Homotopy Technique Research Articles

Homotopy perturbation technique for improving solutions of large quadratic eigenvalue problems: Application to friction-induced vibration

This paper puts forward a projection technique for accurately calculating solutions of large Quadratic Eigenvalue Problem. The aim here is to stabilize the complex eigensolutions whilst reducing residual errors, especially when considering significant damping contribution or asymmetric stiffness matrices. Hence, more confident results can be obtained in the frequency band of interest. To achieve this, high order modes, calculated using the homotopy perturbation technique, are introduced in the projection step of the classical method. This numerical proposal is a generalization of the classical projection, based only on normal modes of the associated undamped problem. To evaluate the efficiency of the suggested method, a finite element application dedicated to a friction-induced vibration problem is investigated.

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.

Iteratively weighted thresholding homotopy method for the sparse solution of underdetermined linear equations

Recently, iteratively reweighted methods have attracted much interest in compressed sensing, outperforming their unweighted counterparts in most cases. In these methods, decision variables and weights are optimized alternatingly, or decision variables are optimized under heuristically chosen weights. In this paper, we present a novel weighted $l_1$-norm minimization problem for the sparsest solution of underdetermined linear equations. We propose an iteratively weighted thresholding method for this problem, wherein decision variables and weights are optimized simultaneously. Furthermore, we prove that the iteration process will converge eventually. Using the homotopy technique, we enhance the performance of the iteratively weighted thresholding method. Finally, extensive computational experiments show that our method performs better in terms of both running time and recovery accuracy compared with some state-of-the-art methods.

Parameter identification of Bouc–Wen type hysteresis models using homotopy optimization

Structural members exhibit hysteretic behavior under cyclic loading. Among the hysteresis models available in the literature, the differential model proposed by Bouc-Wen is most widely used, owing to its robustness. This model involves many parameters that define the shape of the hysteresis loops. Estimating these unknown parameters is an identification problem that can be tackled by optimization algorithms by using prediction error as the objective function. Stochastic methods like simulated annealing and genetic algorithms can help find global minima but at a high computational cost. Here, the homotopy technique is employed to identify the unknown parameters. The efficiency of this technique in identifying the parameters of the Bouc–Wen model is demonstrated with examples. The present approach is then compared with global optimization methods, such as genetic algorithms and particle swarm optimization techniques. Numerical results confirm that the homotopy method is superior in terms of computational effort and convergence efficiency.

Low-thrust spacecraft trajectory optimization via a DNN-based method

Initial solution guess has a significant impact on the convergence of indirect methods, especially for continuous low-thrust trajectory optimization problems. In this study, an intelligent initial solution supplying approach based on deep neural networks (DNNs) is proposed to help achieve the fast generation of optimal trajectories for low-thrust orbit transfers. Energy-optimal and fuel-optimal trajectories with three different terminal conditions are considered. Based on the training dataset obtained by an indirect method, DNNs are constructed to approximate the solutions corresponding to different flight states. Based on the trained DNNs, an intelligent trajectory optimization method named DNN-based method is developed with the help of the homotopy technique. Numerical simulations are conducted to evaluate the performance of the proposed method on success rates and time consumptions. Simulation results demonstrate that the combination of traditional techniques and the new DNN technology can achieve the fast generation of low-thrust optimal trajectories with advantages on computational efficiency and reliability.

Homotopy Optimization of Microwave and Millimeter-Wave Filters Based on Neural Network Model

High-performance microwave and millimeter-wave filters’ design is a challenging task because the filter characteristic is rather sensitive to the variation of geometric dimensions and electrical sizes. A common practice in filter design is to optimize the design variables starting from a set of initial values. However, if the initial values are not sufficiently close to the optimal solution, the optimization often fails to provide any satisfactory result. To deal with this problem, for the first time, the homotopy method is introduced to microwave and millimeter-wave filters’ optimization problems in this article. The homotopy method formulates a series of intermediate optimization problems, which can guide the optimizer to approach the optimal solution for the target filter design. In this article, the artificial neural network (ANN) is adopted as the surrogate model to the time-consuming electromagnetic model to speed up the homotopy filter optimization process. Two design examples are given to demonstrate the homotopy optimization technique based on the ANN model, including an all-pole filter and a generalized Chebyshev filter with a frequency-dependent coupling. Both filters with optimized geometric dimensions are simulated, and the all-pole filter is fabricated and measured. The simulation and measurement results verify the accuracy of the ANN model and validate the homotopy optimization method.

Entropy analysis in nonlinearly convective flow of the Sisko model in the presence of Joule heating and activation energy: the Buongiorno model

This investigation deals with entropy generation optimization and an activation energy mechanism for nonlinear convective flow of a Sisko nanofluid due to a stretchable rotating disk. Heat transfer analysis is accomplished through nonlinear thermal radiation and non-uniform heat generation/absorption. The Joule heating effect is also considered with viscous dissipation. The nanofluid model includes Brownian motion and thermophoresis. Apposite transformations are endorsed to obtain the nonlinear-coupled ordinary differential equation system. The attained system endeavors for series solutions by the homotopy technique. Total entropy generation is analyzed through numerous flow variables. A comparative study of shear-thickening and shear-thinning fluids is also presented for the Bejan number, heat transfer rate, mass transfer Nusselt number, entropy generation, temperature, velocity and concentration. Here the entropy generation rate (irreversibility rate) is enhanced for the thermal radiation parameter while the reverse behavior is noted for the material parameter and Brinkman number. Furthermore, the behavior of the temperature profile and nanoparticle concentration will be displayed through a Brownian motion parameter and an activation energy parameter.

Computational analysis of thermally stratified mixed convective non-Newtonian fluid flow with radiation and chemical reaction impacts

The important goal of this article is to examine the thermally stratified flow of Oldroyd-B liquid induced by a stretchy sheet. Mathematical model of physical problem is situated with radiation and chemical reaction impacts. By help of similarity transformation, PDE’s are converted into ODE’s with dimensionless variables. Homotopy technique is employed for solving nonlinear ODE’s. Graphs are represented to notify the change of flow parameters, temperature and concentration parameters. Also mass and heat transfer rates are demonstrated through graphically. We observe that Thermal stratification constant reduces the heat transfer rate. On the other hand, thermophoretic and chemical reaction constants boost up the mass transfer rate.

Sparse Group Feature Selection by Weighted Thresholding Homotopy Method

In this paper, we investigate the sparse group feature selection problem, in which covariates posses a grouping structure sparsity at the level of both features and groups simultaneously. We reformulate the feature sparsity constraint as an equivalent weighted $l_1$ -norm constraint in the sparse group optimization problem. To solve the reformulated problem, we first propose a weighted thresholding method based on a dynamic programming algorithm. Then we improve the method to a weighted thresholding homotopy algorithm using homotopy technique. We prove that the algorithm converges to an $L$ -stationary point of the original problem. Computational experiments on synthetic data show that the proposed algorithm is competitive with some state-of-the-art algorithms.

The Vibrational Motion of a Dynamical System Using Homotopy Perturbation Technique

This paper outlines the vibrational motion of a nonlinear system with a spring of linear stiffness. Homotopy perturbation technique (HPT) is used to obtain the asymptotic solution of the governing equation of motion. The numerical solution of this equation is obtained using the fourth order Runge-Kutta method (RKM). The comparison between both solutions reveals high consistency between them which confirms that, the accuracy of the obtained solution using aforementioned perturbation technique. The time history of the attained solution is represented through some plots to reveal the good effect of the different parameters of the considered system on the motion at any instant. The conditions of the stability of the attained solution are presented and discussed.

Transportation of heat generation/absorption and radiative heat flux in homogeneous–heterogeneous catalytic reactions of non-Newtonian fluid (Oldroyd-B model)

BackgroundThis study addresses the three-dimensional (3D) stagnation point flow of non-Newtonian material (Oldroyd-B) with magnetohydrodynamics. Furthermore, Ohmic heating and radiative flux are used in the modeling of energy expression. The surface is convectively heated. Equal strengths of diffusions for homogeneous and heterogeneous reactions are counted. Results are computed and presented graphically. Heat transfer rate is numerically discussed through table. MethodHere the nonlinear differential system first converted into ordinary differential equation through implementation of appropriate similarity variables. The obtained ordinary system is tackled through homotopy technique for convergent solutions. The outcomes are presented through different graphs and discussed in section six. OutcomesThe remarkable results of the present communication which is obtained from the semi analytical method i.e., “homotopy method” is summarized as(i) Opposite impact is noticed for velocity components i.e., (f′(ξ), g(ξ)) for rising fluid parameter and rotation parameter.(ii) The temperature is direct relation with Biot number and radiative variable.(iii) Heat transfer rate is more versus Biot number and radiation variable.(iv) The concentration field shows opposite impact versus homogeneous and heterogeneous parameters.

Limiting shifted homotopy in higher-spin theory and spin-locality

Higher-spin vertices containing up to quintic interactions at the LagTangian level are explicitly calculated in the one-form sector of the non-linear unfolded higher-spin equations using a \U0001d6fd →-∞-shifted contracting homotopy introduced in the paper. The problem is solved in a background independent way and for any value of the complex parameter \U0001d702 in the higher-spin equations. All obtained vertices are shown to be spin-local containing a finite number of derivatives in the spinor space for any given set of spins. The vertices proportional to \U0001d7022 and {overline{eta}}^2 are in addition ultra-local, i.e., zero-forms that enter into the vertex in question are free from the dependence on at least one of the spinor variables y or overline{y} . Also the \U0001d7022 and {overline{eta}}^2 vertices are shown to vanish on any purely gravitational background hence not contributing to the higher-spin current interactions on AdS4. This implies in particular that the gravitational constant in front of the stress tensor is positive being proportional to eta overline{eta} . It is shown that the \U0001d6fd-shifted homotopy technique developed in this paper can be reinterpreted as the conventional one but in the \U0001d6fd-dependent deformed star product.

A multigrid–homotopy method for nonlinear inverse problems

In the present contribution, we develop a novel method combining the multigrid idea and the homotopy technique for nonlinear inverse problems, in which the forward problems are modeled by some forms of partial differential equations. The method first attempts to use the multigrid method to decompose the original inverse problem into a sequence of sub-inverse problems which depend on the grid variables and are solved in proper order according to the grid size from the coarsest to the finest, and then carries out the inversion on the coarsest grid by the homotopy method. The strategy may give a rapidly and globally convergent method. As a practical application, this method is used to solve the nonlinear inverse problem of a nonlinear convection–diffusion equation, which is the saturation equation within the two-phase porous media flow. We demonstrate the effectiveness and merits of the multigrid–homotopy method on two actual model problems.

Cattaneo–Christov double-diffusion model for viscoelastic nanofluid with activation energy and nonlinear thermal radiation

PurposeThe purpose of this paper is to explore the novel aspects of activation energy in the nonlinearly convective flow of Walter-B nanofluid in view of Cattaneo–Christov double-diffusion model over a permeable stretched sheet. Features of nonlinear thermal radiation, dual stratification, non-uniform heat generation/absorption, MHD and binary chemical reaction are also evaluated for present flow problem. Walter-B nanomaterial model is employed to describe the significant slip mechanism of Brownian and thermophoresis diffusions. Generalized Fourier’s and Fick’s laws are examined through Cattaneo–Christov double-diffusion model. Modified Arrhenius formula for activation energy is also implemented.Design/methodology/approachSeveral techniques are employed for solving nonlinear differential equations. The authors have used a homotopy technique (HAM) for our nonlinear problem to get convergent solutions. The homotopy analysis method (HAM) is a semi-analytical technique to solve nonlinear coupled ordinary/partial differential equations. The capability of the HAM to naturally display convergence of the series solution is unusual in analytical and semi-analytic approaches to nonlinear partial differential equations. This analytical method has the following great advantages over other techniques:It provides a series solution without depending upon small/large physical parameters and applicable for not only weakly but also strongly nonlinear problems.It guarantees the convergence of series solutions for nonlinear problems.It provides us a great choice to select the base function of the required solution and the corresponding auxiliary linear operator of the homotopy.Brief mathematical description of HAM technique (Liao, 2012; Mabood et al., 2016) is as follows. For a general nonlinear equation:(1)N[u(x)]=0,where N denotes a nonlinear operator, x the independent variables and u(x) is an unknown function, respectively. By means of generalizing the traditional homotopy method, Liao (1992) creates the so-called zero-order deformation equation:(2)(1−q)L[uˆ(x;q)−uo(x)]=qhH(x)N[uˆ(x;q)],here q∈[0, 1] is the embedding parameter, H(x) ≠ 0 is an auxiliary function, h(≠ 0) is a nonzero parameter, L is an auxiliary linear operator, uo(x) is an initial guess of u(x) and is an unknown function, respectively. It is significant that one has great freedom to choose auxiliary things in HAM. Noticeably, when q=0 and q=1, following holds:(3)uˆ(x;0)=uo(x)anduˆ(x;1)=u(x),Expanding in Taylor series with respect to (q), we have:(4)uˆ(x;q)=uo(x)+∑m=1∞um(x)qm,whereum(x)=1m!∂muˆ(x;q)∂qm|q=0.If the initial guess, the auxiliary linear operator, the auxiliary h and the auxiliary function are selected properly, then the series (4) converges at q=1, then we have:(5)u(x)=uo(x)+∑m=1+∞um(x).By defining a vector and differentiating Equation (2) m-times with respect to (q) and then setting q=0, we obtain the mth-order deformation equation:(6)L[uˆm(x)−χmum−1(x)]=hH(x)Rm[u→m−1],where:(7)Rm[u→m−1]=1(m−1)!∂m−1N[u(x;q)]∂qm−1|q=0andχm=|0m⩽11m>1.Applying L−1 on both sides of Equation (6), we get:(8)um(x)=χmum−1(x)+hL−1[H(x)Rm[u→m−1]].In this way, we obtain um for m ⩾ 1, at mth-order, we have:(9)u(x)=∑m=1Mum(x).FindingsIt is evident from obtained results that the nanoparticle concentration field is directly proportional to the chemical reaction with activation energy. Additionally, both temperature and concentration distributions are declining functions of thermal and solutal stratification parameters (P1) and (P2), respectively. Moreover, temperature Θ(Ω1) enhances for greater values of Brownian motion parameter (Nb), non-uniform heat source/sink parameter (B1) and thermophoresis factor (Nt). Reverse behavior of concentration ϒ(Ω1) field is remarked in view of (Nb) and (Nt). Graphs and tables are also constructed to analyze the effect of different flow parameters on skin friction coefficient, local Nusselt number, Sherwood numbers, velocity, temperature and concentration fields.Originality/valueThe novelty of the present problem is to inspect the Arrhenius activation energy phenomena for viscoelastic Walter-B nanofluid model with additional features of nonlinear thermal radiation, non-uniform heat generation/absorption, nonlinear mixed convection, thermal and solutal stratification. The novel aspect of binary chemical reaction is analyzed to characterize the impact of activation energy in the presence of Cattaneo–Christov double-diffusion model. The mathematical model of Buongiorno is employed to incorporate Brownian motion and thermophoresis effects due to nanoparticles.

Double-homotopy technique for fuel optimization of power-limited interplanetary trajectories

A double-homotopy technique is proposed for the fuel optimization problem of power-limited trajectories with consideration of both variable-specific-impulse and dual-specific-impulse. The thruster input power is smoothed using a sigmoid function. Consequently, the derivative of the thruster input power relative to the position vector becomes continuous. In order to deal with the discontinuous specific impulse of the dual-specific-impulse thruster, the logarithmic homotopy technique is applied to connect the dual-specific-impulse trajectory with the variable-specific-impulse trajectory. Therefore, the optimal dual-specific-impulse trajectory can be solved starting from a variable-specific-impulse trajectory. Missions exploiting the variable-specific-impulse thruster are compared to missions exploiting the dual-specific-impulse thruster. The numerical simulations validate the accuracy and effectiveness of the double-homotopy technique for solving power-limited fuel-optimal trajectories.

Stratified flow of sutterby fluid with homogeneous-heterogeneous reactions and Cattaneo-Christov heat flux

Purpose The purpose of this paper is to analyze the Sutterby fluid flow by a rotating disk with homogeneous-heterogeneous reactions. Inspection of heat transfer is through Cattaneo–Christov model. Stratification effect is also considered. Design/methodology/approach Nonlinear equations are solved by the homotopy technique. Findings Sutterby fluid flow by rotating disk is not considered yet. Here the authors intend to analyze it with Cattaneo–Christov heat flux and homogeneous-heterogeneous reactions. Thermal stratification is also taken into consideration. Originality/value No such work is yet done in the literature.

Thermally stratified flow of Jeffrey fluid with homogeneous-heterogeneous reactions and non-Fourier heat flux model

This article investigates the mixed convective flow of Jeffrey fluid near the axisymmetric stagnation point over an inclined permeable stretching cylinder. Analysis subjected to Cattaneo-Christov heat flux, thermal stratification and homogeneous-heterogeneous reactions are accounted. Suitable transformations are employed to obtain nonlinear ordinary differential system. Non-dimensional system is computed by Homotopy technique. Graphs and tables are constructed to analyze the influence of different flow parameters on temperature, concentration and velocity fields. The interpretation of skin friction coefficient is deliberated. It is noticed from obtained results that temperature is a decreasing function of thermal stratification parameter. Reverse behaviour of concentration is witnessed for higher estimations of homogeneous and heterogeneous parameters. Numerical results are compared with previous published results and found to be in good agreement for special casesof the emerging parameters.

Comparing the new fractional derivative operators involving exponential and Mittag-Leffler kernel

In this manuscript, we have proposed a comparison based on newly defined fractional derivative operators which are called as Caputo-Fabrizio (CF) and Atangana-Baleanu (AB). In 2015, Caputo and Fabrizio established a new fractional operator by using exponential kernel. After one year, Atangana and Baleanu recommended a different-type fractional operator that uses the generalized Mittag-Leffler function (MLF). Many real-life problems can be modelled and can be solved by numerical-analytical solution methods which are derived with these operators. In this paper, we suggest an approximate solution method for PDEs of fractional order by using the mentioned operators. We consider the Laplace homotopy transformation method (LHTM) which is the combination of standard homotopy technique (SHT) and Laplace transformation method (LTM). In this study, we aim to demonstrate the effectiveness of the aforementioned method by comparing the solutions we have achieved with the exact solutions. Furthermore, by constructing the error analysis, we test the practicability and usefulness of the method.

Effects of Radiative Electro-Magnetohydrodynamics Diminishing Internal Energy of Pressure-Driven Flow of Titanium Dioxide-Water Nanofluid due to Entropy Generation.

The internal average energy loss caused by entropy generation for steady mixed convective Poiseuille flow of a nanofluid, suspended with titanium dioxide (TiO2) particles in water, and passed through a wavy channel, was investigated. The models of thermal conductivity and viscosity of titanium dioxide of 21 nm size particles with a volume concentration of temperature ranging from 15 °C to 35 °C were utilized. The characteristics of the working fluid were dependent on electro-magnetohydrodynamics (EMHD) and thermal radiation. The governing equations were first modified by taking long wavelength approximations, which were then solved by a homotopy technique, whereas for numerical computation, the software package BVPh 2.0 was utilized. The results for the leading parameters, such as the electric field, the volume fraction of nanoparticles and radiation parameters for three different temperatures scenarios were examined graphically. The minimum energy loss at the center of the wavy channel due to the increase in the electric field parameter was noted. However, a rise in entropy was observed due to the change in the pressure gradient from low to high.

Comparison of Homotopy Perturbation Method with New Homotopy Perturbation Method

In this article, a new homotopy technique is presented for the mathematical analysis of finding the solution of partial differential equation and we compare the decision of using NHPM with those using HPM to view the efficacy and advantageousness of the new capability. The purpose of this study is to show that constructing the proper homotopy in a correct way determines the solution with less computational work than using the existing approach while supplying quantitatively reliable results. To illustrate the capability and reliability of the method examples are provided. Comparison of the results of applying NHPM with those of applying HPM reveal the effectiveness and convenience of the new technique.