Singular perturbations of two-point boundary problems for systems of ordinary differential equations

Asymptotic solutions of linear systems of ordinary differential equations are employed to discuss the relationship of the solution of a certain “complete” boundary problem.

$$\begin{gathered} \left\{ \begin{gathered} {\text{ }}\frac{{d{\text{ }}x_1 }}{{d{\text{ }}t}} = A_{11} (t,\varepsilon ){\text{ }}x_1 (t,\varepsilon ){\text{ }} + \cdots + A_{1p} (t,\varepsilon ){\text{ }}x_p (t,\varepsilon ) \hfill \\ \varepsilon ^{h_2 } \frac{{d{\text{ }}x_2 }}{{d{\text{ }}t}} = A_{21} (t,\varepsilon ){\text{ }}x_1 (t,\varepsilon ){\text{ }} + \cdots + A_{2p} (t,\varepsilon ){\text{ }}x_p (t,\varepsilon ) \hfill \\ {\text{ }} \vdots {\text{ }} \vdots {\text{ }} \vdots \hfill \\ \varepsilon ^{h_p } \frac{{d{\text{ }}x_2 }}{{d{\text{ }}t}} = A_{p1} (t,\varepsilon ){\text{ }}x_1 (t,\varepsilon ){\text{ }} + \cdots + A_{pp} (t,\varepsilon ){\text{ }}x_p (t,\varepsilon ) \hfill \\ \end{gathered} \right\} \hfill \\ {\text{ }}R(\varepsilon ){\text{ }}x(a,{\text{ }}\varepsilon ){\text{ }} + {\text{ }}S(\varepsilon ){\text{ }}x(b,{\text{ }}\varepsilon ) = c(\varepsilon ){\text{ }} \hfill \\ \end{gathered}$$     

Approximating equations of a plane gas flow and their integration

It is known that the nonlinear system of equations of plane steady isentropic potential gas flow can be linearized and transformed to a single equivalent linear differential equation of second order. For the case of a perfect gas this equation has the form [1]

$$\begin{gathered} \frac{{1 - \tau ^2 }}{{\tau ^2 (1 - \alpha \tau ^2 )}} \frac{{\partial ^2 \Phi }}{{\partial \theta ^2 }} + \frac{{\partial ^2 \Phi }}{{\partial \tau ^2 }} + \frac{{\tau (1 - \tau ^2 )}}{{\tau ^2 (1 - \alpha \tau ^2 )}} \frac{{\partial \Phi }}{{\partial \tau }} = 0, \hfill \\ (\tau = w/c_k , w = \sqrt {u^2 + \upsilon ^2 } , \alpha = (\gamma - 1)/(\gamma + 1); \gamma = c_p /c_\upsilon ). (0.1) \hfill \\ \end{gathered} $$     

Turbulent heat transfer in circular tube flows of viscoelastic fluids

The effects of thermal entrance length, polymer degradation and solvent chemistry were found to be critically important in the determination of the drag and heat transfer behavior of viscoelastic fluids in turbulent pipe flow. The minimum heat transfer asymptotic values in the thermally developing and in the fully developed regions were experimentally determined for relatively high concentration solutions of heat transfer resulting in the following correlations: $$\begin{gathered} j_H = 0.13\left( {\frac{x}{d}} \right)^{ - 0.24} \operatorname{Re} _a^{ - 0.45} thermally developing region \hfill \\ x/d< 450 \hfill \\ j_H = 0.03 \operatorname{Re} _a^{ - 0.45} thermally developed region \hfill \\ x/d< 450 \hfill \\ \end{gathered} $$ For dilute polymer solutions the heat transfer is a function ofx/d, the Reynolds number and the polymer concentration. The Reynolds analogy between momentum and heat transfer which has been widely used in the literature for Newtonian fluids is found not to apply in the case of drag-reducing viscoelastic fluids.     

A shock tube study on the thermal decomposition of CS2 based on S(3P) and S(1D) concentration measurements

The thermal decomposition of CS₂ highly diluted in Ar was studied behind reflected shock waves by monitoring time-dependent absorption profiles of S(³P) and S(¹D) using atomic resonance absorption spectroscopy (ARAS). The rate coefficient of the reaction:

Messung des gleichzeitigen Wärme- und Stoffübergangs am horizontalen Zylinder bei freier Konvektion

Simultaneous heat and mass transfer at horizontal cylinders under free convection conditions have been investigated by an electrochemical method. Over all and local measurements of the mass transfer coefficient have been executed. The over all results can be represented by two empirical equations: $$\begin{gathered} Nu = 0,858 \cdot (Gr \cdot Pr)^{0,22} \hfill \\ Sh = 0,23 \cdot [(Gr' \cdot Sc)^{1,07} + 1,49\sqrt {Sc/Pr} \cdot Gr \cdot Sc]^{0,28} \hfill \\ \end{gathered}$$ The ranges of the dimensionsless groups were as follows:Gr: 7,31 · 10² to 8,69 · 10⁵ Gr′: 1,21 · 10³ to 2,79 · 10⁵ Pr: 10 to 7,56Sc: 3440 to 1890     

The existence of solution of a class of two-order quasilinear boundary value problem

Ref. [1] discussed the existence of positive solutions of quasilinear two-point boundary problems: but it restricts O相似文献   

Asymptotic method for singular perturbation problem of ordinary difference equations

This paper is taken up for the following difference equation problem(P,)(L,y)_k≡εy(k 1) a(k,ε)y(k) b(k,ε)y(k-1)=f(k,ε)(1≤k≤N-1),B_1y≡-y(0) c_1y(1)=a,B_2y≡-c_2y(N-1) y(N)=βwhereεis a small parameter,c_1,c_2,a,βconstants and a(k,ε),b(k,ε),f(k,ε)(1≤k≤N)functions of k andε.Firstly,the case with constant coefficients isconsidered.Secondly,a general method based on extended transformation is given tohandle(P.)where the coefficients may be variable and uniform asymptotic expansionsare obtained Finally,a numerical example is provided to illustrate the proposed method.     

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

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Finite-dimensional asymptotic behavior of some semilinear damped hyperbolic problems

We prove that the solution semigroup $$S_t \left[ {u_0 ,v_0 } \right] = \left[ {u(t),u_t (t)} \right]$$ generated by the evolutionary problem $$\left\{ P \right\}\left\{ \begin{gathered} u_{tt} + g(u_t ) + Lu + f(u) = 0, t \geqslant 0 \hfill \\ u(0) = u_0 , u_t (0) = \upsilon _0 \hfill \\ \end{gathered} \right.$$ possesses a global attractorA in the energy spaceE _o=V×L ²(Ω). Moreover,A is contained in a finite-dimensional inertial setA attracting bounded subsets ofE ₁=D(L)×V exponentially with growing time.     

Filtration problems with a piecewise-linear resistance law

The article discusses elementary solutions of problems of nonlinear filtration with a piece-wise-linear resistance law, and analyzes their behavior with a relative increase in the resistance in the region of small velocities, and a transition to the law of filtration with a limiting gradient. The results obtained are applied to a determination of the dimensions of the stagnant zones in stratified strata. The law of filtration with a limiting gradient (0.1) $$\begin{gathered} w = - \frac{k}{\mu }\left( {grad p - G\frac{{grad p}}{{|grad p|}}} \right),|grad p| > G \hfill \\ w = 0,|grad p|< G \hfill \\ \end{gathered}$$ describes motion in some intermediate range of velocities w, but its satisfaction in the region of the smallest velocities, as a rule, remains unverified. It is natural to pose the problem of the degree to which a divergence between the true filtration law and its approximation (0.1) affects the accuracy of calculation of the flow fields, and the significance of a determination of the dimensions of the stagnant zones under such conditions. To answer this problem to some measure, there are considered below several simple exact (elementary) solutions obtained for a more general nonlinear filtration law (0.2) $$\begin{gathered} grad p = - (\mu /k) (w + \lambda )w/w,\lambda = kG/\mu ,w \geqslant w_0 \hfill \\ grad p = - \mu w/k\varepsilon ), w \leqslant w_0 ,\varepsilon = w_0 /w_0 + \lambda \hfill \\ \end{gathered}$$ going over into (0.1) with w₀→ 0. The solutions obtained are applied also to an evaluation of the dimensions of stagnant zones, forming in stratified strata when the effects of the limiting gradient in one of the intercalations are considerable.     

Forced convective mass transfer studies from spheroids

In order to obtain information on the effect of shape on mass transfer, overall mass transfer rates were measured from naphthalene spheroids suspended in a wind tunnel (Schmidt number 2.4). Spheroidal shapes which included spheres, oblate spheroids and spheroids with composite halves were employed for the study. The ratio of the minor to major axes of the spheroids ranged from 1∶1 to 1∶4. The data obtained from one-hundred and fifty six experimental runs were best correlated by the use of Pasternak and Gauvin's characteristic dimension defined as total surface area of the body divided by maximum perimeter normal to flow. The correlations for the ranges 200 < Re < 2000 and 2000 < Re < 32000 are as follows. $$\begin{gathered} Sh = 0.62 (Re)^{0.5} (Sc)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} \hfill \\ Sh = 0.26 (Re)^{0.6} (Sc)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} \hfill \\ \end{gathered}$$ which correlated the data with standard deviation of 3.75% and 3.50% respectively.     

Two-phase flow in heterogeneous porous media I: The influence of large spatial and temporal gradients

In order to capture the complexities of two-phase flow in heterogeneous porous media, we have used the method of large-scale averaging and spatially periodic models of the local heterogeneities. The analysis leads to the large-scale form of the momentum equations for the two immiscible fluids, a theoretical representation for the large-scale permeability tensor, and a dynamic, large-scale capillary pressure. The prediction of the permeability tensor and the dynamic capillary pressure requires the solution of a large-scale closure problem. In our initial study (Quintard and Whitaker, 1988), the solution to the closure problem was restricted to the quasi-steady condition and small spatial gradients. In this work, we have relaxed the constraint of small spatial gradients and developed a dynamic solution to the closure problem that takes into account some, but not all, of the transient effects that occur at the closure level. The analysis leads to continuity and momentum equations for the-phase that are given by

Asymptotic solutions of a nonhomogeneous differential equation with a turning point

A linear second-order differential equation of the form

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

Degenerate solutions of electron-beam equations

In this paper, exact solutions are constructed for stationary election beams that are degenerate in the Cartesian (x,y,z), axisymmetric (r,θ,z), and spiral (in the planes y=const (u,y,v)) coordinate systems. The degeneracy is determined by the fact that at least two coordinates in such a solution are cyclic or are integrals of motion. Mainly, rotational beams are considered. Invariant solutions for beams in which the presence of vorticity resulted in a linear dependence of the electric-field potential ? on the above coordinates were considered in [1], In degenerate solutions, the presence of vorticity results in a quadratic or more complex dependence of the potential on the coordinates that are integrals of motion. In [2] and in a number of papers referred to in [2], the degenerate states of irrotational beams are described. The known degenerate solutions for rotational beams apply to an axisymmetric one-dimensional (r) beam with an azimuthal velocity component [3] and to relativistic conical flow [1]. The equations used below follow from the system of electron hydrodynamic equations for a stationary relativistic beam $$\begin{array}{*{20}c} {\sum\limits_{\beta = 1}^3 {\frac{\partial }{{\partial q^\beta }}\left[ {\sqrt \gamma g^{\beta \beta } g^{\alpha \alpha } \left( {\frac{{\partial A_\alpha }}{{\partial q^\beta }} - \frac{{\partial A_\beta }}{{\partial q^\alpha }}} \right)} \right]} = 4\pi \rho \sqrt \gamma g^{\alpha \alpha } u_\alpha ,} \\ {\sum\limits_{\beta = 1}^3 {\frac{\partial }{{\partial q^\beta }}\left( {\sqrt \gamma g^{\beta \beta } \frac{{\partial \varphi }}{{\partial q^\beta }}} \right)} = 4\pi \rho \sqrt {\gamma u} ,\sum\limits_{\beta = 1}^3 {g^{\beta \beta } u_\beta ^2 + 1 = u^2 } } \\ \begin{gathered} \frac{\eta }{c}u\frac{{\partial \mathcal{E}}}{{\partial q^\alpha }} = \sum\limits_{\beta = 1}^3 {g^{\beta \beta } u_\beta } \left( {\frac{{\partial p_\beta }}{{\partial q^\alpha }} - \frac{{\partial p_\alpha }}{{\partial q^\beta }}} \right), \hfill \\ \begin{array}{*{20}c} {\sum\limits_{\beta = 1}^3 {\frac{\partial }{{\partial q^\beta }}(\sqrt \gamma g^{\beta \beta } \rho u_\beta ) = 0,u \equiv \frac{\eta }{{c^2 }}(\varphi + \mathcal{E}) + 1,} } \\ {cu_\alpha \equiv \frac{\eta }{c}A_\alpha + p_\alpha ,\alpha ,\beta = 1,2,3,\gamma \equiv g_{11} g_{22} g_{33} } \\ \end{array} \hfill \\ \end{gathered} \\ \end{array} $$ where q^β denotes orthogonal coordinates with the metric tensor g^ββ (β=1,2,3); A_α is the magnetic potential; A_α = (u_α/u)c is the electron velocity; ρ is the scalar space-charge density (ρ > 0); is the energy in eV; p_α is the generalized momentum of an electron per unit mass; η is the electron charge-mass ratio.     

Bounded solutions of some nonlinear elliptic equations in cylindrical domains

The existence of a (unique) solution of the second-order semilinear elliptic equation $$\sum\limits_{i,j = 0}^n {a_{ij} (x)u_{x_i x_j } + f(\nabla u,u,x) = 0}$$ withx=(x ₀,x ₁,?,x _n )?(s ₀, ∞)× Ω′, for a bounded domainΩ′, together with the additional conditions $$\begin{array}{*{20}c} {u(x) = 0for(x_1 ,x_2 ,...,x_n ) \in \partial \Omega '} \\ {u(x) = \varphi (x_1 ,x_2 ,...,x_n )forx_0 = s_0 } \\ {|u(x)|globallybounded} \\ \end{array}$$ is shown to be a well-posed problem under some sign and growth restrictions off and its partial derivatives. It can be seen as an initial value problem, with initial value?, in the spaceC ₀ ⁰ $(\overline {\Omega '} )$ and satisfying the strong order-preserving property. In the case thata _ij andf do not depend onx ₀ or are periodic inx ₀, it is shown that the corresponding dynamical system has a compact global attractor. Also, conditions onf are given under which all the solutions tend to zero asx ₀ tends to infinity. Proofs are strongly based on maximum and comparison techniques.     

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

On the existence of periodic solutions to higher dimensional periodic system with delay

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Nonlinear oscillations and boundary value problems for Hamiltonian systems

We prove the existence of solutions of various boundary-value problems for nonautonomous Hamiltonian systems with forcing terms $$\begin{gathered} \dot x(t) = H'_p (t, x(t), p(t)) + g(t), \hfill \\ \dot p(t) = - H'_x (t, x(t), p(t)) - f(t). \hfill \\ \end{gathered} $$ Among these problems is the existence of T-periodic solutions, namely those satisfying x(t+T)=x(t) and p(t+T)+p(t). A special study is made of the classical case, where H(x, p)=1/2 |p|²+V(x). In the case of parametric oscillations, where (f, g)=(0, 0) and t ? H(t, x, p) is T-periodic, we give a lower bound for the true (minimal) period of the T-periodic solution (x, p) produced by our method, and we prove the existence of an infinite number of subharmonics.     

Multiple Solutions for a Class of Nonhomogeneous Fractional Schrödinger Equations in $$\mathbb {R}^{N}$$

This paper is concerned with the following fractional Schrödinger equation

$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s} u+u= k(x)f(u)+h(x) \text{ in } \mathbb {R}^{N}\\ u\in H^{s}(\mathbb {R}^{N}), \, u>0 \text{ in } \mathbb {R}^{N}, \end{array} \right. \end{aligned}$$

where \(s\in (0,1),N> 2s, (-\Delta )^{s}\) is the fractional Laplacian, k is a bounded positive function, \(h\in L^{2}(\mathbb {R}^{N}), h\not \equiv 0\) is nonnegative and f is either asymptotically linear or superlinear at infinity. By using the s-harmonic extension technique and suitable variational methods, we prove the existence of at least two positive solutions for the problem under consideration, provided that \(|h|_{2}\) is sufficiently small.

     

On the stability or instability of the singular solution of the semilinear heat equation with exponential reaction term

We study questions of existence, uniqueness and asymptotic behaviour for the solutions of u(x, t) of the problem $$\begin{gathered} {\text{ }}u_t - \Delta u = \lambda e^u ,{\text{ }}\lambda {\text{ > 0, }}t > 0,{\text{ }}x{\text{ }}\varepsilon B, \hfill \\ (P){\text{ }}u(x,0) = u_0 (x),{\text{ }}x{\text{ }}\varepsilon B, \hfill \\ {\text{ }}u(x,t) = 0{\text{ }}on{\text{ }}\partial B \times (0,\infty ), \hfill \\ \end{gathered} $$ where B is the unit ball $\{ x\varepsilon R^N :|x|{\text{ }} \leqq {\text{ }}1\} {\text{ and }}N \geqq 3$ . Our interest is focused on the parameter λ ₀=2(N?2) for which (P) admits a singular stationary solution of the form $$S(x) = - 2log|x|$$ . We study the dynamical stability or instability of S, which depends on the dimension. In particular, there exists a minimal bounded stationary solution u which is stable if $3 \leqq N \leqq 9$ , while S is unstable. For $N \geqq 10$ there is no bounded minimal solution and S is an attractor from below but not from above. In fact, solutions larger than S cannot exist in any time interval (there is instantaneous blow-up), and this happens for all dimensions.