Solution to Euler’s Gyrodynamics—I

1964 ◽  
Vol 31 (2) ◽  
pp. 325-328 ◽  
Author(s):  
C. F. Harding
Keyword(s):  
Differential Equation ◽  
Angular Velocity ◽  
Formal Solution ◽  
Three Dimensional ◽  
Rotation Group ◽  
Axially Symmetric ◽  
Free Case ◽  
Thrust Vector ◽  
Linear Differential ◽  
Constant Moment

A little used parameterization of the three-dimensional rotation group is taken as basis in deriving an easily integrable kinematic relation (a 4-vector linear differential equation) for the attitude rate, in terms of the present attitude and angular velocity of one reference frame relative to another. If the angular velocity is known and well behaved one obtains the exact solution from an iteration procedure explained in detail. The formal solution to a large class of rigid-body problems is thus implied; a particular one being that of an axially symmetric rocket with variable thrust vector and constant moment-of-inertia tensor which as such generalizes Jacobi’s torque-free case.

A GEOMETRIC ANALYSIS OF STABILITY REGIONS FOR A LINEAR DIFFERENTIAL EQUATION WITH TWO DELAYS

1995 ◽  
Vol 05 (03) ◽  
pp. 779-796 ◽  
Author(s):  
JOSEPH M. MAHAFFY ◽  
KATHRYN M. JOINER ◽  
PAUL J. ZAK
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Stability Region ◽  
Geometric Analysis ◽  
Three Dimensional ◽  
Initial Point ◽  
Two Delays ◽  
Analysis Of Stability ◽  
Linear Differential ◽  
The Stability

We describe an algorithmic approach for determining the geometry of the region of stability for a linear differential equation with two delays. Numerous applications utilize two-delay differential equations and require a framework to assay stability. The imaginary and zero solutions of the characteristic equation, where bifurcations in stability occur, produce an infinite set of surfaces in the coefficient parameter space. A methodology is outlined for identifying which of these surfaces form the boundary of the stability region. For a range of delays, the stability region changes in only three ways, starting at an identified initial point and becoming more complex as one coefficient increases. Detailed graphical analyses, including three-dimensional plots, show the evolution of the stability surface for given ratios of delays, highlighting variations across delays. The results demonstrate that small changes in the delay ratio cause significant changes in the size and shape of the stability region.

scholarly journals Correctness conditions for high-order differential equations with unbounded coefficients

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Kordan N. Ospanov
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Integral Operators ◽  
Order Linear Differential Equation ◽  
Unbounded Coefficients ◽  
Weighted Norms ◽  
Uniqueness Of The Solution ◽  
Linear Differential

AbstractWe give some sufficient conditions for the existence and uniqueness of the solution of a higher-order linear differential equation with unbounded coefficients in the Hilbert space. We obtain some estimates for the weighted norms of the solution and its derivatives. Using these estimates, we show the conditions for the compactness of some integral operators associated with the resolvent.

Asymptotic behavior of solutions of half-linear differential equations and generalized Karamata functions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Kusano Takaŝi ◽  
Jelena V. Manojlović
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Continuous Function ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Regular Variation ◽  
Asymptotic Formulas ◽  
Asymptotic Behavior Of Solutions ◽  
Positive Continuous Function ◽  
Linear Differential

AbstractWe study the asymptotic behavior of eventually positive solutions of the second-order half-linear differential equation(p(t)\lvert x^{\prime}\rvert^{\alpha}\operatorname{sgn}x^{\prime})^{\prime}+q(% t)\lvert x\rvert^{\alpha}\operatorname{sgn}x=0,where q is a continuous function which may take both positive and negative values in any neighborhood of infinity and p is a positive continuous function satisfying one of the conditions\int_{a}^{\infty}\frac{ds}{p(s)^{1/\alpha}}=\infty\quad\text{or}\quad\int_{a}^% {\infty}\frac{ds}{p(s)^{1/\alpha}}<\infty.The asymptotic formulas for generalized regularly varying solutions are established using the Karamata theory of regular variation.

scholarly journals Hyperbolastic Models from a Stochastic Differential Equation Point of View

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1835
Author(s):  
Antonio Barrera ◽  
Patricia Román-Román ◽  
Francisco Torres-Ruiz
Keyword(s):  
Differential Equation ◽  
Simulated Data ◽  
Real Data ◽  
Point Of View ◽  
Likelihood Method ◽  
Numerical Resolution ◽  
The Family ◽  
The Common ◽  
Linear Differential ◽  
Mean Function

A joint and unified vision of stochastic diffusion models associated with the family of hyperbolastic curves is presented. The motivation behind this approach stems from the fact that all hyperbolastic curves verify a linear differential equation of the Malthusian type. By virtue of this, and by adding a multiplicative noise to said ordinary differential equation, a diffusion process may be associated with each curve whose mean function is said curve. The inference in the resulting processes is presented jointly, as well as the strategies developed to obtain the initial solutions necessary for the numerical resolution of the system of equations resulting from the application of the maximum likelihood method. The common perspective presented is especially useful for the implementation of the necessary procedures for fitting the models to real data. Some examples based on simulated data support the suitability of the development described in the present paper.

Applications of Erdélyi-Kober fractional integral for solving time-fractional Tricomi-Keldysh type equation

2020 ◽  
Vol 23 (5) ◽  
pp. 1381-1400 ◽  
Author(s):  
Kangqun Zhang
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Integral Operator ◽  
Nonlinear Equations ◽  
Existence And Uniqueness ◽  
Fractional Integral ◽  
Formal Solution ◽  
Type Equation ◽  
Multiplier Theorem ◽  
Problem Of Time

Abstract In this paper we consider Cauchy problem of time-fractional Tricomi-Keldysh type equation. Based on the theory of a Erdélyi-Kober fractional integral operator, the formal solution of the inhomogeneous differential equation involving hyper-Bessel operator is presented with Mittag-Leffler function, then nonlinear equations are considered by applying Gronwall-type inequalities. At last, we establish the existence and uniqueness of L p -solution of time-fractional Tricomi-Keldysh type equation by use of Mikhlin multiplier theorem.

scholarly journals Steady Motion of Ice Sheets

1980 ◽  
Vol 25 (92) ◽  
pp. 229-246 ◽  
Author(s):  
L. W. Morland ◽  
I. R. Johnson
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Power Law ◽  
Perturbation Expansion ◽  
Ice Sheet ◽  
Leading Order ◽  
Plane Flow ◽  
General Law ◽  
Non Linear ◽  
Linear Differential

AbstractSteady plane flow under gravity of a symmetric ice sheet resting on a horizontal rigid bed, subject to surface accumulation and ablation, basal drainage, and basal sliding according to a shear-traction-velocity power law, is treated. The surface accumulation is taken to depend on height, and the drainage and sliding coefficient also depend on the height of overlying ice. The ice is described as a general non-linearly viscous incompressible fluid, with illustrations presented for Glen’s power law, the polynomial law of Colbeck and Evans, and a Newtonian fluid. Uniform temperature is assumed so that effects of a realistic temperature distribution on the ice response are not taken into account. In dimensionless variables a small paramter ν occurs, but the ν = 0 solution corresponds to an unbounded sheet of uniform depth. To obtain a bounded sheet, a horizontal coordinate scaling by a small factor ε(ν) is required, so that the aspect ratio ε of a steady ice sheet is determined by the ice properties, accumulation magnitude, and the magnitude of the central thickness. A perturbation expansion in ε gives simple leading-order terms for the stress and velocity components, and generates a first order non-linear differential equation for the free-surface slope, which is then integrated to determine the profile. The non-linear differential equation can be solved explicitly for a linear sliding law in the Newtonian case. For the general law it is shown that the leading-order approximation is valid both at the margin and in the central zone provided that the power and coefficient in the sliding law satisfy certain restrictions.

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