Bending of Plates on Thin Elastomeric Foundations

1989 ◽  
Vol 56 (2) ◽  
pp. 382-386 ◽  
Author(s):  
D. A. Dillard
Keyword(s):  
Differential Equation ◽  
Closed Form ◽  
Fourth Order ◽  
Order Equation ◽  
Decay Rates ◽  
Series Solutions ◽  
Rigid Substrate ◽  
Winkler Elastic Foundation ◽  
Governing Differential Equation ◽  
Oscillation Decay

Closed-form and series solutions are presented for the bending of plates bonded to a thin elastomeric foundation which is in turn bonded to a rigid substrate. The standard fourth-order governing differential equation of a classical Winkler elastic foundation becomes a sixth-order equation for the case of an incompressible foundation. Oscillation decay rates are shown to be significantly different from those of the Winkler solution due to the incompressibility of the elastomer.

A Closed-Form Numerical Algorithm for the Periodic Response of High-Speed Elastic Linkages

1979 ◽  
Vol 101 (1) ◽  
pp. 154-162 ◽  
Author(s):  
A. Midha ◽  
A. G. Erdman ◽  
D. A. Frohrib
Keyword(s):  
Differential Equation ◽  
Closed Form ◽  
High Speed ◽  
Order Equation ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Time Period ◽  
Governing Differential Equation ◽  
Constant Coefficients ◽  
Spring System

A numerical closed-form algorithm, easily adaptable for computer simulation, is developed to solve for the periodic solutions of vibrating systems, and in particular, the high-speed elastic linkage. The algorithm is first introduced to solve the single degree-of-freedom mass-dashpot-spring system, the governing differential equation of which is a linear, second-order equation with constant coefficients. This algorithm is utilized as a basic tool and extended to solve a single degree-of-freedom mass-dashpot-spring system whose governing differential equation of motion is a linear, second-order equation with time-dependent and periodic coefficients. The system is excited by a periodic forcing function and solution is made possible by discretizing the forcing time period into a number of time intervals, the system parameters remaining constant over the duration of each interval. During each interval, the solution form is assumed to be that of the differential equation with “constant” coefficients. Constraint equations result from imposing the conditions of “compatibility” of response at the discrete time nodes and the conditions of “periodicity” of response at the end nodes of the time period. Also, the sum of the integration required is over one forcing time period only. This closed-form nature of the computational procedure results in large savings in computer time to acquire the periodic solution. The suggested numerical algorithm is then employed to solve an elastic linkage problem.

26.—The Strong Limit-2 Case of Fourth-order Differential Equations

1974 ◽  
Vol 71 (4) ◽  
pp. 297-304
Author(s):  
V. Krishna Kumar
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Order Equation ◽  
Complex Numbers ◽  
Strong Limit ◽  
Fourth Order Equation ◽  
Linearly Independent ◽  
Image Position ◽  
Adjoint Operators ◽  
Bounded Below

SynopsisThe fourth-order equation considered isConditions are given on the coefficients r, p and q which ensure that this differential equation (*) is in the strong limit-2 case at ∞, i.e. is limit-2 at ∞. This implies that (*) has exactly two linearly independent solutions which are in the integrable-square space ℒ2(0, ∞) for all complex numbers λ with im [λ] ≠ 0. Additionally the conditions imply that self-adjoint operators generated by M[·] in ℒ2(0, ∞) are semi-bounded below. The results obtained are applied to the case when the coefficients r, p and q are powers of x ∈ [0, ∞).

XXIII.—On an Extension to an Integro-differential Inequality of Hardy, Littlewood and Polya

1972 ◽  
Vol 69 (4) ◽  
pp. 295-333 ◽  
Author(s):  
W. N. Everitt
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Differential Inequality ◽  
Order Differential Equation ◽  
Order Equation ◽  
Second Order ◽  
Image Position ◽  
Fourth Order Differential Equation

SynopsisThis paper considers an extension of the following inequality given in the book Inequalities by Hardy, Littlewood and Polya; let f be real-valued, twice differentiable on [0, ∞) and such that f and f are both in the space fn, ∞), then f′ is in L,2(0, ∞) andThe extension consists in replacing f′ by M[f] wherechoosing f so that f and M[f] are in L2(0, ∞) and then seeking to determine if there is an inequality of the formwhere K is a positive number independent of f.The analysis involves a fourth-order differential equation and the second-order equation associated with M.A number of examples are discussed to illustrate the theorems obtained and to show that the extended inequality (*) may or may not hold.

scholarly journals Euler case for a general fourth-order differential equation

2004 ◽  
Vol 2004 (51) ◽  
pp. 2705-2717
Author(s):  
A. S. A. Al-Hammadi
Keyword(s):  
Differential Equation ◽  
General Formula ◽  
Fourth Order ◽  
Asymptotic Form ◽  
Order Differential Equation ◽  
Order Equation ◽  
Fourth Order Equation ◽  
Fourth Order Differential Equation

We deal with an Euler case for a general fourth-order equation and under this case, we obtain the general formula for the asymptotic form of the solutions.

The asymptotic form of the Titchmarsh–Weyl coefficient for a fourth order differential equation

1984 ◽  
Vol 97 ◽  
pp. 109-123
Author(s):  
D. B. Hinton ◽  
J. K. Shaw
Keyword(s):  
Differential Equation ◽  
Asymptotic Formula ◽  
Real Axis ◽  
Fourth Order ◽  
Asymptotic Form ◽  
Order Differential Equation ◽  
Order Equation ◽  
Fourth Order Equation ◽  
The Matrix ◽  
Fourth Order Differential Equation

SynopsisThis paper considers the asymptotic form, as λ tends to infinity in sectors omitting the real axis, of the matrix Titchmarsh-Weyl coefficient M(λ) for the fourth order equation y(4) + q(x)y = λy, where q(x) is real and locally absolutely integrable. By letting M0(λ) denote the m-coefficient for the Fourier case y(4) = λy, the asymptotic formula M(λ) = M0(λ) + 0(1) is established.

scholarly journals Existence of Positive Solutions for a Pertubed Fourth-Order Equation

2021 ◽  
Vol 45 (4) ◽  
pp. 623-633
Author(s):  
MOHAMMAD REZA HEIDARI TAVANI ◽  
◽  
ABDOLLAH NAZARI ◽  
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Weak Solutions ◽  
Nonlinear Term ◽  
Point Theory ◽  
Order Equation ◽  
Fourth Order Equation ◽  
Equation Problem ◽  
Existence Of Positive Solutions ◽  
Fourth Order Problems

In this paper, a special type of fourth-order differential equations with a perturbed nonlinear term and some boundary conditions is considered which is very important in mechanical engineering. Therefore, the existence of a non-trivial solution for such equations is very important. Our goal is to ensure at least three weak solutions for a class of perturbed fourth-order problems by applying certain conditions to the functions that are available in the differential equation (problem (??)). Our approach is based on variational methods and critical point theory. In fact, using a fundamental theorem that is attributed to Bonanno, we get some important results. Finally, for some results, an example is presented.

scholarly journals Fourth-Order Differential Equation with Deviating Argument

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
M. Bartušek ◽  
M. Cecchi ◽  
Z. Došlá ◽  
M. Marini
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Order Equation ◽  
Asymptotically Linear ◽  
Middle Term ◽  
Deviating Argument ◽  
Necessary And Sufficient ◽  
Fourth Order Differential Equation

We consider the fourth-order differential equation with middle-term and deviating argumentx(4)(t)+q(t)x(2)(t)+r(t)f(x(φ(t)))=0, in case when the corresponding second-order equationh″+q(t)h=0is oscillatory. Necessary and sufficient conditions for the existence of bounded and unbounded asymptotically linear solutions are given. The roles of the deviating argument and the nonlinearity are explained, too.

On linear wave motions in magnetic-velocity shears

1977 ◽  
Vol 285 (1328) ◽  
pp. 607-636 ◽  
Keyword(s):  
Differential Equation ◽  
Critical Level ◽  
Velocity Shear ◽  
Linear Wave ◽  
Wave Amplification ◽  
Isothermal Fluid ◽  
Governing Differential Equation ◽  
Boussinesq Fluid ◽  
Propagation Properties ◽  
Background State

The propagation properties of linear wave motions in magnetic and/or velocity shears which vary in one coordinate z (say) are usually governed by a second order linear ordinary differential equation in the independent variable z. It is proved that associated with any such differential equation there always exists a quantity A which is independent of z. By employing A a measure of the intensity of the wave, this result is used to investigate the general propagation properties of hydromagnetic-gravity waves (e.g. critical level absorption, valve effects and wave amplification) in magnetic and/or velocity shears, using a full wave treatment. When variations in the basic state are included, the governing differential equation usually has more singularities than it has in the W.K.B.J. approximation, which neglects all variations in the background state. The study of a wide variety of models shows that critical level behaviour occurs only at the singularities predicted by the W.K.B.J. approximation. Although the solutions of the differential equation are necessarily singular at the irregularities whose presence is solely due to the inclusion of variations in the basic state, the intensity of the wave (as measured by A) is continuous there. Also the valve effect is found to persist whatever the relation between the wavelength of the wave and the scale of variations of the background state. In addition, it is shown that a hydromagnetic-gravity wave incident upon a finite magnetic and/or velocity shear can be amplified (or over-reflected) in the absence of any critical levels within the shear layer. In a Boussinesq fluid rotating uniformly about the vertical, wave amplification can occur if the horizontal vertically sheared flow and magnetic field are perpendicular. In a compressible isothermal fluid, on the other hand, wave amplification not only occurs in both magnetic-velocity and velocity shears but also in a magnetic shear acting alone.

THE FORWARD PDE FOR EUROPEAN OPTIONS ON STOCKS WITH FIXED FRACTIONAL JUMPS

2005 ◽  
Vol 08 (02) ◽  
pp. 239-253 ◽  
Author(s):  
PETER CARR ◽  
ALIREZA JAVAHERI
Keyword(s):  
Differential Equation ◽  
Closed Form ◽  
Stock Price ◽  
Arrival Rate ◽  
European Options ◽  
Standard Assumption ◽  
Integro Differential Equation ◽  
Local Variance ◽  
Calendar Dates

We derive a partial integro differential equation (PIDE) which relates the price of a calendar spread to the prices of butterfly spreads and the functions describing the evolution of the process. These evolution functions are the forward local variance rate and a new concept called the forward local default arrival rate. We then specialize to the case where the only jump which can occur reduces the underlying stock price by a fixed fraction of its pre-jump value. This is a standard assumption when valuing an option written on a stock which can default. We discuss novel strategies for calibrating to a term and strike structure of European options prices. In particular using a few calendar dates, we derive closed form expressions for both the local variance and the local default arrival rate.

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