The Perturbation Method is an approximation scheme with a solvable leading order. The standard way is to choose a non-interacting sector for the leading order. The adaptive Perturbation Method improves the solvable part by using all diagonal elements for a Fock state. We consider the harmonic oscillator with the interacting term, λ1x4/6 + λ2x6/120, where λ1 and λ2 are coupling constants, and x is the position operator. The spectrum shows a quantitative result from the second-order, less than 1 percent error, compared to a numerical solution when turning off the λ2. When we turn on the λ2, more deviation occurs, but the error is still less than 2 percent. We show a quantitative result beyond a weak-coupling region. Our study should provide interest in the holographic principle and strongly coupled boundary theory.

In this article, the Homotopy Perturbation Method (HPM) is employed to approximate solutions of a modified Lotka–Volterra equation. HPM has been introduced by He to solve approximately linear or nonlinear differential equations. Approximate polynomials have also been constructed to find approximate solutions of a modified Lotka-Volterra system. Numerical comparisons are made between HPM and maple numerical results.

This paper applies the homotopy Perturbation Method (HPM) to find the exact solution of Poisson equation with the Dirichlet and Neumann boundary conditions. To illustrate the simplicity and reliability of the Method, two examples are provided. The results obtained reveal that the Method is capable and easy to apply.

This paper applies homotopy Perturbation Method (HPM) to find the exact solution of Poisson equation with Dirichlet and Neumann boundary conditions. To illustrate the simplicity and reliability of the Method, two examples are provided. The results obtained reveal that the Method is capable and easy to apply.

More recently, Wazwaz [An analytic study of Fisher’s equation by using Adomian decomposition Method, Appl. Math. Comput.154 (2004) 609–620] employed the Adomian decomposition Method (ADM) to obtain exact solutions to Fisher’s equation and to a nonlinear diffusion equation of the Fisher type. In this paper, He’s homotopy Perturbation Method is employed for the generalized Fisher’s equation to overcome the difficulty arising in calculating Adomian polynomials.