General approach on the best fitted linear operator and basis function for homotopy methods and application to strongly nonlinear oscillators

Abstract

In this study, we propose a universal mathematical guideline for the selection of auxiliary linear operator L(u) and the basis function u0(x), which are the principal parts of Homotopy methods: Homotopy Perturbation Method (HPM) and Homotopy Analysis Method (HAM). We assume the coefficients of derivatives involved in L(u) as a function of auxiliary roots of L(u)=0. Based on the residual error minimization we compute unknown roots and thereby obtain the optimal linear operator at an order of approximation. Also, an effort is made to explore the role of the number of unknowns embedded into L(u) (for different choices of real roots of L(u)=0) on the rate of converges of the solution and CPU uses. Hence the best fitted linear-operator and corresponding basis function are proposed for a particular problem. Also, we propose to discretize the exact square residual using Simpson’s 1/3 algorithms, and compare its accessibility in comparison to other existing residual calculation methods. Initially, we applied our algorithms to three IVPs (Logistic, Stiff system and Riccati system) and three singular BVPs, and then finally to a strongly nonlinear oscillator (cubic-quintic Duffing equation). We compare our technique’s accuracy, efficiency and simplicity with other contemporary analytical and numerical methods. It is demonstrated that our optimal linear operator, which is updated at every order of approximation, is much more efficient, important (than the artificial parameters and functions of developed HAM), and self-sufficient for the convergence of series solutions. Our approach is more effective, straightforward and easy to use when applied to many nonlinear problems arising in mechanics and engineering. It is seen for the initial value problems that our technique provides the globally convergent series solution, while the solutions by HPM and optimal HAM are locally convergent. This novel development will make homotopy methods more powerful.