamics are considered to explore the dependencies of the optimal La and the convergence of the solution on various parameters. The results show that OMHPM is highly accurate and efficient compared to HPM, optimal HAM, and Domain decomposition optimal HAM. Additionally, OMHPM is a simple approach applicable to singular/non-singular highly nonlinear ordinary differential equations without the need for decomposition, linearization, artificial controlling parameters, or discretization.