Higher accuracy analytical approximations to a nonlinear oscillator with discontinuity by He's homotopy perturbation method

He's homotopy perturbation method is used to calculate higher-order approximate periodic solutions of a nonlinear oscillator with discontinuity for which the elastic force term is proportional to sgn(x). We find He's homotopy perturbation method works very well for the whole range of initial amplitudes, and the excellent agreement of the approximate frequencies and periodic solutions with the exact ones has been demonstrated and discussed. Only one iteration leads to high accuracy of the solutions with a maximal relative error for the approximate period of less than 1.56% for all values of oscillation amplitude, while this relative error is 0.30% for the second iteration and as low as 0.057% when the third-order approximation is considered. Comparison of the result obtained using this method with those obtained by different harmonic balance methods reveals that He's homotopy perturbation method is very effective and convenient.     

Application of energy balance method and variational iteration method to an oscillation of a mass attached to a stretched elastic wire

This letter applies energy balance method and variational iteration method to a nonlinear oscillation of a mass attached to a stretched wire. Comparison of the period of oscillation and obtained solutions with the exact one shows that both methods are very effective and convenient and quite accurate to nonlinear engineering problems.     

Analysis of modified Van der Pol’s oscillator using He’s parameter-expanding methods

In this work, a powerful analytical method, called He’s parameter-expanding methods (HPEM) is used to obtain the exact solutions of non-linear modified Van der Pol’s oscillator. The classical Van der Pol equation with delayed feedback and a modified equation where a delayed term provides the damping are considered. It is shown that one term in series expansions is sufficient to obtain a highly accurate solution, which is valid for the whole solution domain. Comparison of the obtained solution with those obtained using perturbation method shows that this method is effective and convenient to solve this problem. This method introduces a capable tool to solve this kind of non-linear problems.     

The residue harmonic balance for fractional order van der Pol like oscillators

When seeking a solution in series form, the number of terms needed to satisfy some preset requirements is unknown in the beginning. An iterative formulation is proposed so that when an approximation is available, the number of effective terms can be doubled in one iteration by solving a set of linear equations. This is a new extension of the Newton iteration in solving nonlinear algebraic equations to solving nonlinear differential equations by series. When Fourier series is employed, the method is called the residue harmonic balance. In this paper, the fractional order van der Pol oscillator with fractional restoring and damping forces is considered. The residue harmonic balance method is used for generating the higher-order approximations to the angular frequency and the period solutions of above mentioned fractional oscillator. The highly accurate solutions to angular frequency and limit cycle of the fractional order van der Pol equations are obtained analytically. The results that are obtained reveal that the proposed method is very effective for obtaining asymptotic solutions of autonomous nonlinear oscillation systems containing fractional derivatives. The influence of the fractional order on the geometry of the limit cycle is investigated for the first time.     

Efficient solutions to robust, semi-implicit discretizations of the immersed boundary method

The immersed boundary method is a versatile tool for the investigation of flow-structure interaction. In a large number of applications, the immersed boundaries or structures are very stiff and strong tangential forces on these interfaces induce a well-known, severe time-step restriction for explicit discretizations. This excessive stability constraint can be removed with fully implicit or suitable semi-implicit schemes but at a seemingly prohibitive computational cost. While economical alternatives have been proposed recently for some special cases, there is a practical need for a computationally efficient approach that can be applied more broadly. In this context, we revisit a robust semi-implicit discretization introduced by Peskin in the late 1970s which has received renewed attention recently. This discretization, in which the spreading and interpolation operators are lagged, leads to a linear system of equations for the interface configuration at the future time, when the interfacial force is linear. However, this linear system is large and dense and thus it is challenging to streamline its solution. Moreover, while the same linear system or one of similar structure could potentially be used in Newton-type iterations, nonlinear and highly stiff immersed structures pose additional challenges to iterative methods. In this work, we address these problems and propose cost-effective computational strategies for solving Peskin’s lagged-operators type of discretization. We do this by first constructing a sufficiently accurate approximation to the system’s matrix and we obtain a rigorous estimate for this approximation. This matrix is expeditiously computed by using a combination of pre-calculated values and interpolation. The availability of a matrix allows for more efficient matrix–vector products and facilitates the design of effective iterative schemes. We propose efficient iterative approaches to deal with both linear and nonlinear interfacial forces and simple or complex immersed structures with tethered or untethered points. One of these iterative approaches employs a splitting in which we first solve a linear problem for the interfacial force and then we use a nonlinear iteration to find the interface configuration corresponding to this force. We demonstrate that the proposed approach is several orders of magnitude more efficient than the standard explicit method. In addition to considering the standard elliptical drop test case, we show both the robustness and efficacy of the proposed methodology with a 2D model of a heart valve.     

A kernel-free boundary integral method for elliptic boundary value problems

This paper presents a class of kernel-free boundary integral (KFBI) methods for general elliptic boundary value problems (BVPs). The boundary integral equations reformulated from the BVPs are solved iteratively with the GMRES method. During the iteration, the boundary and volume integrals involving Green’s functions are approximated by structured grid-based numerical solutions, which avoids the need to know the analytical expressions of Green’s functions. The KFBI method assumes that the larger regular domain, which embeds the original complex domain, can be easily partitioned into a hierarchy of structured grids so that fast elliptic solvers such as the fast Fourier transform (FFT) based Poisson/Helmholtz solvers or those based on geometric multigrid iterations are applicable. The structured grid-based solutions are obtained with standard finite difference method (FDM) or finite element method (FEM), where the right hand side of the resulting linear system is appropriately modified at irregular grid nodes to recover the formal accuracy of the underlying numerical scheme. Numerical results demonstrating the efficiency and accuracy of the KFBI methods are presented. It is observed that the number of GMRES iterations used by the method for solving isotropic and moderately anisotropic BVPs is independent of the sizes of the grids that are employed to approximate the boundary and volume integrals. With the standard second-order FEMs and FDMs, the KFBI method shows a second-order convergence rate in accuracy for all of the tested Dirichlet/Neumann BVPs when the anisotropy of the diffusion tensor is not too strong.     

Approximate analytical solutions to a conservative oscillator using global residue harmonic balance method

In the current research paper, a conservative system comprising of a mass grounded by linear and nonlinear springs in series connection is studied. The equation of motion for the aforementioned system has been derived as a nonlinear ordinary differential equation with inertia and static–type cubic nonlinearities. The global residue harmonic balance method is applied to obtain an approximate analytical frequency and periodic solution of the problem. Using the obtained analytical expressions, the influences of the hardening and softening nonlinear spring on the non–dimensional frequency are investigated. The results show that developing the system nonlinearity leads the displacement of the mass and the deflection of linear spring to approach each other. Moreover, comparison of the results obtained using the proposed procedure with those achieved by other methods such as numerical method, variational iteration method and harmonic balance approach demonstrates the accuracy and advantages of the current approach.     

Compacton and solitary pattern solutions for nonlinear dispersive KdV-type equations involving Jumarie?s fractional derivative

In this Letter, the fractional variational iteration method using He?s polynomials is implemented to construct compacton solutions and solitary pattern solutions of nonlinear time-fractional dispersive KdV-type equations involving Jumarie?s modified Riemann-Liouville derivative. The method yields solutions in the forms of convergent series with easily calculable terms. The obtained results show that the considered method is quite effective, promising and convenient for solving fractional nonlinear dispersive equations. It is found that the time-fractional parameter significantly changes the soliton amplitude of the solitary waves.     

Sub-iteration leads to accuracy and stability enhancements of semi-implicit schemes for the Navier–Stokes equations

We present an iterative semi-implicit scheme for the incompressible Navier–Stokes equations, which is stable at CFL numbers well above the nominal limit. We have implemented this scheme in conjunction with spectral discretizations, which suffer from serious time step limitations at very high resolution. However, the approach we present is general and can be adopted with finite element and finite difference discretizations as well. Specifically, at each time level, the nonlinear convective term and the pressure boundary condition – both of which are treated explicitly in time – are updated using fixed-point iteration and Aitken relaxation. Eigenvalue analysis shows that this scheme is unconditionally stable for Stokes flows while numerical results suggest that the same is true for steady Navier–Stokes flows as well. This finding is also supported by error analysis that leads to the proper value of the relaxation parameter as a function of the flow parameters. In unsteady flows, second- and third-order temporal accuracy is obtained for the velocity field at CFL number 5–14 using analytical solutions. Systematic accuracy, stability, and cost comparisons are presented against the standard semi-implicit method and a recently proposed fully-implicit scheme that does not require Newton’s iterations. In addition to its enhanced accuracy and stability, the proposed method requires the solution of symmetric only linear systems for which very effective preconditioners exist unlike the fully-implicit schemes.     

Application of a modified rational harmonic balance method for a class of strongly nonlinear oscillators

An analytical approximate technique for conservative nonlinear oscillators is proposed. This method is a modification of the rational harmonic balance method in which analytical approximate solutions have rational form. This approach gives us the frequency of the motion as a function of the amplitude of oscillation. We find that this method works very well for the whole range of parameters, and excellent agreement of the approximate frequencies with the exact one has been demonstrated and discussed. The most significant features of this method are its simplicity and its excellent accuracy for the whole range of oscillation amplitude values and the results reveal that this technique is very effective and convenient for solving conservative truly nonlinear oscillatory systems with complex nonlinearities.     

Folded novel accurate analytical and semi-analytical solutions of a generalized Calogero–Bogoyavlenskii–Schiff equation

This paper studies the analytical and semi-analytic solutions of the generalized Calogero–Bogoyavlenskii–Schiff(CBS) equation. This model describes the(2 + 1)–dimensional interaction between Riemann-wave propagation along the y-axis and the x-axis wave. The extended simplest equation(ESE) method is applied to the model, and a variety of novel solitarywave solutions is given. These solitary-wave solutions prove the dynamic behavior of soliton waves in plasma. The accuracy of the obtained solution is verified using a variational iteration(VI) semi-analytical scheme. The analysis and the match between the constructed analytical solution and the semi-analytical solution are sketched using various diagrams to show the accuracy of the solution we obtained. The adopted scheme's performance shows the effectiveness of the method and its ability to be applied to various nonlinear evolution equations.     

非线性孤子方程的齐次平衡法

齐次平衡法是求非线性孤子方程孤波解的一种十分有效的方法.对齐次平衡法的一些关键步骤进行拓广使用，获得了非线性孤子方程一批新的具有更为丰富形式的精确解，孤波解仅是其中的一种特殊情形，使得结果更加完美. 关键词：     

A nonextensive maximum entropy approach to a family of nonlinear reaction–diffusion equations

We consider a monoparametric family of reaction–diffusion equations endowed with both a nonlinear diffusion term and a nonlinear reaction one that possess exact time-dependent particular solutions of the Tsallis’ maximum entropy (MaxEnt) form. The evolution of these solutions is governed by a system of three coupled nonlinear ordinary differential equations that are integrated numerically. A simple population dynamics interpretation provides a qualitative understanding of the behaviour of the q-MaxEnt solutions. When the reaction term vanishes the time-dependent distributions studied here reduce to the previously known Tsallis’ MaxEnt solutions for the nonlinear diffusion equation.     

Modified variational iteration method

In this Letter, we introduce a modified variational iteration method by inserting some unknown parameters into the correctional functional. The main advantage of this method is that one can avoid the uncontrollability problems, of the nonzero endpoint conditions, encountered in the traditional variational iteration method. Moreover, the method is applied to some nonlinear equations and the numerical solutions reveal that the modified method is accurate and efficient to solve a large class of nonlinear differential equations. Furthermore, the method does not share the drawbacks of the conventional variational iteration method, namely the restriction of the order of the nonlinearity term or even the form of the boundary conditions.     

Application of variational iteration method and homotopy–perturbation method for nonlinear heat diffusion and heat transfer equations

Perturbation methods depend on a small parameter which is difficult to be found for real-life nonlinear problems. To overcome this shortcoming, two new but powerful analytical methods are introduced to solve nonlinear heat transfer problems in this Letter; one is He's variational iteration method (VIM) and the other is the homotopy–perturbation method (HPM). Nonlinear convective–radiative cooling equations are used as examples to illustrate the simple solution procedures. These methods are useful and practical for solving the nonlinear heat diffusion equation, which is associated with variable thermal conductivity condition. Comparison of the results obtained by both methods with exact solutions reveals that both methods are tremendously effective.     

Nonlinear Oscillations Analysis of the Elevator Cable in a Drum Drive Elevator System

This paper aims to investigate nonlinear oscillations of an elevator cable in a drum drive. The governing equation of motion of the objective system is developed by virtue of Lagrangian's method. A complicated term is broached in the governing equation of the motion of the system owing to existence of multiplication of a quadratic function of velocity with a sinusoidal function of displacement in the kinetic energy of the system. The obtained equation is an example of a well-known category of nonlinear oscillators, namely, non-natural systems. Due to the complex terms in the governing equation, perturbation methods cannot directly extract any closed form expressions for the natural frequency. Unavoidably, different non-perturbative approaches are employed to solve the problem and to elicit a closed-form expression for the natural frequency. Energy balance method, modified energy balance method and variational approach are utilized for frequency analyzing of the system. Frequency-amplitude relationships are analytically obtained for nonlinear vibration of the elevator's drum. In order to examine accuracy of the obtained results, exact solutions are numerically obtained and then compared with those obtained from approximate closed-form solutions for several cases. In a parametric study for different nonlinear parameters, variation of the natural frequencies against the initial amplitude is investigated. Accuracy of the three different approaches is then discussed for both small and large amplitudes of the oscillations.     

Construction of solitary solutions for nonlinear dispersive equations by variational iteration method

Variational iteration method is implemented to construct solitary solutions for nonlinear dispersive equations. In this scheme the solution takes the form of a convergent series with easily computable components. The chosen initial solution or trial function plays a major role in changing the physical structure of the solution. Many models are approached and the obtained results reveal that the method is very effective and convenient for constructing solitary solutions.     

Accurate analytic presentation of solution of the Schrödinger equation with arbitrary physical potential

High precision approximate analytic expressions of the ground state energies and wave functions for the arbitrary physical potentials are found by first casting the Schrödinger equation into the nonlinear Riccati form and then solving that nonlinear equation analytically in the first iteration of the quasilinearization method (QLM). In the QLM the nonlinear differential equation is treated by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The choice of zero iteration is based on general features of exact solutions near the boundaries. The approach is illustrated on the examples of the Yukawa, Woods-Saxon and funnel potentials. For the latter potential, solutions describing charmonium, bottonium and topponium are analyzed. Comparison of our approximate analytic expressions for binding energies and wave functions with the exact numerical solutions demonstrates their high accuracy in the wide range of physical parameters. The accuracy ranging between 10⁻⁴ and 10⁻⁸ for the energies and, correspondingly, 10⁻² and 10⁻⁴ for the wave functions is reached. The derived formulas enable one to make accurate analytical estimates of how variation of different interactions parameters affects correspondent physical systems.     

Accurate analytic presentation of solution for the spiked harmonic oscillator problem

High precision approximate analytic expressions of the ground state energies and wave functions for the spiked harmonic oscillator are found by first casting the correspondent Schrödinger equation into the nonlinear Riccati form and then solving that nonlinear equation analytically in the first iteration of the quasilinearization method (QLM). In the QLM the nonlinear differential equation is treated by approximating the nonlinear terms with a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The choice of zero iteration is based on general features of exact solutions near the boundaries. Comparison of our approximate analytic expressions for binding energies and wave functions with the exact numerical solutions demonstrates their high accuracy in the wide range of parameters. The accuracy ranging between 10⁻³ and 10⁻⁷ for the energies and, correspondingly, 10⁻² and 10⁻⁷ for the wave functions in the regions, where they are not extremely small is reached. The derived formulas enable one to make accurate analytical estimates of how variation of different interactions parameters affects the correspondent physical systems.     

Generalized Lagrange Jacobi Gauss-Lobatto (GLJGL) Collocation Method for Solving Linear and Nonlinear Fokker-Planck Equations

In this study, we have constructed a new numerical approach for solving the time-dependent linear and nonlinear Fokker-Planck equations. In fact, we have discretized the time variable with Crank-Nicolson method and for the space variable, a numerical method based on Generalized Lagrange Jacobi Gauss-Lobatto(GLJGL) collocation method is applied. It leads to in solving the equation in a series of time steps and at each time step, the problem is reduced to a problem consisting of a system of algebraic equations that greatly simplifies the problem. One can observe that the proposed method is simple and accurate. Indeed, one of its merits is that it is derivative-free and by proposing a formula for derivative matrices, the difficulty aroused in calculation is overcome, along with that it does not need to calculate the General Lagrange basis and matrices; they have Kronecker property. Linear and nonlinear Fokker-Planck equations are given as examples and the results amply demonstrate that the presented method is very valid, effective,reliable and does not require any restrictive assumptions for nonlinear terms.