scholarly journals XXVI.—On a Group of Linear Differential Equations of the 2nd Order, including Professor Chrystal's Seiche-equations

1906 ◽  
Vol 41 (3) ◽  
pp. 651-676 ◽  
Author(s):  
J. Halm
Keyword(s):  
Differential Equations ◽  
Stokes Equation ◽  
Mathematical Theory ◽  
General Type ◽  
The Other ◽  
Special Cases ◽  
Linear Differential ◽  
Image Position ◽  
Special Case ◽  
The Stokes Equation

It is readily seen that the two differential equationswhich play an important rôle in Professor Chrystal's mathematical theory of the Seiches, are special cases of the more general typeWith regard to the first, the Seiche-equation, this becomes at once apparent by writing a= − ½. Equation (2), on the other hand, which we may briefly call the Stokes equation [see Professor Chrystal's paper on “Some further Results in the Mathematical Theory of Seiches,” Proc. Roy. Soc. Edin., vol. xxv.] will be recognised as a special case (a = + 1) of the equationwhich is transformed into (3) by the substitution .

scholarly journals On some results concerning integral equations

1913 ◽  
Vol 32 ◽  
pp. 19-29
Author(s):  
Pierre Humbert
Keyword(s):  
Integral Equation ◽  
Differential Equations ◽  
Large Class ◽  
Integral Equations ◽  
Singular Integral ◽  
General Type ◽  
Singular Integral Equations ◽  
General Function ◽  
Linear Differential ◽  
Image Position

It is proposed in this paper to show now the well-known Laplace's transformation,which is of great help in finding the solution of linear differential equations, gives also interesting results concerning the theory of integral equations. In §2 we shall study its application to certain differential equations, and find a large class of equations which remain unchanged by this transformation. Then, (§3), taking instead of eazt, a more general function of the product zt, we shall find a solution for some homogeneous integral equations ; in § 4 we shall describe a method of solving a very general type of integral equation of the first kind, namely,a further extension to integral equations with the kernel ef(z)f(t) the object of §5. Then, studying an extension of Euler's transformation, we shall (§ 6) consider equations such aswhich will prove to be singular; and finally, in §7, we shall give other examples of singular integral equations.

Modeling, Simulation, and Modal Analysis Hydraulic Valve Lifter With Oil Compressibility Effects

1989 ◽  
Author(s):  
T. Hatch ◽  
A. P. Pisano
Keyword(s):  
Differential Equations ◽  
Modal Analysis ◽  
Equations Of Motion ◽  
Check Valve ◽  
Linear Differential Equations ◽  
Third Order ◽  
Mode Behavior ◽  
Special Cases ◽  
Linear Differential ◽  
Hydraulic Valve

Abstract A two-degree-of-freedom (2-DOF), analytical model of a hydraulic valve lifter is derived. Special features of the model include the effects of bulk oil compressibility, multi-mode behavior due to plunger check valve modeling, and provision for the inclusion of third and fourth body displacements to aid In the use of the model in extended, multi-DOF systems. It is shown that motion of the lifter plunger and body must satisfy a coupled system of third-order, non-linear differential equations of motion. It is also shown that the special cases of zero oil compressibility and/or 1-DOF motion of lifter plunger can be obtained from the general third-order equations. For the case of zero oil compressibility, using Newtonian fluid assumptions, the equations of motion are shown to reduce to a system of second-order, linear differential equations. The differential equations are numerically integrated in five scenarios designed to test various aspects of the model. A modal analysis of the 2-DOF, compressible model with an external contact spring is performed and is shown to be in excellent agreement with simulation results.

Solution Space Decompositions for nth Order Linear Differential Equations

1975 ◽  
Vol 27 (3) ◽  
pp. 508-512
Author(s):  
G. B. Gustafson ◽  
S. Sedziwy
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Solution Space ◽  
Decomposition Theorem ◽  
Linear Differential Equations ◽  
Direct Sum ◽  
Direct Sum Decomposition ◽  
Linear Differential ◽  
Image Position

Consider the wth order scalar ordinary differential equationwith pr ∈ C([0, ∞) → R ) . The purpose of this paper is to establish the following:DECOMPOSITION THEOREM. The solution space X of (1.1) has a direct sum Decompositionwhere M1 and M2 are subspaces of X such that(1) each solution in M1\﹛0﹜ is nonzero for sufficiently large t ﹛nono sdilatory) ;(2) each solution in M2 has infinitely many zeros ﹛oscillatory).

The Schwarzian Derivative and Disconjugacy of nth order Linear Differential Equations

1969 ◽  
Vol 21 ◽  
pp. 235-249 ◽  
Author(s):  
Meira Lavie
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Schwarzian Derivative ◽  
Order Linear ◽  
Second Order Differential Equations ◽  
Number Of Zeros ◽  
Linearly Independent ◽  
Basic Relationship ◽  
Linear Differential ◽  
Image Position

In this paper we deal with the number of zeros of a solution of the nth order linear differential equation1.1where the functions pj(z) (j = 0, 1, …, n – 2) are assumed to be regular in a given domain D of the complex plane. The differential equation (1.1) is called disconjugate in D, if no (non-trivial) solution of (1.1) has more than (n – 1) zeros in D. (The zeros are counted by their multiplicity.)The ideas of this paper are related to those of Nehari (7; 9) on second order differential equations. In (7), he pointed out the following basic relationship. The function1.2where y1(z) and y2(z) are two linearly independent solutions of1.3is univalent in D, if and only if no solution of equation(1.3) has more than one zero in D, i.e., if and only if(1.3) is disconjugate in D.

scholarly journals Discussion of Christofides' Conjecture Regarding Wang's Premium Principle

1999 ◽  
Vol 29 (2) ◽  
pp. 191-195 ◽  
Author(s):  
Virginia R. Young
Keyword(s):  
Distribution Function ◽  
Proportional Hazards ◽  
Random Variable ◽  
Reliable Measure ◽  
Scale Family ◽  
Special Cases ◽  
Parametric Families ◽  
Image Position ◽  
Special Case ◽  
Family Of Distributions

Christofides (1998) studies the proportional hazards (PH) transform of Wang (1995) and shows that for some parametric families, the PH premium principle reduces to the standard deviation (SD) premium principle. Christofides conjectures that for a parametric family of distributions with constant skewness, the PH premium principle reduces to the SD principle. I will show that this conjecture is false in general but that it is true for location-scale families and for certain other families.Wang's premium principle has been established as a sound measure of risk in Wang (1995, 1996), Wang, Young, and Panjer (1997), and Wang and Young (1998). Determining when the SD premium principle is a special case of Wang's premium principle is important because it will help identify circumstances under which the more easily applied SD premium principle is a reliable measure of risk.First, recall that a distortion g is a non-decreasing function from [0, 1] onto itself. Wang's premium principle, with a fixed distortion g, associates the following certainty equivalent with a random variable X, (Wang, 1996) and (Denneberg, 1994):in which Sx is the decumulative distribution function (ddf) of X, Sx(t) = Pr(X > t), t ∈ R. If g is a power distortion, g(p) = pc, then Hg is the proportional hazards (PH) premium principle (Wang, 1995).Second, recall that a location-scale family of ddfs is , in which Sz is a fixed ddf. Alternatively, if Z has ddf Sz, then {X = μ + σZ: μ∈ R, σ > 0} forms a location-scale family of random variables, and the ddf of . Examples of location-scale families include the normal, Cauchy, logistic, and uniform families (Lehmann, 1991, pp. 20f). In the next proposition, I show that Wang's premium principle reduces to the SD premium principle on a location-scale family. Christofides (1998) observes this phenomenon in several special cases.

scholarly journals The Elliptic Cylinder Functions of the Second Kind

1914 ◽  
Vol 33 ◽  
pp. 2-13 ◽  
Author(s):  
E. Lindsay Ince
Keyword(s):  
Differential Equation ◽  
General Solution ◽  
Differential Equations ◽  
Elliptic Cylinder ◽  
Linear Differential Equations ◽  
Periodic Functions ◽  
Linear Differential ◽  
Image Position

The differential equation of Mathieu, or the equation of the elliptic cylinder functionsis known by the theory of linear differential equations to have a general solution of the typeφ and ψ being periodic functions of z, with period 2π.

Trajectory Generation for Three-Degree-of-Freedom Cable-Suspended Parallel Robots Based on Analytical Integration of the Dynamic Equations

2016 ◽  
Vol 8 (4) ◽  
Author(s):  
Xiaoling Jiang ◽  
Clément Gosselin
Keyword(s):  
Differential Equations ◽  
Trajectory Generation ◽  
Parallel Robots ◽  
Degree Of Freedom ◽  
Dynamic Equations ◽  
Special Cases ◽  
Linear Differential ◽  
Pass Through ◽  
Cable Forces ◽  
Three Degree Of Freedom

This paper proposes a trajectory generation technique for three degree-of-freedom (3-dof) planar cable-suspended parallel robots. Based on the kinematic and dynamic modeling of the robot, positive constant ratios between cable tensions and cable lengths are assumed. This assumption allows the transformation of the dynamic equations into linear differential equations with constant coefficients for the positioning part, while the orientation equation becomes a pendulum-like differential equation for which accurate solutions can be found in the literature. The integration of the differential equations is shown to yield families of translational trajectories and associated special frequencies. This result generalizes the special cases previously identified in the literature. Combining the results obtained with translational trajectories and rotational trajectories, more general combined motions are analyzed. Examples are given in order to demonstrate the results. Because of the initial assumption on which the proposed method is based, the ratio between cable forces and cable lengths is constant and hence always positive, which ensures that all cables remain under tension. Therefore, the acceleration vector remains in the column space of the Jacobian matrix, which means that the mechanism can smoothly pass through kinematic singularities. The proposed trajectory planning approach can be used to plan dynamic trajectories that extend beyond the static workspace of the mechanism.

Oscillatory and asymptotic behaviour of all solutions of differential equations with deviating arguments

1978 ◽  
Vol 81 (3-4) ◽  
pp. 195-210 ◽  
Author(s):  
Ch. G. Philos
Keyword(s):  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Unknown Function ◽  
Continuous Functions ◽  
Linear Differential Equations ◽  
Deviating Arguments ◽  
All Solutions ◽  
Non Linear ◽  
Linear Differential ◽  
Image Position

SynopsisThis paper deals with the oscillatory and asymptotic behaviour of all solutions of a class of nth order (n > 1) non-linear differential equations with deviating arguments involving the so called nth order r-derivative of the unknown function x defined bywhere r1, (i = 0,1,…, n – 1) are positive continuous functions on [t0, ∞). The results obtained extend and improve previous ones in [7 and 15] even in the usual case where r0 = r1 = … = rn–1 = 1.

scholarly journals Stiff systems of ordinary differential equations

1981 ◽  
Vol 23 (2) ◽  
pp. 136-172 ◽  
Author(s):  
J. J. Mahony ◽  
J. J. Shepherd
Keyword(s):  
Differential Equations ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Singular Points ◽  
Operator Norm ◽  
Existence Theory ◽  
Stiff Systems ◽  
Stiff System ◽  
Linear Differential ◽  
Image Position

AbstractSolutions of the stiff system of linear differential equationsare obtained in a form yielding tight estimates of their properties, and conditions are obtained under which the operator norm of the map from r to the solution x does not become exponentially large for small values of ε. When these conditions are satisfied, the solutions are shown to be close to those of Ax + r = 0, save at any singular points of A, and in boundary layers. The behaviour of solutions near admissible singular points is also obtained.The results are used to characterize those boundary-value problems for the above system in which the solution defines maps from the data that are of “moderate” operator norm. This leads to a constructive existence theory for a limited class of boundary-value problems for the nonlinear systemIt is suggested that the treatment of more general classes of boundary-value problems may be simplified using these results. By the use of simple examples, the problems involving large operator norms are shown to be related to the stability properties of the possible branches of the outer solutions close to those of

Periodic Solutions by Picard's Approximations

1968 ◽  
Vol 11 (5) ◽  
pp. 743-745 ◽  
Author(s):  
T.A. Burton
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
General Framework ◽  
Floquet Theory ◽  
Linear Differential Equations ◽  
All Solutions ◽  
Private Correspondence ◽  
Linear Differential ◽  
Image Position

In [1] Demidovic considered a system of linear differential equationswith A(t) continuous, T-periodic, odd, and skew symmetric. He proved that all solutions of (1) are either T-periodic or 2T-periodic0 In [2] Epstein used Floquet theory to prove that all solutions of (1) are T-periodic without the skew symmetric hypothesis. Epstein's results were then generalized by Muldowney in [7] using Floquet theory. Much of the above work can also be interpreted as being part of the general framework of autosynartetic systems discussed by Lewis in [5] and [6]. According to private correspondence with Lewis it seems that he was aware of these results well before they were published. However, it appears that these theorems were neither stated nor suggested in the papers by Lewis.

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