Third-order Approximate Solution Research Articles

A modified global error minimization method for solving nonlinear Duffing-harmonic oscillators

<abstract> <p>In this paper, a third-order approximate solution of strongly nonlinear Duffing-harmonic oscillators is obtained by extending and improving an analytical technique called the global error minimization method (GEMM). We have made a comparison between our results, those obtained from the other analytical methods and the numerical solution. Consequently, we notice a better agreement with the numerical solution than other known analytical methods. The results are valid for both small and large oscillation amplitude. The obtained results demonstrate that the present method can be easily extended to strongly nonlinear problems, as indicated in the presented applications.</p> </abstract>

The Approximate Solution of the Nonlinear Exact Equation of Deflection of an Elastic Beam with the Galerkin Method

Calculating the large deflection of a cantilever beam is one of the common problems in engineering. The differential equation of a beam under large deformation, or the typical elastica problem, is hard to approximate and solve with the known solutions and techniques in Cartesian coordinates. The exact solutions in elliptic functions are available, but not the explicit expressions in elementary functions in expectation. This paper attempts to solve the nonlinear differential equation of deflection of an elastic beam with the Galerkin method by successfully solving a series of nonlinear algebraic equations as a novel approach. The approximate solution based on the trigonometric function is assumed, and the coefficients of the trigonometric series solution are fitted with Chebyshev polynomials. The numerical results of solving the nonlinear algebraic equations show that the third-order approximate solution is highly consistent with the exact solution of the elliptic function. The effectiveness and advantages of the Galerkin method in solving nonlinear differential equations are further demonstrated.

Semi-analytical method for propagation of harmonic waves in nonlinear magneto-thermo-elasticity

In this paper, we develop and apply a semi-analytical method called the Method of Directly Defining inverse Mapping (MDDiM) to obtain a series solution for the propagation of harmonic waves in a nonlinear generalized thermo-elasticity with relaxation time, heating, rotation, and magnetic field. We obtained third-order approximate solutions for the displacement and the temperature fields of the waves with variations in the magnetic field, the relaxation time, and the rotation. The obtained results, with minimum errors, are presented graphically and discussed. Our new extended MDDiM out performs the existing Optimal Homotopy Analysis Method (OHAM) with minimum error and a faster convergent rate. The method can be applied to various systems of nonlinear partial differential equations arising in science and engineering.

On two approaches to the third-order solution of surface gravity waves

A third-order approximate solution for surface gravity waves in the finite water depth is studied in the context of potential flow theory. This solution corresponds to the steady-state part of a multidirectional irregular wave field and provides explicit expressions for the surface elevation, free-surface velocity potential, and velocity potential. The amplitude dispersion relation is also provided. Two approaches are used to derive the third-order analytical solution, resulting in two types of approximate solutions: the perturbation solution and the Hamiltonian solution. The perturbation solution is obtained by a classical perturbation technique in which the time variable is expanded in multiscale to eliminate secular terms. The Hamiltonian solution is derived from the canonical transformation in the Hamiltonian theory of water waves. By comparing the two types of solutions, it is found that they are completely equivalent for the first- and second-order solutions and the nonlinear dispersion, but for the third-order part only the sum–sum terms are the same. Due to the canonical transformation that could completely separate the dynamic and bound harmonics, the Hamiltonian solutions break through the difficulty that the perturbation theory breaks down due to singularities in the transfer functions when the quartet resonance criterion is satisfied. Furthermore, it is also found that some time-averaged quantities based on the Hamiltonian solution, such as mean potential energy and mean kinetic energy, are equal to those in the initial state in which the sea surface is assumed to be a Gaussian random process. This is because there are associated conserved quantities in the Hamiltonian form. All of these show that the Hamiltonian solution is more reasonable and accurate to describe the third-order steady-state wave field. Finally, based on the Hamiltonian solution, some statistics are given such as the volume flux, skewness, excess kurtosis, and non-uniqueness of the induced mean flow and mean surface.

A Modified Energy Balance Method to Obtain Higher-order Approximations to the Oscillators with Cubic and Harmonic Restoring Force

This article analyzes a strongly nonlinear oscillator with cubic and harmonic restoring force and proposes an efficient analytical technique based on the modified energy balance method (MEBM). The proposed method incorporates higher-order approximations. After applying the proposed MEBM, a set of complicated higher-order nonlinear algebraic equations are obtained. Higher-order nonlinear algebraic equations are cumbersome to investigate especially in the case of a large initial oscillation amplitude. This limitation is overcome in the proposed method by using the iterative procedure based on the homotopy perturbation method. The approximated results agree well with the numerically obtained exact solutions. These third-order approximate solutions are found to be almost the same as exact solutions, which cannot be found using the existing energy balance method. Highly accurate result and simple solution procedure are advantages of this proposed method, which could be applied to other nonlinear oscillatory problems arising in nonlinear science and engineering.

Cubic–quintic nonlinear parametric resonance of a simply supported beam

We investigated the cubic–quintic nonlinear response in a parametrically excited simply supported beam subjected to a spring force in the axial direction. Taking into account the cubic and quintic geometric nonlinearities of curvature of the beam, the governing equation of the parametrically excited beam was derived based on Hamilton’s principle. The fifth-order approximate solution was analytically obtained using the method of multiple scales; with this calculation, the third-order nonlinear normal mode was also obtained. Its associated amplitude revealed saddle-node bifurcation and a hysteresis in the frequency response curve, which could not be predicted using the third-order approximate solution for the governing equation that included only cubic nonlinearity. Experimental results taken using a simple apparatus qualitatively verify the theoretically predicted nonlinear features in the parametric resonance caused by the cubic–quintic geometric nonlinearity of the beam.

High-order approximate solutions of strongly nonlinear cubic-quintic Duffing oscillator based on the harmonic balance method

In this paper, a new reliable analytical technique has been introduced based on the Harmonic Balance Method (HBM) to determine higher-order approximate solutions of the strongly nonlinear cubic-quintic Duffing oscillator. The application of the HBM leads to very complicated sets of nonlinear algebraic equations. In this technique, the high-order nonlinear algebraic equations are approximated in the form of a power series solution, and this solution produces desired results even for small as well as large amplitudes of oscillation. Moreover, a suitable truncation formula is found in which the solution measures better results than existing results and it saves a lot of calculation. It is highly noteworthy that using the proposed technique, the third-order approximate solutions gives an excellent agreement as compared with the numerical solutions (considered to be exact). The proposed technique is applied to the strongly nonlinear cubic-quintic Duffing oscillator to reveals its novelty, reliability and wider applicability.

Mathematical study of fractional-order biological population model using optimal homotopy asymptotic method

In this paper, we study the fractional-order biological population models (FBPMs) with Malthusian, Verhulst, and porous media laws. The fractional derivative is defined in Caputo sense. The optimal homotopy asymptotic method (OHAM) for partial differential equations (PDEs) is extended and successfully implemented to solve FBPMs. Third-order approximate solutions are obtained and compared with the exact solutions. The numerical results unveil that the proposed extension in the OHAM for fractional-order differential problems is very effective and simple in computation. The results reveal the effectiveness with high accuracy and extremely efficient to handle most complicated biological population models.

A third-order approximate solution of the reaction–diffusion process in an immobilized biocatalyst particle

A third-order polynomial model was suggested to describe the reaction–diffusion process of substrate in the immobilized biocatalyst particles and further to estimate their effectiveness factors. Both the numerical solution of the differential equation using the shooting method and the third-order approximate solution have been obtained. They were in good agreement for describing the substrate concentration distribution inside the particles when β= C s/ K m was larger than or equal to 0.75 φ 2−1 and φ=(R/3) r m /DK m . The relative errors of effectiveness factors obtained by the two methods were found less than 10% in a wide range of φ 2 and β except for the high values of φ 2. Some published practical systems were also examined. The results have indicated that the method proposed here could be of value for a quick estimation of substrate distribution and effectiveness factor of some systems such as dynamic oxygen transfer into natural mycelial pellets in submerged cultures.

Remarks on the Perturbation Methods in Solving the Second-Order Delay Differential Equations

The paper presents a study on the validity of perturbation methods, suchas the method of multiple scales, the Lindstedt–Poincare method and soon, in seeking for the periodic motions of the delayed dynamic systemsthrough an example of a Duffing oscillator with delayed velocityfeedback. An important observation in the paper is that the method ofmultiple scales, which has been widely used in nonlinear dynamics, worksonly for the approximate solutions of the first two orders, and givesrise to a paradox for the third-order approximate solutions of delaydifferential equations. The same problem appears when theLindstedt–Poincare method is implemented to find the third-orderapproximation of periodic solutions for delay differential equations,though it is effective in seeking for any order approximation ofperiodic solutions for nonlinear ordinary differential equations. Apossible explanation to the paradox is given by the results obtained byusing the method of harmonic balance. The paper also indicates thatthese perturbation methods, despite of some shortcomings, are stilleffective in analyzing the dynamics of a delayed dynamic system sincethe approximate solutions of the first two orders already enable one togain an insight into the primary dynamics of the system.

Laminar boundary layer between two planes perpendicular to each other

In this paper, we obtain a third-order approximate solution for the laminar boundary layer between two planes perpendicular to each other. In boundary layer equations, the viscous and the inertial terms have the same quantity step. In this paper, at first, supposing that the inertial terms are bigger than the viscous terms, we solve the boundary layer equations, and then we suppose that the viscous terms are bigger than the inertial terms. At last, we take the mean value as the valid solution of the boundary layer equations. The first- and the second-order approximate solutions obtained in this paper coincide with the results in ref. [1], while the third-order solution obtained in this paper is better than that in ref. [1].

Uniform Asymptotic Technique for Analyzing Wave Propagation in Inhomogeneous Slab Waveguides

The guided modes of inhomogeneous dielectric slab waveguides are analyzed by a uniform asymptotic technique based on the related equation method. This technique gives highly accurate solutions in the sense of asymptotic expansion. The algorithm for calculating the guided modes of slab waveguides with an even polynomial refractive-index medium is presented. As an example, we calculate the third-order approximate solutions for the guided modes in an analytic form. The results show that the WKB solutions for higher order modes are more accurate than for the lower order modes and the correction to the WKB solutions is significant for the lower order modes. The numerical result for eigenvalues and modal fields confirms that the third-order asymptotic solution is accurate for all the guided modes of the near-parabolic profile waveguides and for higher order modes in the case of the quasi-Gaussian profile.