Residue Regulating Homotopy Method for Strongly Nonlinear Oscillator

It is highly desired yet challenging to obtain analytical approximate solutions to strongly nonlinear oscillators accurately and efficiently. Here we propose a new approach, which combines the homtopy concept with a “residue-regulating” technique to construct a continuous homotopy from an initial guess solution to a high-accuracy analytical approximation of the nonlinear problems, namely the residue regulating homotopy method (RRHM). In our method, the analytical expression of each order homotopy-series solution is associated with a set of base functions which are pre-selected or generated during the previous order of approximations, while the corresponding coefficients are solved from deformation equations specified by the nonlinear equation itself and auxiliary residue functions. The convergence region, rate and final accuracy of the homotopy are controlled by a residue-regulating vector and an expansion threshold. General procedures of implementing RRHM are demonstrated using the Duffing and Van der Pol-Duffing oscillators, where approximate solutions containing abundant frequency components are successfully obtained, yielding significantly better convergence rate and performance stability compared to the other conventional homotopy-based methods.

PENYELESAIAN PERSAMAAN POISSON MENGGUNAKAN METODE HOMOTOPI PERTUBASI

In this paper, we discuss the solution of the Poisson equation with some initial condition. We use the homotopy pertubation method to get the solution.. The homotopy pertubation method is a combination of the homotopy method and the pertubation method. The solution of the equation is assumed to be in the form of a power series. The result is by using the homotopy pertubation method for the diffution equation, the solution is the same with the exact solution.

Optimization for Far-distance Cooperative Rendezvous with Multiple Direction-fixed Thrusts

The continuous-thrust far-distance cooperative rendezvous problem between two spacecraft is investigated. The indirect optimization method, based on Pontryagin’s maximum principle (PMP), is applied to optimize fuel consumption. To overcome the difficulty in nonsmooth integration caused by the bang-bang control, the homotopy method is adopted to solve the fuel-optimal problem from a related energy-optimal problem. The quantum-behaved particle swarm optimization (QPSO) algorithm is used to obtain the energy-optimal solutions. The energy-optimal solutions are used as the initial values for the homotopic procedure to obtain the fuel-optimal solutions and optimal bang-bang control law. A hybrid algorithm combined homotopy method and sequential quadratic programming (SQP) algorithm is proposed. This hybrid algorithm can effectively obtain feasible optimal solutions even though the indirect optimization method exhibits a narrow convergence domain. Simulations of high-thrust and low-thrust rendezvous problems are provided and the proposed hybrid algorithm is verified. Moreover, the necessity of radial thrust is investigated.

Correction to: A sequential homotopy method for mathematical programming problems

A correction to this paper has been published: https://doi.org/10.1007/s10107-021-01668-5

SIR Model with Homotopy to Predict Corona Cases

In this chapter, we will discuss SIR model to study the spread of COVID-2019 pandemic of India. We will give the prediction of corona cases using homotopy method. The HM is a method for solving the ordinary differential equations. The SIR model consists of three ordinary differential equations. In this study, we have used the data of COVID-2019 Outbreak of India on 20 Jan 2021. In this data, Recovered is 102656163, Active cases are 189245 Susceptible persons are 189347782 for the experimental purpose. Data about a wide variety of infectious diseases has been analyzed with the help of SIR model. Therefore, this model has been already well tested for infectious diseases by various scientists and researchers.