Numerical study of a nonlinear integrodifferential equation

I.IntroductionItiswell-knobal.nthatKorteweg-deVriesequationisacanonicalmodeltodescribethebalanceofthenonlineareffectandthedispersiveeffectofaphysicalsystem.Thisequationpossessestheso-called'soliton"solution,whichhasbeenfoundnumericallybyZabuskyandKruskall'].Ho-c'Jlever,sometimesthebalanceofnonlinearityanddispersionofasystemmayleadtoa,integroditTerentialequationinsteadofadifferentialequation.Forinstance,inthestudyofvortexbreakdownofanunboundedrotatingfluidLeibovich12]derivedfollowingnonline…     

Asymptotic solutions of a nonhomogeneous differential equation with a turning point

A linear second-order differential equation of the form

$$ d^{2 } U/d t^{2 } + \left[ {\lambda ^{2 } \varphi (t) + \lambda \chi (t,\lambda )} \right]U = \lambda ^{2 } \psi (t,\lambda ) $$     

Initial value problems for a class of nonlinear integro-partial differential equations

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Lower Semicontinuity for Integral Functionals in the Space of Functions of Bounded Deformation Via Rigidity and Young Measures

We establish a general weak* lower semicontinuity result in the space BD(Ω) of functions of bounded deformation for functionals of the form

Nonlinear Differential Equations with Quasinilpotent Operator and Unstable Trivial Solution

Let E be a Banach space. We prove the instability of the trivial solution of the differential equation

Oscillating Solutions and Large ω-limit Sets in a Degenerate Parabolic Equation

The paper deals with positive solutions of the initial-boundary value problem for with zero Dirichlet data in a smoothly bounded domain . Here is positive on (0,∞) with f(0) = 0, and λ₁ is exactly the first Dirichlet eigenvalue of −Δ in Ω. In this setting, (*) may possess oscillating solutions in presence of a sufficiently strong degeneracy. More precisely, writing , it is shown that if then there exist global classical solutions of (*) satisfying and . Under the additional structural assumption , s > 0, this result can be sharpened: If then (*) has a global solution with its ω-limit set being the ordered arc that consists of all nonnegative multiples of the principal Laplacian eigenfunction. On the other hand, under the above additional assumption the opposite condition ensures that all solutions of (*) will stabilize to a single equilibrium.      

On nonlinear hyperbolic equation in unbounded domain

The following nonlinear hyperbolic equation is discussed in this paper: where The model comes from the transverse deflection equation of an extensible beam. We prove that there exists a unique local solution of the above equation as M depends on x.     

Modified dissipativity for a non-linear evolution equation arising in turbulence

We are concerned with the regularity properties for all times of the equation $$\frac{{\partial U}}{{\partial t}}\left( {t,x} \right) = - \frac{{\partial ^2 }}{{\partial x^2 }}\left[ {U\left( {t,{\text{0}}} \right) - U\left( {t,x} \right)} \right]^2 - v\left( { - \frac{{\partial ^2 }}{{\partial x^2 }}} \right)^\alpha U\left( {t,x} \right)$$ which arises, with α=1, in the theory of turbulence. Here U(t,·) is of positive type and the dissipativity α is a non-negative real number. It is shown that for arbitrary ν≧0 and ?>0, there exists a global solution in \(L^\infty [0,\infty ;H^{\tfrac{3}{2} - \varepsilon } (\mathbb{R})]\) . If ν>0 and \(\alpha > \alpha _{cr} = \tfrac{1}{2}\) , smoothness of initial data persists indefinitely. If 0≦α<α _cr, there exist positive constants ν₁(α) and ν₂(α), depending on the data, such that global regularity persists for ν>ν₁(α), whereas, for 0≦ν<ν₂(α), the second spatial derivative at the origin blows up after a finite time. It is conjectured that with a suitable choice of α _cr, similar results hold for the Navier-Stokes equation.     

Generalized diagonalization of matrices over quaternion field

Ｉｎｒｅｃｅｎｔｙｅａｒｓ,ａｐｐｌｉｃａｔｉｏｎｓｏｆｑｕａｔｅｒｎｉｏｎｍａｔｒｉｃｅｓａｒｅｂｅｃｏｍｉｎｇｍｏｒｅａｎｄｍｏｒｅｉｍｐｏｒｔａｎｔａｎｄｅｘｔｅｎｓｉｖｅｉｎｒｉｇｉｄｍｅｃｈａｎｉｃｓ,ｑｕａｎｔｕｍｍｅｃｈａｎｉｃｓ,ｃｏｎｔｒｏｌｔｈｅｏｒｙａｎｄｈｅｌｉｃａｌｔｅｃｈｎｏｌｏｇｙ［１～３］．Ｗｉｔｈｔｈｅｒａｐｉｄｄｅｖｅｌｏｐｍｅｎｔｏｆｔｈｅａｂｏｖｅｄｉｓｃｉｐｌｉｎｅｓ,ｉｔｉｓｇｅｔｔｉｎｇｍｏｒｅａｎｄｍｏｒｅｎｅｃｅｓｓａｒｙｆｏｒｕｓｔｏｆｕｒｔｈ…     

The extended Jordan's lemma and the relation between laplace transform and fourier transform

I.IntroductionItiswell4n0wnthatthecontourintergrationofcomp1exvariableftinctioncanmakealotintegrationveryconvenient.Jordan'slemmahasaveryimportantstatusintheonec0mplexvariableintegration,anditisveryusefulforavarityofintegration.Withthetheoryoffunctionsofo…     

Detailed asymptotics for a convex Hamiltonian system with two degrees of freedom

The system of ordinary differential equations

A Nonlinear Elliptic PDE with Two Sobolev–Hardy Critical Exponents

In this paper, we consider the following PDE involving two Sobolev–Hardy critical exponents,

Asymptotic and oscillation of the solutions for a class of the first order neutral differential equations

ＩｎｔｒｏｄｕｃｔｉｏｎＯｗｉｎｇｔｏｔｈｅｅｘｔｅｎｓｉｖｅａｐｐｌｉｃａｔｉｏｎｏｆｎｅｕｔｒａｌｅｑｕａｔｉｏｎｓ,ｍｏｒｅａｎｄｍｏｒｅｓｔｕｄｉｅｓｈａｖｅｂｅｎｍａｄｅｏｎｔｈｅｂｅｈａｖｉｏｒｏｆｔｈｅｓｏｌｕｔｉｏｎｓ［１,２］．Ｆｏ...     

Regularity for the stationary Navier-Stokes equations in bounded domain

Let u, p be a weak solution of the stationary Navier-Stokes equations in a bounded domain ^N, 5N . If u, p satisfy the additional conditions

Piecewise Implicit Differential Systems

In this article we deal with non-smooth dynamical systems expressed by a piecewise first order implicit differential equations of the form

$$\begin{aligned} \dot{x}=1,\quad \left( \dot{y}\right) ^2=\left\{ \begin{array}{lll} g_1(x,y) \quad \text{ if }\quad \varphi (x,y)\ge 0 \\ g_2(x,y) \quad \text{ if }\quad \varphi (x,y)\le 0 \end{array},\right. \end{aligned}$$

where \(g_1,g_2,\varphi :U\rightarrow \mathbb {R}\) are smooth functions and \(U\subseteq \mathbb {R}^2\) is an open set. The main concern is to study sliding modes of such systems around some typical singularities. The novelty of our approach is that some singular perturbation problems of the form

$$\begin{aligned} \dot{x}= f(x,y,\varepsilon ) ,\quad (\varepsilon \dot{ y})^2=g ( x,y,\varepsilon ) \end{aligned}$$

arise when the Sotomayor–Teixeira regularization is applied with \((x, y) \in U\) , \(\varepsilon \ge 0\), and f, g smooth in all variables.

     

On the Damped Harmonic Oscillator with Time Dependent Damping Coefficient

Conditions guaranteeing asymptotic stability for the differential equation

$$\begin{aligned} x''+h(t)x'+\omega ^2x=0 \qquad (x\in \mathbb {R}) \end{aligned}$$

are studied, where the damping coefficient \(h:[0,\infty )\rightarrow [0,\infty )\) is a locally integrable function, and the frequency \(\omega >0\) is constant. Our conditions need neither the requirement \(h(t)\le \overline{h}<\infty \) (\(t\in [0,\infty )\); \(\overline{h}\) is constant) (“small damping”), nor \(0< \underline{h}\le h(t)\) (\(t\in [0,\infty )\); \(\underline{h}\) is constant) (“large damping”); in other words, they can be applied to the general case \(0\le h(t)<\infty \) (\(t\in [0,\infty \))). We establish a condition which combines weak integral positivity with Smith’s growth condition

$$\begin{aligned} \int ^\infty _0 \exp [-H(t)]\int _0^t \exp [H(s)]\,\mathrm{{d}}s\,\mathrm{{d}}t=\infty \qquad \left( H(t):=\int _0^t h(\tau )\,\mathrm{{d}}\tau \right) , \end{aligned}$$

so it is able to control both the small and the large values of the damping coefficient simultaneously.

     

The Extinction Behavior of the Solutions for a Class of Reaction-Diffusion Equations

1　ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＬｅｍｍａｓＴｈｅｒｅａｒｅｍａｎｙｒｅｓｕｌｔｓａｂｏｕｔｅｘｉｓｔｅｎｃｅ (ｇｌｏｂａｌｏｒｌｏｃａｌ)ａｎｄａｓｙｍｐｔｏｔｉｃｂｅｈａｖｉｏｒｏｆｓｏｌｕｔｉｏｎｓｆｏｒｒｅａｃｔｉｏｎ＿ｄｉｆｆｕｓｉｏｎｅｑｕａｔｉｏｎｓ[1- 9].Ｂｙｔｈｅａｉｄｓｏｆｒｅｓｕｌｔｓ[2 ,3]ｏｆｅｑｕａｔｉｏｎ ｕ/ ｔ=Δｕ-λ｜ｕ｜γ- 1ｕｗｉｔｈｉｎｉｔｉａｌ＿ｂｏｕｎｄａｒｙｖａｌｕｅｓ,ｐａｐｅｒ [4 ]ｓｔｕｄｉｅｄｔｈｅｐｒｏｂｌｅｍｏｆ ｕ/ ｔ=Δｕ-λ｜ｅβｔｕ｜γ- …     

Iterative approximation with errors of fixed point for a class of nonlinear operator with a bounder range

Let X be a uniformly smooth real Banach space. Let T:X → X be continuos and strongly accretive operator. For a given f ε X, define S: X → X by Sx =f−Tx+x, for all x ε X. Let {a_n} _n=0 ^∞ , {β_n} _n=0 ^∞ be two real sequences in (0, 1) satisfying:

Stability of a class of neural network models with delay

ＩｎｔｒｏｄｕｃｔｉｏｎＲｅｃｅｎｔｌｙ,ｔｈｅｏｒｅｔｉｃａｌａｎｄａｐｌｉｅｄｓｔｕｄｉｅｓｏｆｎｅｕｒａｌｎｅｔｗｏｒｋｍｏｄｅｌｈａｖｅｂｅｎｔｈｅｎｅｗｆｏｃｕｓｏｆｓｔｕｄｉｅｓｉｎｔｈｅｗｏｒｌｄ．Ｉｔｉｓｗｅｌ＿ｋｎｏｗｎｔｈａｔｑｕ...