Nth Order Ordinary Differential Equations Research Articles

Dynamic Analysis of Rotating Nonuniform Timoshenko Beams With an Elastically Restrained Root

A systematic solution procedure for studying the dynamic response of a rotating nonuniform Timoshenko beam with an elastically restrained root is presented. The partial differential equations are transformed into the ordinary differential equations by taking the Laplace transform. The two coupled governing differential equations are uncoupled into two complete fourth-order differential equations with variable coefficients in the flexural displacement and in the angle of rotation due to bending, respectively. The general solution and the generalized Green function of the uncoupled system are derived. They are expressed in terms of the four corresponding linearly independent homogenous solutions, respectively. The shifting relations of the four homogenous solutions of the uncoupled governing differential equation with constant coefficients are revealed. The generalized Green function of an nth order ordinary differential equation can be obtained by using the proposed method. Finally, the influence of the elastic root restraints, the setting angle, and the excitation frequency on the steady response of a beam is investigated.

Modeling BOD Removal in Constructed Wetlands with Mixing Cell Method

A new concept, transport detention time, is proposed in this paper to describe solute-transport processes. Using this concept, a new mathematical model was developed to describe biochemical oxygen demand removal in constructed wetlands. By treating a constructed wetland as a series of continuous stir tank reactors, an nth-order ordinary differential equation was derived based on the principle of mass balance and convective-dispersive equation and by introducing transfer function and Laplace transform. The number of continuous stir tank reactors of a particular wetland was determined by the parameters, such as dispersion coefficient and flow velocity, occurring in the wetland. Two examples were presented to illustrate the applications of the model. Moment method and a combination of moment and optimization methods were used to estimate the model parameters from tracer experiment data. A comparison between the model presented in this paper and the currently used plug-flow-constructed wetland model indicated that the former was more accurate. Additionally, this model can be applied to transient conditions, is theoretically sound, and represents a theoretical advance in constructed wetland research.

An Elementary Demonstration of the Existence of $s\ell(3,R)$ Symmetry for all Second-Order Linear Ordinary Differential Equations

Previous article Next article An Elementary Demonstration of the Existence of $s\ell(3,R)$ Symmetry for all Second-Order Linear Ordinary Differential EquationsK. S. Govinder and P. G. L. LeachK. S. Govinder and P. G. L. Leachhttps://doi.org/10.1137/S0036144597316838PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractAll second-order linear ordinary differential equations are shown in an elementary way to possess the symmetry algebra ${s}\ell(2,R)$.[1] M. Aguirre and , J. Krause, SL(3,R) as the group of symmetry transformations for all one‐dimensional linear systems, J. Math. Phys., 29 (1988), 9–15 10.1063/1.528139 89g:58184 CrossrefISIGoogle Scholar[2] Robert Anderson and , Suzanne Davison, A generalization of Lie’s “counting” theorem for second‐order ordinary differential equations, J. Math. Anal. Appl., 48 (1974), 301–315 51:971 CrossrefISIGoogle Scholar[3] Nedialko Nedialkov and , Kenneth Jackson, An interval Hermite‐Obreschkoff method for computing rigorous bounds on the solution of an initial value problem for an ordinary differential equation, Kluwer Acad. Publ., Dordrecht, 1999, 289–310 1744280 Google Scholar[4] L. M. Berković, Glyden–Mes˘c˘erskii problem, Celest. Mech., 24 (1981), pp. 407–429. csm CLMCAV 0008-8714 Celest. Mech. CrossrefISIGoogle Scholar[5] Google Scholar[6] F. Mahomed and , P. Leach, The Lie algebra sl(3,R) and linearization, Quaestiones Math., 12 (1989), 121–139 91d:58221 CrossrefGoogle Scholar[7] F. Mahomed and , P. Leach, Symmetry Lie algebras of nth order ordinary differential equations, J. Math. Anal. Appl., 151 (1990), 80–107 91h:58091 CrossrefISIGoogle Scholar[8] C. Wulfman and , B. Wybourne, The Lie group of Newton’s and Lagrange’s equations for the harmonic oscillator, J. Phys. A, 9 (1976), 507–518 10.1088/0305-4470/9/4/007 54:479 CrossrefGoogle ScholarKeywordslinear differential equations$s\ell(3R)$ Previous article Next article FiguresRelatedReferencesCited byDetails Analysis of modified Painlevé–Ince equations28 May 2020 | Ricerche di Matematica, Vol. 71, No. 1 Cross Ref Maximally superintegrable systems in flat three-dimensional space are linearizableJournal of Mathematical Physics, Vol. 62, No. 1 Cross Ref Symmetry analysis of the Klein–Gordon equation in Bianchi I spacetimesInternational Journal of Geometric Methods in Modern Physics, Vol. 12, No. 03 Cross Ref Symmetry group classification of ordinary differential equations: Survey of some results1 January 2007 | Mathematical Methods in the Applied Sciences, Vol. 30, No. 16 Cross Ref Lie symmetry algebra of one-dimensional nonconservative dynamical systems11 September 2007 | Chinese Physics, Vol. 16, No. 9 Cross Ref Integrability analysis of the Emden-Fowler equation21 January 2013 | Journal of Nonlinear Mathematical Physics, Vol. 14, No. 3 Cross Ref Symmetries of First Integrals and Their Associated Differential EquationsJournal of Mathematical Analysis and Applications, Vol. 235, No. 1 Cross Ref Volume 40, Issue 4| 1998SIAM Review History Published online:02 August 2006 InformationCopyright © 1998 Society for Industrial and Applied MathematicsKeywordslinear differential equations$s\ell(3R)$MSC codes34A30PDF Download Article & Publication DataArticle DOI:10.1137/S0036144597316838Article page range:pp. 945-946ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics

On Periodic-Solutions of Certain nth Order Differential Equations

We present two theorems on periodic solutions of nth-order ordinary differential equations with the right-hand side T-periodic with respect to the time variable. We apply a topological method based on the Lefschetz Fixed Point Theorem and the Wazewski Principle.

A Reduction Algorithm for an Ordinary Differential Equation Admitting a Solvable Lie Group

An iterative algorithm is presented for reducing an nth-order ordinary differential equation to an $( {n - r} )$th-order ordinary differential equation (ODE) plus r quadratures when it admits an r-parameter solvable Lie group of transformations. The procedure is automatic. The reduced $( {n - r} )$th-order ODE is obtained without determining intermediate ODES of orders $n - r + 1, \cdots n - 1$. This reduced ODE and the r quadratures are deduced directly after iteratively computing $2r$ invariant coordinates $\{ {x_{( i )} ,y_{( i )} } \}$ and $3( {r - 1} )$ coefficients $\{ {\alpha _j ,\beta _j ,\gamma _j } \}$ of infinitesimal generators associated with an admitted r-parameter solvable Lie group.The reduction algorithm is illustrated by several examples including the third-order Blasius equation which admits a two-parameter group and a fourth-order ODE admitting a three-parameter solvable group which arises in studying the group properties of the wave equation in an inhomogeneous medium.

Symmetry Lie algebras of nth order ordinary differential equations

We show that an nth ( n ⩾ 3) order linear ordinary differential equation has exactly one of n + 1, n + 2, or n + 4 (the maximum) point symmetries. The Lie algebras corresponding to the respective numbers of point symmetries are obtained. Then it is shown that a necessary and sufficient conditon for an nth ( n ⩾ 3) order equation to be linearizable via a point transformation is that it must admit the n dimensional Abelian algebra nA 1 = A 1 ⊕ A 1 ⊕ … ⊕ A 1. We discuss in detail the symmetry realizations of ( n − 1) A 1 ⊕ s A 1. Finally, we prove that an nth ( n ⩾ 3) order equation q ( n) = H( t, q, …, q n − 1 ) cannot admit exactly an n + 3 dimensional algebra of point symmetries which is a subalgebra of nA 1 ⊕, gl(2, R ).

Method of chasing for multipoint boundary value problems

The method of chasing developed by Gel'fand and Lokutsiyevskii is completely formulated here for nth order ordinary linear differential equations subject to separated multipoint boundary conditions.

Lyapunov-type functions for nth order ordinary differential equations

Lyapunov-type functions for nth order ordinary differential equations

The Construction of Finite Difference Approximations to Ordinary Differential Equations

Finite difference approximations of the form Σ^(si)_(i=-rj)d_(j,i)u_(j+i)=Σ^(mj)_(i=1) e_(j,if)(z_(j,i)) for the numerical solution of linear nth order ordinary differential equations are analyzed. The order of these approximations is shown to be at least r_j + s_j + m_j - n, and higher for certain special choices of the points Z_(j,i). Similar approximations to initial or boundary conditions are also considered and the stability of the resulting schemes is investigated.

On the boundedness of solutions of an nth-order time-lag system

A boundedness criterion is established between the solutions of an nth-order nonlinear functional differential equation and an nth-order nonlinear ordinary differential equation.

An existence and uniqueness criterion for solutions of boundary value problems

In this paper a technique is developed for the study of the existence and uniqueness of solutions to nth order ordinary differential equations satisfying n-point boundary conditions. Liapunov-like functions are employed to determine the existence and uniqueness of solutions to linear equations satisfying the boundary conditions, and these solutions are in turn used to determine existence for the general nonlinear case. A by-product of this technique is a matching technique for linear equations by which solutions of certain k-point boundary value problems ( k < n) can be matched to extend the interval of existence for solutions to the n-point problem.

On the reduction of differential equations to the normal form by an analytic transformation: PMM vol. 38, n≗ 5, 1974, pp. 788–790

We treat the problem of reducing a system of nth-order ordinary differential equations to normal form in a neighborhood of a singular point in the presence or absence of resonances. We have shown that such a reduction is possible in the class of analytic transformations if the original system admits of an analytic symmetry group of specified dimension.

Lyapunov function having the stability matrix of Hermite in its time derivative

A method developed to generate Lyapunov functionals for partial differential equations is applied to an nth-order ordinary linear differential equation. The resulting function V has the Hermite stability matrix in its derivative V. The function is efficient as the sufficient stability conditions obtained from V are also necessary.

Extension of the Factorization Method

We show that it is possible to extend the formalism of the factorization method for any displacement in the spectrum space of any second-order differential equation. Following this, we show that we can extend, at least formally, the formalism for some nth-order ordinary differential equations.