scholarly journals Extreme Analysis of a Random Ordinary Differential Equation

2014 ◽  
Vol 51 (4) ◽  
pp. 1021-1036 ◽  
Author(s):  
Jingchen Liu ◽  
Xiang Zhou
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Elliptic Equation ◽  
Gaussian Process ◽  
Stochastic System ◽  
Random Coefficient ◽  
Asymptotic Approximations ◽  
Tail Probabilities ◽  
One Dimensional ◽  
Spatially Varying

In this paper we consider a one dimensional stochastic system described by an elliptic equation. A spatially varying random coefficient is introduced to account for uncertainty or imprecise measurements. We model the logarithm of this coefficient by a Gaussian process and provide asymptotic approximations of the tail probabilities of the derivative of the solution.

scholarly journals Extreme Analysis of a Random Ordinary Differential Equation

2014 ◽  
Vol 51 (04) ◽  
pp. 1021-1036
Author(s):  
Jingchen Liu ◽  
Xiang Zhou
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Elliptic Equation ◽  
Gaussian Process ◽  
Stochastic System ◽  
Random Coefficient ◽  
Asymptotic Approximations ◽  
Tail Probabilities ◽  
One Dimensional ◽  
Spatially Varying

In this paper we consider a one dimensional stochastic system described by an elliptic equation. A spatially varying random coefficient is introduced to account for uncertainty or imprecise measurements. We model the logarithm of this coefficient by a Gaussian process and provide asymptotic approximations of the tail probabilities of the derivative of the solution.

scholarly journals A reaction infiltration problem: classical solutions

1997 ◽  
Vol 40 (2) ◽  
pp. 275-291 ◽  
Author(s):  
John Chadam ◽  
Xinfu Chen ◽  
Roberto Gianni ◽  
Riccardo Ricci
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Parabolic Equation ◽  
Elliptic Equation ◽  
Boundary Conditions ◽  
Classical Solution ◽  
Existence And Uniqueness ◽  
Classical Solutions ◽  
Bounded Domains ◽  
Global Classical Solution

In this paper, we consider a reaction infiltration problem consisting of a parabolic equation for the concentration, an elliptic equation for the pressure, and an ordinary differential equation for the porosity. Existence and uniqueness of a global classical solution is proved for bounded domains Ω⊂RN, under suitable boundary conditions.

Nonlinear stationary magnetoconvection in a rotating fluid

1997 ◽  
Vol 58 (3) ◽  
pp. 395-408 ◽  
Author(s):  
S. G. TAGARE
Keyword(s):  
Magnetic Field ◽  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Pitchfork Bifurcation ◽  
Time Dependent ◽  
Vertical Magnetic Field ◽  
Stationary Convection ◽  
Rotating Fluid ◽  
Order Ordinary Differential Equation ◽  
One Dimensional

We investigate finite-amplitude magnetoconvection in a rotating fluid in the presence of a vertical magnetic field when the axis of rotation is parallel to a vertical magnetic field. We derive a nonlinear, time-dependent, one-dimensional Landau–Ginzburg equation near the onset of stationary convection at supercritical pitchfork bifurcation whenformula hereand a nonlinear time-dependent second-order ordinary differential equation when Ta=T*a (from below). Ta=T*a corresponds to codimension-two bifurcation (or secondary bifurcation), where the threshold for stationary convection at the pitchfork bifurcation coincides with the threshold for oscillatory convection at the Hopf bifurcation. We obtain steady-state solutions of the one-dimensional Landau–Ginzburg equation, and discuss the solution of the nonlinear time-dependent second-order ordinary differential equation.

On the method of matched asymptotic expansions

1969 ◽  
Vol 65 (1) ◽  
pp. 233-261 ◽  
Author(s):  
L. E. Fraenkel
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Value Problem ◽  
Turning Point ◽  
Formal Method ◽  
Asymptotic Approximations ◽  
Restricted Form ◽  
Uniform Approximations ◽  
Asymptotic Matching ◽  
Composite Series

AbstractThe prescription for forming composite series, and a modified form of that procedure, are used systematically and extensively to derive uniform asymptotic approximations to various given functions which cannot be approximated uniformly by expansions of classical (Poincaré) type. The formal method of matched expansions is then applied to a boundary-value problem for an ordinary differential equation with a turning point. It is proved with the help of the uniform approximations found earlier that in this problem a restricted form of the asymptotic matching principle is valid, even when it is applied to truncated inner and outer expansions which do not overlap to the order of the terms being matched.

scholarly journals Multiple positive solutions to mixed boundary value problems for singular ordinary differential equations on the whole line

2012 ◽  
Vol 17 (4) ◽  
pp. 460-480 ◽  
Author(s):  
Yuji Yuji
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Nonnegative Function ◽  
Multiple Positive Solutions ◽  
Mixed Boundary Value Problem ◽  
One Dimensional ◽  
Singular Ordinary Differential Equation ◽  
The One

This paper is concerned with the mixed boundary value problem of the second order singular ordinary differential equation[Φ(ρ(t)x'(t))]' + f(t, x(t), x'(t)) = 0,   t ∈ R,limt→−∞ x(t) = ∫−∞+∞ g(s, x(s), x'(s)) ds,limt→+∞ ρ(t)x'(t) =  ∫−∞+∞h(s, x(s), x' (s)) ds.Sufficient conditions to guarantee the existence of at least one positive solution are established. The emphasis is put on the one-dimensional p-Laplacian term [Φ(ρ(t)x'(t))]' involved with the nonnegative function ρ satisfying ∫−∞+∞1/ρ(s) ds = +∞.

scholarly journals Axisymmetric bending analysis of functionally graded one-dimensional hexagonal piezoelectric quasi-crystal circular plate

2020 ◽  
Vol 476 (2241) ◽  
pp. 20200301
Author(s):  
Yang Li ◽  
Yuan Li ◽  
Qinghua Qin ◽  
Lianzhi Yang ◽  
Liangliang Zhang ◽  
...  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
State Space ◽  
Circular Plate ◽  
Hankel Transform ◽  
Functionally Graded ◽  
The State ◽  
One Dimensional ◽  
Present Solution ◽  
Axisymmetric Bending

Within a framework of the state space method, an axisymmetric solution for functionally graded one-dimensional hexagonal piezoelectric quasi-crystal circular plate is presented in this paper. Applying the finite Hankel transform onto the state space vector, an ordinary differential equation with constant coefficients is obtained for the circular plate provided that the free boundary terms are zero and an exponential function distribution of material properties is assumed. The ordinary differential equation is then used to obtain the stress, displacement and electric components in the physical domain of the elastic simply supported circular plate through the use of the propagator matrix method and the inverse Hankel transform. The numerical studies are carried out to show the validity of the present solution and reveal the influence of material inhomogeneity on the axisymmetric bending of the circular plate with different layers and loadings, which provides guidance for the design and manufacture of functionally graded one-dimensional hexagonal piezoelectric quasi-crystal circular plate.

Optimal shape of slowly rising gas bubbles

2005 ◽  
Vol 83 (7) ◽  
pp. 761-766
Author(s):  
Alexei M Frolov
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Incompressible Fluid ◽  
Three Dimensional ◽  
Gas Bubbles ◽  
Ideal Incompressible Fluid ◽  
Optimal Shape ◽  
Dimensional Problem ◽  
Optimal Form ◽  
One Dimensional

The variational optimal shape of slowly rising gas bubbles in an ideal incompressible fluid is determined. It is shown that the original three-dimensional problem can be reduced to a relatively simple one-dimensional (i.e., ordinary) differential equation. The solution of this equation allows one to obtain the variational optimal form of slowly rising gas bubbles. PACS No.: 47.55.Dz

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