Homotopy Analysis Method for Ablowitz–Ladik Lattice

In this note, the hybrid method (combination of the homotopy perturbation method (HPM) and the Gauss elimination method (GEM)) is developed as a semi-analytical solution for the first kind system of Cauchy-type singular integral equations (CSIEs) with constant coefficients. Before applying the HPM, we have to first reduce the system of CSIEs into a triangle system of algebraic equations using GEM, which is then carried out using the HPM. Using the theory of the bounded, unbounded and semi-bounded solutions of CSIEs, we are able to find inverse operators for the system of CSIEs of the first kind. A stability analysis and convergent of the proposed method has been conducted in the weighted Lp space. Moreover, the proposed method is proven to be exact in the Holder class of functions for the system of characteristic SIEs for any type of initial guess. For each of the four cases, several examples are provided and examined to demonstrate the proposed method’s validity and accuracy. Obtained results are compared with the Chebyshev collocation method and modified HPM (MHPM). Example 3 reveals that the error term of the MHPM is slightly superior to that of the HPM. One of the features of the proposed method is that it can be solved as a complex-valued system of CSIEs. Numerical results revealed that the hybrid method dominates others.

Semi-exact solution of elastic non-uniform thickness and density rotating disks by homotopy perturbation and Adomian's decomposition methods. Part I: Elastic solution

An extend of He's homotopy perturbation method (HPM) is used for finding a new approximate and exact solutions of nonlinear difference differential equations arising in mathematical physics. To illustrate the effectiveness and the advantage of the proposed method, two models of nonlinear difference equations of special interest in physics are chosen, namely, Ablowitz–Ladik lattice equations and Relativistic Toda lattice difference equations. Comparisons are made between the results of the proposed method and exact solutions. The results show that the HPM is a attracted method in solving the differential difference equations (DDEs). The proposed method will become a much more interesting method for solving nonlinear DDEs in science and engineering.

This paper investigates the parametric resonance of electrostatically actuated MicroElectroMechanicalSystems (MEMS) cantilever resonators. The electrostatic force is modeled to include fringe effect. The MEMS consists of a cantilever over a parallel ground plate and an AC voltage between them. The actuation frequency is near first natural frequency of the cantilever beam. This leads to parametric resonance. It is of interest to investigate the amplitude frequency response of MEMS cantilever resonators. This paper uses the Homotopy Analysis Method (HAM), which is able to capture nonlinear behaviors for higher amplitudes, large parameters, and strong nonlinearities. The base method used for comparison in this work is the method of multiple scales (MMS). MMS is a perturbation method. It requires a relatively short computational time for simulations. Although the CPU time is advantageous, MMS is only accurate for weak nonlinearities and low amplitudes. It is in the interest to compare how well HAM captures the softening behavior of this system as opposed to MMS. In this paper the influences of Casimir forces and Van der Waals effects are included. Electrostatic, Van der Waals and Casimir forces are nonlinear. HAM is a deformation technique that continuously deforms the initial guess, provided to the procedure, to the exact solution. In this work the first and second order deformation equations are constructed for the equation of motion governing the behavior of the MEMS cantilever beam. In the first order deformation, HAM deviates from the solution obtained by MMS. This deviation demonstrates the power of the method to capture the softening behavior more accurately than MMS even at the 1st order deformation HAM. In the second order deformation construction, the HAM’s solution softens more than the previous, demonstrating that higher order deformation approximations result in higher accuracy. In the second order deformation, HAM contains the convergence control parameter. This parameter is chosen via the c0 curve approach. Up to 2nd order HAM deformations are evaluated for this paper. These higher order homotopy deformation solutions were developed and automated symbolically in the software Mathematica and tested numerically using Matlab software.

Purpose – The purpose of this paper is to consider analytical solution of wave system in Rn with coupling controllers by using the homotopy perturbation method (HPM).Design/methodology/approach – HPM is applied to the system of linear partial differential equations, i.e. the system of waves in the two‐dimensional version of system equations (1) and (2). This problem is motivated by an analogous problem in ordinary differential equations for coupled oscillators and has potential application in isolating a vibrating object from the outside disturbances. For example, rubber or rubber‐like materials can be used to either absorb or shield a structure from vibration. As an approximation, these materials can be modeled as distributed springs.Findings – In this paper, HPM was used to obtain analytical solution of wave system in with coupling controllers. The method provides the solutions in the form of a series with easily computable terms. Unlike other common methods for solving any physical problem, linear or n...

In this work, we consider the second order nonlinear integro-differential Equation (IDEs) of the Volterra-Fredholm type. One of the popular methods for solving Volterra or Fredholm type IDEs is the method of quadrature while the problem of consideration is a linear problem. If IDEs are nonlinear or integral kernel is complicated, then quadrature rule is not most suitable; therefore, other types of methods are needed to develop. One of the suitable and effective method is homotopy analysis method (HAM) developed by Liao in 1992. To apply HAM, we firstly reduced the IDEs into nonlinear integral Equation (IEs) of Volterra-Fredholm type; then the standard HAM was applied. Gauss-Legendre quadrature formula was used for kernel integrations. Obtained system of algebraic equations was solved numerically. Moreover, numerical examples demonstrate the high accuracy of the proposed method. Comparisons with other methods are also provided. The results show that the proposed method is simple, effective and dominated other methods.

Nonlinear dynamic equation is a common engineering model.There is not precise analytical solution for most of nonlinear differential equations.These nonlinear differential equations should be solved by using approximate methods.Classical perturbation methods such as LP method,KBM method,multi-scale method and the averaging method on weakly nonlinear vibration system is effective,while the strongly nonlinear system is difficult to apply.Approximate solutions of primary resonance for forced Duffing equation is investigated by means of homotopy analysis method (HAM).Different from other approximate computational method,the HAM is totally independent of small physical parameters,and thus is suitable for most nonlinear problems.The HAM provides a great freedom to choose base functions of solution series,so that a nonlinear problem may be approximated more effectively.The HAM provides us a simple way to adjust and control the convergence region of the series solution by means of an auxiliary parameter and the auxiliary function.Therefore,HAM not only may solve the weakly non-linear problems but also may be suitable for the strong non-linear problem.Through the approximate solution of forced Duffing equation with cubic non-linearity,the HAM and fourth order Runge-Kutta method of numerical solution were compared,the results show that the HAM not only can solve the steady state solution,but also can calculate the unsteady state solution,and has the good computational accuracy.

Homotopy perturbation method (HPM) is a semi-analytical technique for solving linear as well as nonlinear ordinary/partial differential equations. The method may also be used to solve a system of coupled linear and nonlinear differential equations. This chapter first illustrates the basic idea of the HPM. It uses the present method to solve a nonlinear ordinary differential equation, a nonlinear partial differential equation, and a coupled nonlinear system of partial differential equations. It is worth mentioning that the HPM can also be used for handling linear ordinary and partial differential equations. The test problems demonstrated in the chapter confirm that the HPM is an efficient method for solving linear/nonlinear ordinary, partial, and coupled differential equations. The series usually converges with an increase in the number of terms, but one may not expect the compact form solution always.

PurposeThe purpose of this paper is to use the generalized differential transform method (GDTM) and homotopy perturbation method (HPM) for solving time‐fractional Burgers and coupled Burgers equations. The fractional derivatives are described in the Caputo sense.Design/methodology/approachIn these schemes, the solutions takes the form of a convergent series. In GDTM, the differential equation and related initial conditions are transformed into a recurrence relation that finally leads to the solution of a system of algebraic equations as coefficients of a power series solution. HPM requires a homotopy with an embedding parameter which is considered as a small parameter.FindingsThe paper extends the application and numerical comparison of the GDTM and HPM to obtain analytic and approximate solutions to the time‐fractional Burgers and coupled Burgers equations.Research limitations/implicationsBurgers and coupled Burgers equations with time‐fractional derivative used.Practical implicationsThe implications include traffic flow, acoustic transmission, shocks, boundary layer, the steepening of the waves and fluids, thermal radiation, chemical reaction, gas dynamics and many other phenomena.Originality/valueThe numerical results demonstrate the significant features, efficiency and reliability of the two approaches. The results show that HPM is more promising, convenient, and computationally attractive than GDTM.

In this paper, we study on the initial guess in homotopy analysis method. To illustrate our point, we applied to homotopy analysis method to solving the (2+1)dimensional boussinesq equation and (3+1) dimensional KP equation. Comparisons are made between the exact solution and the approximation of homotopy analysis method. Both of the analytic solutions agree well with the exact solutions. The results show that homotopy analysis method is a potential analytic method in solving nonlinear problems.

In this paper, a combined form of the Laplace transform method with the homotopy perturbation method (HPTM) is applied to solve nonlinear systems of partial differential equations viz. the system of third order KdV Equations and the systems of coupled Burgers’ equations in one- and two- dimensions. The nonlinear terms can be easily handled by the use of He’s polynomials. The results shows that the HPTM is very efficient, simple and avoids the round-off errors. Four test examples are considered to illustrate the present scheme. Further the results are compared with Homotopy perturbation method (HPM) which shows that this method is a suitable method for solving systems of partial differential equations.

In this article, He’s homotopy perturbation method was applied in a variant way to solve the system of Volterra integral equations of first kind. The results reveal that the proposed approach is very efficient for handling such system of integral equations. Some examples are given to show the ability of the proposed modification.   Key words: Integral equations, Volterra integral equations of first kind, homotopy perturbation method.

In this paper, the nonlinear free vibrations of thin symmetric and non-symmetric cross-ply composite plates subjected to biaxial initial stresses are investigated. Because of their excellent properties such as specific strength and specific stiffness, composite plates have wide applications in aerospace and mechanical structures. Based on Von-Karman's strain-displacement relations and using Galerkin method, the nonlinear differential equation of free vibrations of initially stressed composite plate is obtained. This nonlinear equation is solved using two different analytical perturbation methods, namely method of multiple scales (MTS) and homotopy perturbation method (HPM), to analyze the nonlinear vibrations of initially stressed cross-ply composite plates. Effects of tensile and compressive biaxial initial stresses, initial vibration amplitude, thickness, and aspect ratios of the composite plates on the frequency behavior are investigated. The validity of the results is confirmed by making a comparison with those reported in the literature. According to the results, both analytical solutions show increasing trends for natural frequency parameters by increasing normal initial stresses. Regardless of the value of initial biaxial stresses, for both symmetric and non-symmetric plates, the results of MTS and HPM are in close agreement for the smallest initial amplitude. However, for compressive initial stresses, by increasing initial amplitude ratios, the discrepancies between the results of HPM and MTS increase for symmetric and non-symmetric plates. Although HPM includes less computational effort (smaller length of formulation) than MTS, the linear-to-nonlinear frequency ratios obtained using MTS method become closer to those obtained by HPM as initial vibration amplitude is decreased and initial stress is increased.

The aim of this work is to carry out a comparative analysis of a more recent homotopy perturbation method and the method of integral manifolds for solving the problem of thermal explosion in a combustible mixture containing vaporizing fuel droplets of different radii, i.e., polydisperse fuel spray. The model under consideration, known as parcel approximation, is a system of nonlinear ordinary differential equations. Our results include a comparative analysis between the following models: (1) the parcel model, which is solved numerically, by applying the method of integral manifolds for zero-order and third-order approximation and by applying the homotopy perturbation method, and (2) the continuous model, which is solved by Runge--Kutta methods. We compare these methods applied to experimental fuel spray data such as $n$-Butanol, $n$-Decane, and $n$-Heptane. Although these methods are in agreement with each other, application of the homotopy perturbation method to these systems of equations shows rapid con...

In this article we have introduced the mathematical theory of multistep optimal homotopy asymptotic method (MOHAM). The proposed method is implemented to different models having system of partial differential equations (PDEs). The results obtained by proposed method are compared with Homotopy Analysis Method (HAM) and closed form solutions. The comparisons of these results show that MOHAM is simpler in applicability, effective, explicit, control the convergence through optimal constants, involve less computational work. The MOHAM is independent of the assumption of initial conditions and small parameters like Homotopy Perturbation Method (HPM), HAM, Variational Iteration Method (VIM), Adomian Decomposition Method (ADM) and Perturbation Method (PM).Â