Homotopy Equation Research Articles

A GOOD INITIAL GUESS FOR APPROXIMATING NONLINEAR OSCILLATORS BY THE HOMOTOPY PERTURBATION METHOD

A good initial guess and an appropriate homotopy equation are two main factors in applications of the homotopy perturbation method. For a nonlinear oscillator, a cosine function is used in an initial guess. This article recommends a general approach to construction of the initial guess and the homotopy equation. Duffing oscillator is adopted as an example to elucidate the effectiveness of the method.

On free abelian categories for theorem proving

We give a computational approach to theorem proving in homological algebra . This approach is based on computations in the free abelian category of an additive category A . We show that the free abelian category is amenable to explicit computations whenever we can decide homotopy equations in A . As some consequences of our investigations, we recover Dowker's explicit formula for the connecting homomorphism ∂ in the snake lemma, we find a universal sense in which ∂ is unique, and we give a refined version of the 5-lemma.

Explicit Solutions to Large Deformation of Cantilever Beams by Improved Homotopy Analysis Method I: Rotation Angle

An improved homotopy analysis method (IHAM) is proposed to solve the nonlinear differential equation, especially for the case when nonlinearity is strong in this paper. As an application, the method was used to derive explicit solutions to the rotation angle of a cantilever beam under point load at the free end. Compared with the traditional homotopy method, the derivation includes two steps. A new nonlinear differential equation is firstly constructed based on the current nonlinear differential equation of the rotation angle and the auxiliary quadratic nonlinear differential equation. In the second step, a high-order non-linear iterative homotopy differential equation is established based on the new non-linear differential equation and the auxiliary linear differential equation. The analytical solution to the rotation angle is then derived by solving this high-order homotopy equation. In addition, the convergence range can be extended significantly by the homotopy–Páde approximation. Compared with the traditional homotopy analysis method, the current improved method not only speeds up the convergence of the solution, but also enlarges the convergence range. For the large deflection problem of beams, the new solution for the rotation angle is more approachable to the engineering designers than the implicit exact solution by the Euler–Bernoulli law. It should have significant practical applications in the design of long bridges or high-rise buildings to minimize the potential error due to the extreme length of the beam-like structures.

Explicit Solution to Large Deformation of Cantilever Beam by Improved Homotopy Analysis Method II: Vertical and Horizontal Displacements

Explicit solutions to vertical and horizontal displacements are derived for large deformation of a cantilever beam under point load at the free end by an improved homotopy analysis method (IHAM). Quadratic and cubic nonlinear differential equations are adopted to construct more proficient nonlinear equations for vertical and horizontal displacements respectively combined with their currently available nonlinear displacement equations. Higher-order nonlinear iterative homotopy equations are established to solve the vertical and horizontal displacements by combining simultaneous equations of the constructed nonlinear equations and the auxiliary linear equations. The convergence range of vertical displacement is extended by the homotopy-Páde approximation. The explicit solutions to the vertical and horizontal displacements are in favorable agreements with the respective exact solutions. The convergence ranges for a relative error of 1% by the improved homotopy analysis method for vertical and horizontal displacements increases by 60% and 7%, respectively. These explicit formulas are helpful in practical engineering design for very slender structures, such as high-rise buildings and long bridges.

A Reliable Treatment for Nonlinear Differential Equations

In this paper, we use the concept of homotopy, Laplace transform, and He’s polynomials, to propose the auxiliary Laplace homotopy parameter method (ALHPM). We construct a homotopy equation consisting on two auxiliary parameters for solving nonlinear differential equations, which switch nonlinear terms with He’s polynomials. The existence of two auxiliary parameters in the homotopy equation allows us to guarantee the convergence of the obtained series. Compared with numerical techniques, the method solves nonlinear problems without any discretization and is capable to reduce computational work. We use the method for different types of singular Emden–Fowler equations. The solutions, constructed in the form of a convergent series, are in excellent agreement with the existing solutions.

Thermophysical properties of pure gases and mixtures at temperatures of 300–30 000 K and atmospheric pressure: thermodynamic properties and solution of equilibrium compositions

The equilibrium compositions and thermodynamic properties (density, enthalpy, etc at constant pressure) of plasma of pure gases and mixtures under local thermodynamic nonequilibrium have been calculated in this paper. The homotopy Levenberg–Marquardt algorithm was proposed to accurately solve nonlinear equations with singular Jacobian matrices, and is constructed by the Saha equation and Guldberg–Waage equation combined with mass conservation, the electric neutrality principle and Dalton’s partial pressure law, to solve the problem of dependence on the initial value in the process of iteration calculation. In this research, the equations at a higher temperature were solved and used as the auxiliary equations, and the homotopy control parameters’ sequence of the homotopy equations was selected by equal ratios. For auxiliary equations, the iterative initial value was obtained by assuming that there were only the highest-valence atomic cations and electrons at this temperature, and the plasma equilibrium composition distribution with the required accuracy was ultimately solved under the current conditions employing the Levenberg–Marquardt algorithm. The control parameter sequence was arranged according to the geometric sequence and the homotopy step was gradually shortened to ensure continuity of the homotopy process. Finally, the equilibrium composition and thermodynamic properties of pure N2, Mg(30%)–CO2(70%) and Mg(40%)–CO(50%)–N2(10%) mixture plasma at atmospheric pressure were calculated and the calculation process of some specified temperatures was shown and analyzed. The calculation accuracy of equilibrium composition is higher than other findings in the literature. The results for the thermodynamic properties are in good agreement with data reported by the literature.

Design of anti-intrusion storage system for agricultural products sales data based on multi layer cross entropy

In view of the existing system in the process of agricultural product sales data storage at this stage, data intrusion detection is not carried out, which leads to the reduction of data storage efficiency and the increase of CPU occupancy. This paper proposes an anti-intrusion storage system of agricultural products sales data based on multi-layer cross entropy. The system the regional chain of agricultural products through the fabric alliance framework. Combined with the actual needs of the system, it carries out membership registration and management through go language. The error function in the original algorithm is modified, and the multi-layer cross entropy function is used to replace the variance function to ensure the convergence of the algorithm and optimize the convergence process. At the same time, the homotopy equation and momentum term are used to improve the weights once and twice to realize the anti-intrusion detection of agricultural product sales data and ensure the storage security of agricultural product sales data. The Comparison results of storage efficiency of different systems shows 98.5% of accuracy. Simulation results show that the proposed method can effectively improve the storage efficiency and reduce the CPU occupancy. • Proposed an anti-intrusion storage system of agricultural products sales data based on multi-layer cross entropy. • The regional chain of agricultural products is constructed through fabric alliance framework. • Comparison results of storage efficiency of different systems shows 98.5% of accuracy. • Simulation results show the proposed method effectively improve the storage efficiency and reduce CPU occupancy.

A new modification of the homotopy perturbation method

This study proposes a new modification of the homotopy perturbation method. A new parameter alpha is introduced into the homotopy equation in order to improve the results and accuracy. An optimal analysis identifies the parameter alpha, aimed at improving the solutions. A comparative analysis of the proposed method reveals that the new method presents results with higher degree of accuracy and precision than the classic homotopy perturbation method. Absolute error analysis shows the convenience of the proposed method, providing much smaller errors. Two examples are presented: Duffing and Van der pol’s nonlinear oscillators to demonstrate the efficiency, accuracy, and applicability of the new method.

Probing Homotopy Continuation Method for Solving Nonlinear Magnetic Network Equations

Nonlinear magnetic network is an effective model to solve the magnetic field of the motor. For the strong nonlinearity of magnetic network parameters, it is difficult to solve the nonlinear magnetic network equations. To overcome the divergent issue for solving the nonlinear magnetic network equations, a new method, the homotopy continuation method, is proposed. Based on the benchmark model of Testing Electromagnetic Analysis Methods Problem 20 (TEAM P20), the two-dimensional nonlinear magnetic network model is established. Next, homotopy equations of nonlinear magnetic network equations are derived, and they are solved by the homotopy continuation method. So that solving the nonlinear magnetic network equations is transformed into solving a set of homotopy equations. Then, the magnetic flux density of iron core with different saturation levels is calculated using the simple iteration method, the relaxation iteration method, Newton-Raphson method, and the homotopy continuation method, respectively. The results and convergence performance of the four methods are compared. It is proved that the proposed method can effectively expand the convergence domain and simplify the calculation process. The magnetic flux density calculated by the homotopy continuation method is compared with the finite element method in the supersaturated state, and the results are in good agreement. The relative error of the magnetic flux density of the core is less than 5%, which verifies the correctness of the proposed algorithm.

EFFECT OF FRACTIONAL DERIVATIVE PROPERTIES ON THE PERIODIC SOLUTION OF THE NONLINEAR OSCILLATIONS

A periodic solution of the time-fractional nonlinear oscillator is derived based on the Riemann–Liouville definition of the fractional derivative. In this approach, the particular integral to the fractional perturbed equation is found out. An enhanced perturbation method is developed to study the forced nonlinear Duffing oscillator. The modified homotopy equation with two expanded parameters and an additional auxiliary parameter is applied in this proposal. The basic idea of the enhanced method is to apply the annihilator operator to construct a simplified equation freeness of the periodic force. This method makes the solution process for the forced problem much simpler. The resulting equation is valid for studying all types of possible resonance states. The outcome shows that this alteration method overcomes all shortcomings of the perturbation method and leads to obtain a periodic solution.

Stability Analysis of a Damped Nonlinear Wave Equation

The current manuscript is concerned with extracting an analytical approximate periodic solution of a damped cubic nonlinear Klein-Gordon equation. The Riemann-Liouville fractional calculus is utilized to obtain an analytic approximate solution. The Homotopy technique is absorbed in the multiple time-spatial scales. The approved scheme yields a generalization of the Homotopy equation; whereas, two different small parameters are adapted. The first parameter concerns with the temporal perturbation, simultaneously, the second one is accompanied by the spatial one. Therefore, the analytic approximate solution needs the two perturbation expansions. This approach conducts more advantages in handling the classical multiple scales method. Furthermore, the initial conditions are included throughout the multiple scale method to achieve a special solution of the governing equation of motion. The analysis ends up deriving two first-order equations within the extended variables and their actual solution is achieved. The procedure adopted here is very promising and powerful in managing similar numerous nonlinear problems arising in physics and engineering. Furthermore, the linearized stability of the corresponding ordinary Duffing differential equation is analyzed. Additionally, some phase portraits are shown.

Li–He’s modified homotopy perturbation method coupled with the energy method for the dropping shock response of a tangent nonlinear packaging system

This paper couples Li–He’s homotopy perturbation method with the energy method to obtain an approximate solution of a tangent nonlinear packaging system. A higher order homotopy equation is constructed by adopting the basic idea of the Li–He’s homotopy perturbation method. The energy method is used to improve the maximal displacement and the frequency of the system to an ever higher accuracy. Comparison with the numerical solution obtained by the Runge–Kutta method shows that the shock responses of the system solved by the new method are more effective with a relative error of 0.15%.

A homotopy continuation method for nonlinear electric field analysis

Electric field analysis is an important basis for the insulation design of high-voltage direct-current equipment. The resistivity is a strong function of temperature and electric field, which may lead to the divergence of a numerical solution, and the initial approximations are the key to convergence. Therefore, a novel approach for nonlinear electric field indirect coupling analysis is developed. This algorithm converts the original nonlinear electric field into homotopy equations under a specific parameter, and uses the numerical continuation method to gradually approximate the true solution. Taking a fictive ±200 kV direct-current cable extrusion molded joint as an example, two homotopy equations are constructed to successfully calculate the nonlinear electric field based on the numerical continuation method combined with Steffensen iterative formula. On the contrary, divergence occurs in the conventional indirect coupling calculation method, indicating the superiority of this approach. Subsequently, some factors affecting the convergence of the homotopy continuation method are discussed. Finally, the calculated results and the solution times using the presented approach are compared with the direct coupling method by COMSOL. The proposed algorithm can enlarge the region of convergence in nonlinear electric field analysis based on the indirect coupling method and effectively overcome the divergent issue due to the inappropriate choice of initial approximations.

Asymptotic analysis to free-convective boundary-layer problem by homotopy renormalization method

In this paper, the homotopy renormalization (HTR) method is applied to investigate a free-convective boundary-layer problem arising from the sheet (or fiber) manufacturing, which is modeled by a system of nonlinear ordinary differential equations. By taking the inhomogeneous variable coefficient homotopy equations, the explicit asymptotic solutions satisfying the boundary conditions are given. Moreover, the comparisons with the numerical results show that our explicit function solutions have good performances in both local and large scales.

Homotopy method for solving generalized Nash equilibrium problem with equality and inequality constraints

In this paper, we utilize a new homotopy method to solve generalized Nash equilibrium problem with equality and inequality constraints on unbounded sets. Based on the existing homotopy method, we establish a new homotopy equation by introducing a suitable perturbation on the equality constraint, the existence and the global convergence of homotopy path under certain assumptions have also been proved. In the proposed method, the initial point only needs to satisfy the inequality constraints. Compared with the existing homotopy method, this method expands the scope of the initial points and provides the convenience for solving generalized Nash equilibrium problem. The numerical results illustrate the effectiveness of this method.

A New Proof for Global Convergence of a Smoothing Homotopy Method for the Nonlinear Complementarity Problem

In this paper, we propose a new proof for smoothing homotopy method based on the Fischer–Burmeister function to solve the nonlinear complementarity problem under a nonmonotone solution condition. Under this assumption condition imposed on the defined mapping [Formula: see text], global convergence of a smooth curve determined by the referred homotopy equation is established for almost all initial points in [Formula: see text] and it is actually regarded as an interior point method. Besides, if the initial point is expanded to [Formula: see text], the global convergence of the homotopy method is ensured under a similar condition. The numerical results are reported and illustrate that the method is efficient for some nonlinear complementarity problems.

Global convergence analysis of the aggregate constraint homotopy method for nonlinear programming problems with both inequality and equality constraints

In this paper, we construct appropriate aggregate mappings and a new aggregate constraint homotopy (ACH) equation by converting equality constraints to inequality constraints and introducing two variable parameters. Then, we propose an ACH method for nonlinear programming problems with inequality and equality constraints. Under suitable conditions, we obtain the global convergence of this ACH method, which makes us prove the existence of a bounded smooth path that connects a given point to a Karush–Kuhn–Tucker point of nonlinear programming problems. The numerical tracking of this path can lead to an implementable globally convergent algorithm. A numerical procedure is given to implement the proposed ACH method, and the computational results are reported.

Exact solutions and asymptotic solutions of one-dimensional domain walls in nonlinearly coupled system

We use homotopy renormalization method to investigate one-dimensional radial domain wall modes in the system of two nonlinear Schrodinger or Gross–Pitaevskii equations, which are coupled by linear mixing and by nonlinear XPM (cross-phase-modulation). The asymptotic solutions satisfying the boundary conditions are given by taking the inhomogeneous variable coefficient homotopy equations. To our knowledge, this is the first time that the inhomogeneous variable coefficient homotopy equations are taken. Moreover, the comparisons with numerical results show that our explicit function solutions have good performances in both local and large scale. In particular, in some conditions with special parameters, the asymptotic solutions are actually the exact solutions.

Application of New Homotopy Perturbation Method for Solving Partial Differential Equations

In this paper, a novel technique, that is to read, the New Homotopy Perturbation Method (NHPM) is utilized for solving a linear and non-linear differential equations and integral equations. The two most important steps in the application of the new homotopy perturbation method are to invent a suitable homotopy equation and to choose a suitable initial conditions. Comparing between the effects of the method (NHPM), is given exact solution, and the method (HPM), is given approximate solution, in this paper, we make some instances are provided to prove the ability of the method (NHPM). Show that the method (NHPM) is valid and effective, easy and accurate in solving linear and nonlinear differential equations, compared with the Homotopy Perturbation Method (HPM).

Asymptotic Analysis of a Nonlinear Problem on Domain Boundaries in Convection Patterns by Homotopy Renormalization Method

Abstract In this article, using the homotopy renormalization method, the asymptotic analysis to a nonlinear problem on domain boundaries in convection patterns are given. In particular, by taking a variable coefficient homotopy equation, the global asymptotic solutions satisfying boundary conditions are obtained. These results are better than the existing analytic approximation solutions.