scholarly journals SIMILARITY SOLUTIONS AND CONSERVATION LAWS FOR THE BEAM EQUATIONS: A COMPLETE STUDY

2020 ◽  
Vol 60 (2) ◽  
pp. 98-110
Author(s):  
Amlan Kanti Halder ◽  
Andronikos Paliathanasis ◽  
Peter Gavin Lawrence Leach
Keyword(s):  
Conservation Laws ◽  
Fourth Order ◽  
Singularity Analysis ◽  
Travelling Wave ◽  
Similarity Solutions ◽  
Third Order ◽  
The Third ◽  
Beam Equations ◽  
Complete Study ◽  
Wave Reduction

We study the similarity solutions and we determine the conservation laws of various forms of beam equations, such as Euler-Bernoulli, Rayleigh and Timoshenko-Prescott. The travelling-wave reduction leads to solvable fourth-order odes for all the forms. In addition, the reduction based on the scaling symmetry for the Euler-Bernoulli form leads to certain odes for which there exists zero symmetries. Therefore, we conduct the singularity analysis to ascertain the integrability. We study two reduced odes of second and third orders. The reduced second-order ode is a perturbed form of Painlevé-Ince equation, which is integrable and the third-order ode falls into the category of equations studied by Chazy, Bureau and Cosgrove. Moreover, we derived the symmetries and its corresponding reductions and conservation laws for the forced form of the abovementioned beam forms. The Lie Algebra is mentioned explicitly for all the cases.

scholarly journals Conservation Laws for a Generalized Coupled Korteweg-de Vries System

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Daniel Mpho Nkwanazana ◽  
Ben Muatjetjeja ◽  
Chaudry Masood Khalique
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Fourth Order ◽  
Nonlinear Partial Differential Equations ◽  
Order System ◽  
Conserved Quantities ◽  
Third Order ◽  
The Third ◽  
New System

We construct conservation laws for a generalized coupled KdV system, which is a third-order system of nonlinear partial differential equations. We employ Noether's approach to derive the conservation laws. Since the system does not have a Lagrangian, we make use of the transformationu=Ux,v=Vxand convert the system to a fourth-order system inU,V. This new system has a Lagrangian, and so the Noether approach can now be used to obtain conservation laws. Finally, the conservation laws are expressed in theu,vvariables, and they constitute the conservation laws for the third-order generalized coupled KdV system. Some local and infinitely many nonlocal conserved quantities are found.

Children’s Recall of Approximations to English

1973 ◽  
Vol 16 (2) ◽  
pp. 201-212 ◽  
Author(s):  
Elizabeth Carrow ◽  
Michael Mauldin
Keyword(s):  
Language Development ◽  
Fourth Order ◽  
Third Order ◽  
Memory Processes ◽  
The Third ◽  
General Index ◽  
General Linguistic

As a general index of language development, the recall of first through fourth order approximations to English was examined in four, five, six, and seven year olds and adults. Data suggested that recall improved with age, and increases in approximation to English were accompanied by increases in recall for six and seven year olds and adults. Recall improved for four and five year olds through the third order but declined at the fourth. The latter finding was attributed to deficits in semantic structures and memory processes in four and five year olds. The former finding was interpreted as an index of the development of general linguistic processes.

scholarly journals Asymptotic formulae for linear equations

1972 ◽  
Vol 13 (2) ◽  
pp. 147-152 ◽  
Author(s):  
Don B. Hinton
Keyword(s):  
Asymptotic Behaviour ◽  
Fourth Order ◽  
Linear Equations ◽  
Order Equation ◽  
Third Order ◽  
Fourth Order Equation ◽  
The Third ◽  
Asymptotic Formulae ◽  
Image Position

Numerous formulae have been given which exhibit the asymptotic behaviour as t → ∞solutions ofwhere F(t) is essentially positive and Several of these results have been unified by a theorem of F. V. Atkinson [1]. It is the purpose of this paper to establish results, analogous to the theorem of Atkinson, for the third order equationand for the fourth order equation

scholarly journals Volume derivatives of the Grüneisen parameter in the limit of infinite pressure

2019 ◽  
Vol 97 (1) ◽  
pp. 114-116 ◽  
Author(s):  
A. Dwivedi
Keyword(s):  
Boundary Condition ◽  
Fourth Order ◽  
Fundamental Theorem ◽  
High Pressures ◽  
Grüneisen Parameter ◽  
Third Order ◽  
Gruneisen Parameter ◽  
The Third ◽  
Properties Of Materials ◽  
Derivatives Of

Expressions have been obtained for the volume derivatives of the Grüneisen parameter, which is directly related to the thermal and elastic properties of materials at high temperatures and high pressures. The higher order Grüneisen parameters are expressed in terms of the volume derivatives, and evaluated in the limit of infinite pressure. The results, that at extreme compression the third-order Grüneisen parameter remains finite and the fourth-order Grüneisen parameter tends to zero, have been used to derive a fundamental theorem according to which the volume derivatives of the Grüneisen parameter of different orders, all become zero in the limit of infinite pressure. However, the ratios of these derivatives remain finite at extreme compression. The formula due to Al’tshuler and used by Dorogokupets and Oganov for interpolating the Grüneisen parameter at intermediate compressions has been found to satisfy the boundary condition at infinite pressure obtained in the present study.

WENO-TVD schemes for hyperbolic conservation laws

Analysis ◽  
2007 ◽  
Vol 27 (1) ◽  
Author(s):  
Yousef Hashem Zahran
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Building Blocks ◽  
Second Order ◽  
Third Order ◽  
Weno Schemes ◽  
Hyperbolic Conservation ◽  
Systematic Assessment ◽  
The Third ◽  
Seventh Order

The purpose of this paper is twofold. Firstly we carry out a modification of the finite volume WENO (weighted essentially non-oscillatory) scheme of Titarev and Toro [14] and [15].This modification is done by using two fluxes as building blocks in spatially fifth order WENO schemes instead of the second order TVD flux proposed by Titarev and Toro [14] and [15]. These fluxes are the second order TVD flux [19] and the third order TVD flux [20].Secondly, we propose to use these fluxes as a building block in spatially seventh order WENO schemes. The numerical solution is advanced in time by the third order TVD Runge–Kutta method. A way to extend these schemes to general systems of nonlinear hyperbolic conservation laws, in one and two dimension is presented. Systematic assessment of the proposed schemes shows substantial gains in accuracy and better resolution of discontinuities, particularly for problems involving long time evolution containing both smooth and non-smooth features.

New bounds on effective elastic moduli of two-component materials

1982 ◽  
Vol 380 (1779) ◽  
pp. 305-331 ◽  
Keyword(s):  
Shear Modulus ◽  
Bulk Modulus ◽  
Fourth Order ◽  
Geometrical Parameters ◽  
Effective Moduli ◽  
Third Order ◽  
Effective Shear Modulus ◽  
The Third ◽  
Two Component ◽  
Order Bounds

Based on the perturbation solution, we derive new bounds on the effective moduli of a two-component composite material which are exact up to fourth order in δ μ = μ 1 — μ 2 and δ K = K 1 — K 2 , where μ i and K i , i = 1, 2, are the shear and bulk modulus, respectively, of the phases. The bounds on the effective bulk modulus involve three microstructural parameters whereas eight parameters are needed in the bounds on the effective shear modulus. For engineering calculations, we recommend the third-order bounds on the effective shear modulus which require only two geometrical parameters. We show in detail how Hashin-Shtrikman’s bounds can be extended and how Walpole’s bounds can be improved using two inequalities on the two geometrical parameters that appear in the third-order bounds on the effective shear modulus. The third- and fourth-order bounds on the effective moduli are shown to be more restrictive than, or at worst, coincident with, existing bounds due to Hashin and Shtrikman, McCoy, Beran and Molyneux and Walpole.

STANDARD NOMOGRAPHIC FORMS FOR EQUATIONS IN THREE VARIABLES

1935 ◽  
Vol 12 (1) ◽  
pp. 14-40 ◽  
Author(s):  
F. M. Wood
Keyword(s):  
Unit Circle ◽  
Fourth Order ◽  
Third Order ◽  
Rectangular Hyperbola ◽  
Practical Utility ◽  
The Third ◽  
Cubic Curve ◽  
Final Adjustment ◽  
Straight Lines ◽  
Suitable Location

Equations of the third and fourth nomographic order in three variables have been dealt with and classified. Equations of the third order may be reduced to one of two standard forms, α + β + γ = 0 and α + βγ = 0, which give alignment charts composed of three straight lines. Equations of the fourth order may also be reduced to one of two standard forms, resulting in charts composed of (a) two straight lines and a curve, or (b) two scales on a conic, and the third on another curve. Transformations of these four standard forms are given which permit of rapid and easy adjustment of the position and length of the scales for any given example, resulting in a chart of practical utility. Although the underlying theory has been studied by other writers, notably Soreau and Clark, it has possibly never appeared before in such a neat form. On this account, and also because of the standard transformations, it is felt that this article is of particular value.Standard forms have also been developed for third order equations leading to charts composed of two scales on a conic and a third straight scale, and in conclusion a third type of chart, in which all three scales appear on a single cubic curve, has been standardized. The practical value of the last type is questionable, but the conic charts are of use since we may arbitrarily choose the unit circle, or the rectangular hyperbola, for our conic scales. Final adjustment forms which permit suitable location of the scales in particular examples have been obtained in every case.

scholarly journals Conservation Laws for a Variable Coefficient Variant Boussinesq System

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Ben Muatjetjeja ◽  
Chaudry Masood Khalique
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Fourth Order ◽  
Variable Coefficient ◽  
Boussinesq System ◽  
Order System ◽  
Conserved Quantities ◽  
Third Order ◽  
Partial Differential

We construct the conservation laws for a variable coefficient variant Boussinesq system, which is a third-order system of two partial differential equations. This system does not have a Lagrangian and so we transform it to a system of fourth-order, which admits a Lagrangian. Noether’s approach is then utilized to obtain the conservation laws. Lastly, the conservation laws are presented in terms of the original variables. Infinite numbers of both local and nonlocal conserved quantities are derived for the underlying system.

The mechanism of the Cannizzaro reaction of Formaldehyde

1954 ◽  
Vol 7 (4) ◽  
pp. 335 ◽  
Author(s):  
RJL Martin
Keyword(s):  
Fourth Order ◽  
Kinetic Data ◽  
Order Reaction ◽  
Rate Equations ◽  
Third Order ◽  
The Third ◽  
Hydride Ion ◽  
Cannizzaro Reaction ◽  
Wide Range ◽  
Mechanism Of The Reaction

For a wide range of concentrations of formaldehyde and alkali, the Cannizzaro reaction of formaldehyde can be described as the sum of a third and a fourth order reaction. However, the concentrations which are used for the rate equations must be corrected for the amount of methylene glycol anion present. The dissociation constant of methylene glycol as determined from the kinetic data is the same magnitude as that derived electrometrically. The mechanism of the reaction is interpreted as a reaction between formaldehyde and the hydride ion donors CH2(O-)(OH) and CH2(O-)(O-) It is shown why the third order reaction proposed by previous workers is not always applicable.

scholarly journals Lie symmetries and singularity analysis for generalized shallow-water equations

2020 ◽  
Vol 21 (7-8) ◽  
pp. 739-747
Author(s):  
Andronikos Paliathanasis
Keyword(s):  
Lie Algebra ◽  
Three Dimensional ◽  
Singularity Analysis ◽  
Complete Classification ◽  
Similarity Solutions ◽  
Lie Symmetries ◽  
Reduced Equations ◽  
Complete Study ◽  
The One

AbstractWe perform a complete study by using the theory of invariant point transformations and the singularity analysis for the generalized Camassa-Holm (CH) equation and the generalized Benjamin-Bono-Mahoney (BBM) equation. From the Lie theory we find that the two equations are invariant under the same three-dimensional Lie algebra which is the same Lie algebra admitted by the CH equation. We determine the one-dimensional optimal system for the admitted Lie symmetries and we perform a complete classification of the similarity solutions for the two equations of our study. The reduced equations are studied by using the point symmetries or the singularity analysis. Finally, the singularity analysis is directly applied on the partial differential equations from where we infer that the generalized equations of our study pass the singularity test and are integrable.

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