Sixth-order Ordinary Differential Equation Research Articles

Performance of GRKM-method for solving classes of ordinary and partial differential equations of sixth-orders

Abstract A general quasilinear sixth-order ordinary differential equation (ODE) is an important class of ODEs. The primary objective of this study is to establish a numerical method for solving a general class of quasilinear sixth-order partial differential equations (PDEs) and ODEs. However, the Runge–Kutta method (RKM) approach for solving special classes of ODEs has been generalized as an effort to solve the general class of ODEs. Nonlinear algebraic order condition (OCs) equations have been obtained up to the tenth order using the Taylor-series expansion methodology which is used to derive the novel generalized Runge–Kutta method (GRKM). In this study, a GRKM integrator has been derived for solving a general class of quasilinear sixth-order ODEs and then this method is modified subsequently to solve a class of PDEs. Accordingly, the proposed GRKM is modified to solve a quasilinear sixth-order PDE by converting it to a system of sixth-ODEs using the method of lines. Nine problems have been implemented to prove the efficiency and accuracy of the proposed method. Simulation results of these problems showed that the proposed numerical GRKM is an accurate and efficient method. In contrast, by comparing the proposed GRKM numerical approach with the classical RK method, the numerical results demonstrate that the direct integrator outperforms the indirect classical RK method in terms of algorithm complexity and function evaluations, proving that the numerical GRKM is efficient.

Construction of Lyapunov Functions for the Stability of Sixth Order Ordinary Differential Equation

This study employed Lyapunov function method to investigate the stability of nonlinear ordinary differential equations. Using Lyapunov direct method, we constructed Lyapunov function to investigate the stability of sixth order nonlinear ordinary differential equations. We find $ V(x) $, a quadratic form, positive definite and $ U(x) $ which is also positive definite was chosen such that the derivative of $ V(x) $ with respect to time was equal to the negative value of $ U(x) $.

Numerical Solution to Sixth Order Ordinary Differential Equation Using Three Stage Eighth Order Runge-Kutta Type Method

The present paper provides an innovative approach in developing numerical structure with minimal processing and maximum precision for sixth order ordinary differential equation satisfying underlying associated initial conditions. The algebraic order conditions for the three stage eighth order Runge-Kutta methods (RKSD8) been well designed for evaluating local and global truncated errors towards sixth order ordinary differential equation having form vvi = f (u,v,v',v''). The convergence of a numerical method is well explained and proved beneficial in evaluating norms of error and zero stability of RKSD8 with effective values of weights and nodes in the form of Butcher tableau.

Existence of Nontrivial Solutions for Sixth-Order Differential Equations

We show the existence of at least one nontrivial solution for a nonlinear sixth-order ordinary differential equation is investigated. Our approach is based on critical point theory.

$$C^2$$ Tension Splines Construction Based on a Class of Sixth-Order Ordinary Differential Equations

$$C^2$$ Tension Splines Construction Based on a Class of Sixth-Order Ordinary Differential Equations

Efficient Approximate Analytic Solution for the Problem of Stability of a Three-Layer Conic Shell Under Combined Loading

We propose a solution of the linear problem of behavior of an elastic three-layer conic shell under the combined action of several external force factors (uniform pressure, axial compression, and torque), which may lead to the loss of stability of the analyzed structure. The problem is reduced to the integration of the resolving singular sixth-order ordinary differential equation with variable coefficients. The solutions obtained by the methods of phase integrals (WKB method), hybrid WKB–Galerkin method, and finite-difference method are compared. We demonstrate the advantages of the asymptotic hybrid approach to the solution of the analyzed class of equations. The rational relationships between the thicknesses and elasticity moduli of layers of the shell required for attainment of the highest stability of the investigated structure under given external loads are analyzed. The problem of construction of the boundary surfaces separating the regions of stability and instability of the shell is discussed. The efficiency of three-layer shells as load-bearing elements aimed at improving stability is confirmed.

Generalized RK integrators for solving class of sixth-order ordinary differential equations

In this paper, the classical RK integrators for solving quasi-linear(QL) ordinary differential equations (ODEs) up to 5th-order have been generalized to solve class of 6th-order ODEs. The novel contribution of this work is the derivation of two efficient direct integrators for solving class of ODEs. Non-linear equations of order conditions (OCs) for the proposed RKM methods have derived up to the 7th-order using Taylor expansion. Two RKM integrators of three-stages, 5th-order and four-stages, 6th-order have been derived depend on these OCs. The proposed integrates have been tested using some implementations in order to compare them with classical methods which show that the proposed integrators are less than classical RK methods in term of time complexity and function evaluations.

An investigation of free vibrations of a strain gradient Timoshenko beams with the method of initial values

Investigated herein is the free vibrations of beams based on the strain gradient Timoshenko beam theory with the method of initial values. For the vibration of strain-gradient Timoshenko beam (SGTB), the sixth-order ordinary differential equation and three boundary conditions at each end have been obtained by using the Hamilton principle. The effect of the characteristic length on the frequencies of free vibrations is shown. The frequencies of the SGTB are compared to the frequencies of the strain gradient Euler beam (SGEB), classical Timoshenko beam (CTB) and classical Euler beam (CEB). It has been observed that the high-frequency values of conventional and strain-gradient beams are very different. This result can be used to determine the value of the material characteristic length for a nanobeam for which lengthscale effects are believed to be dominant.

Simplified variational iteration method for solving ordinary differential equations and eigenvalue problems

A simplified variational iteration method is proposed to solve high-order homogeneous or nonhomogeneous linear ordinary differential equation and ordinary differential equation eigenvalue problems more efficiently and conveniently. The simplification includes two aspects: (1) explicitly deducing the general form of the differential equation for the identification of the general Lagrange multiplier while avoiding the complexity of variational calculations during identification and (2) simplifying the iterative expressions to reduce the computational work of each iteration. Three ordinary differential equations in mechanics are solved by this simplified variational iteration method, which proves that it is valid and more concise than traditional methods. To make the method more practical, it is suggested that some complicated analytical derivations be executed numerically, thereby achieving a simplified semi-analytical variational iteration method that can be easily implemented by computer programs. The method is then used to numerically solve two complex ordinary differential equation problems derived from the continuum analysis of tall building structures: a sixth-order nonhomogeneous ordinary differential equation with complex boundary conditions and a sixth-order ordinary differential equation eigenvalue problem. Numerical computer programs are developed for these two problems, and corresponding examples are provided to verify the accuracy and efficiency of the simplified variational iteration method in solving complex ordinary differential equation problems.

Axisymmetric analysis of a thermoelastic isotropic half-space under buried sources in displacement and temperature potentials

ABSTRACTAnalytical and numerical investigations have been made to determine the solution of a thermoelastic isotropic half-space under buried mechanical load and heat flux based on Biot’s coupled thermoelasticity. Using a scalar potential function, the coupled governing equations have been uncoupled and a sixth order partial differential equation that governs the potential function is received. Applying the Hankel transform to suppress the radial variable, a sixth order ordinary differential equation with respect to depth is received. Solving that equation and utilizing the boundary and continuity conditions, the potential function is derived and displacements, temperature, and stress components are derived in the transformed domain in two regions. Applying the inverse Hankel transform, responses are derived in the physical domain as line integrals. The line integrals are calculated numerically through Mathematica software. Numerical results are depicted graphically for different cases of vertical load and heat flux. Effects of elastic and thermal properties as well as the depth of buried loads and frequency of excitation are investigated. Numerical evaluation shows that the agreement between the results of this study for surface excitation and the results reported in the literature is excellent.

Latitude dependence of amplitude factor δ for degree 2 tides

The study updates the theory of tides by specifying latitude dependence of the gravimetric factor (amplitude factor δ) for diurnal and semidiurnal tidal waves in the oceanless elastic Earth. The strain of an elastic flattened gravitating sphere with latitude-dependent geopotential, density, and Lamé constants is described by a system of sixth-order ordinary differential equations. The elliptical sphere is presented as consisting of layers, with density and elasticity varying as a function of latitude along the sphere surface as mean radiuses of equidensity and equielasticity ellipsoids that cross this surface. Dissipation is taken into account as a logarithmic creep function. Integration of the derived equations allows avoiding the use of approximate methods. The predicted δ factors for degree 2 tides depend on latitude and increase from the equator to the pole by 0.12 to 0.18% for different Earth models. The obtained latitude dependences are compared with superconducting gravimeter (SG) data after including effects associated with the inertial and Coriolis forces, as well as dynamic resonance. The theoretical predictions and the observations show good fit, to hundredths of percent. The average δ factors predicted in this study for the PREM model are intermediate between those computed with the hydrostatic and nonhydrostatic tidal gravity models of Dehant, Defraigne, and Wahr (DDW/H and DDW/NH), while the estimates obtained with reference to the IASP91 model coincide with the DDW/NH results to the fifth decimal digit.

An analytical solution for out-of-plane deflection of a curved Timoshenko beam with strong nonlinear boundary conditions

This paper presents a simple method for finding the analytical solution for nonlinear boundary problems. The shifting function method is applied to developing the static deflection of an out-of-plane curved Timoshenko beam with nonlinear boundary conditions. Considering the out-of-plane motion of a uniform curved Timoshenko beam of constant radius R and a doubly symmetric cross section, three coupled governing differential equations are derived via Hamilton’s principle. After some simple arithmetic operations, the curved beam system can be decomposed into a complete sixth-order ordinary differential characteristic equation and the associated boundary conditions. An example is given to illustrate the analysis and show that the proposed method performs very well for problems with strong nonlinearity.

Exact Static Analysis of In-Plane Curved Timoshenko Beams with Strong Nonlinear Boundary Conditions

Analytical solutions have been developed for nonlinear boundary problems. In this paper, the shifting function method is applied to develop the static deflection of in-plane curved Timoshenko beams with nonlinear boundary conditions. Three coupled governing differential equations are derived via the Hamilton’s principle. The mathematical modeling of the curved beam system can be decomposed into a complete sixth-order ordinary differential characteristic equation and the associated boundary conditions. It is shown that the proposed method is valid and performs well for problems with strong nonlinearity.

Exact Solutions for the Free Vibration of Extensional Curved Non-uniform Timoshenko Beams

The three coupled governing differential equations for the in-plane vibrations of curved non-uniform Timoshenko beams are derived via the Hamilton’s principle. Three physical parameters are introduced to simplify the analysis. By eliminating all the terms with the axial displacement parameter, then reducing the order of differential operator acting on the flexural displacement parameter, one uncouples the three governing characteristic differential equations with variable coefficients and reduces them into a sixth-order ordinary differential equation with variable coefficients in term of the angle of the rotation due to bending for the first time. The explicit relations between the axial and the flexural displacements and the angle of the rotation due to bending are also revealed. It is shown that if the material and geometric properties of the beam are in arbitrary polynomial forms, then the exact solutions for the in-plane vibrations of the beam can be obtained. Several limiting studies are illustrated. Finally, limiting cases are studied and the influence of the taper ratio and the arc length on the first two natural frequencies of the beams is explored.

Stability analysis of gradient elastic beams by the method of initial value

The buckling of a bar is studied analytically on the basis of a simple linear theory of gradient elasticity in the frame of the method of initial values. The method of initial values provides the values of the displacements and stress resultants throughout the bar once the initial displacements and initial stress resultants are known. We use probably for the first time the method of initial values to get critical loads of a strain gradient beam under completely different boundary conditions at the two end faces of the beam. Exact carryover matrix is presented for the classical beam and gradient beam analytically. The first mode shapes of classical beam and gradient beam are plotted. The method of initial values is also applied to the beams with variable cross-section. The priorities of the method of initial values are depicted. The variational approach gives a sixth-order ordinary differential equation for a beam in buckling. The additional boundary conditions are used to obtain critical loads. It is observed that critical loads increase dramatically for increasing values of the gradient coefficient.

Sixth order differential operators with eigenvalue dependent boundary conditions

We consider eigenvalue problems for sixth-order ordinary differential equations. Such differential equations occur in mathematical models of vibrations of curved arches. With suitably chosen eigenvalue dependent boundary conditions, the problem is realized by a quadratic operator pencil. It is shown that the operators in this pencil are self-adjoint, and that the spectrum of the pencil consists of eigenvalues of finite multiplicity in the closed upper half-plane, except for finitely many eigenvalues on the negative imaginary axis.

Frequency Domain Analysis of an Axisymmetric Thermoelastic Transversely Isotropic Half-Space

By virtue of a new complete scalar potential function, an analytical formulation is presented to determine displacements, stresses, and temperature in an axisymmetric linear thermoelastic transversely isotropic half-space affected by time harmonic surface axisymmetric vertical traction and/or surface heat flux. With the use of only one scalar potential function in a cylindrical coordinate system, the coupled partial differential equations in thermoelasticity are transformed to a sixth-order partial differential equation governing the potential function. Then, by using the Hankel integral transforms, a sixth-order ordinary differential equation is received, which after solving, results in the displacements, stresses, and temperature fields. To prove the validation of the analytical solution presented in this paper, the solutions are reduced to the case of elastodynamics in transversely isotropic half-space and in a quasi-static thermoelastic isotropic half-space, where exact agreements with the solutions reported in the literature are achieved. Because of complexity of the integrands, numerical evaluations of the integrals involved in this paper are carried out using a suitable quadrature scheme by Mathematica software. To show the accuracy and efficiency of the numerical algorithm, the solution of this study is compared with the results of an existing elastodynamic transversely isotropic half-space, where an excellent agreement is achieved.

Dynamic Green's Functions of an Axisymmetric Thermoelastic Half-Space by a Method of Potentials

AbstractWith the aid of a new complete scalar potential function, an analytical formulation for thermoelastic Green's functions of an axisymmetric linear elastic isotropic half-space is presented within the theory of Biot's coupled thermoelasticity. By using the potential function, the governing equations of coupled thermoelasticity are uncoupled into a sixth-order partial differential equation in a cylindrical coordinate system. Then, by using Hankel integral transforms to suppress the radial variable, a sixth-order ordinary differential equation is received. By solving this equation and considering boundary conditions, displacements, stresses, and temperature are derived in the Hankel integral transformed domain. By applying the theorem of inverse Hankel transforms, the solution is obtained generally for arbitrary surface time-harmonic traction and heat distribution. Subsequently, point-load Green's functions for the displacements, temperature, and stresses are given in the form of some improper line in...

Free vibrations of a strain gradient beam by the method of initial values

We extend the application of the method of initial values (also known as the transfer matrix method) to find frequencies of free vibrations of a strain-gradient-dependent Euler–Bernoulli beam (EBB) under different boundary conditions at the two end faces of the beam. For the classical EBB, we find the exact matricant or the carry-over matrix but it is numerically evaluated for the strain-gradient-dependent EBB. For the numerically evaluated matricant, it is found that ten iterations give converged values of the first six frequencies for the classical and the strain-gradient-dependent EBB. For the strain-gradient EBB, the sixth-order ordinary differential equation for the lateral deflection and three boundary conditions at each end have been derived by using the Hamilton principle. The material characteristic length is found to noticeably affect frequencies of free vibrations. Thus, the difference between frequencies of the classical and the strain-gradient-dependent EBB can be used to determine the value of the material characteristic length for a nanobeam for which length scale effects are believed to be dominant.

Investigation of solvability condition for sixth-order boundary value problem

This paper is concerned with the solvability condition for nonhomogenous linear boundary value problem for sixth-order ordinary differential equation. Throughout this study, we observed that, when the homogenous problem have nontrivial solution,then the nonhomogenous boundary value problem have a solution in case of nonhomogenous term that satisfied the solvability condition. We justified our results through the given example. Keywords :Sixth-order boundary value problem, self-adjoint problem.