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1.
In this paper, the authors consider the high-frequency asymptoticsof the phase s() of acoustic waves scattered by an obstacleRn with fractal boundary. Under certain conditions, it is provedthat if is –Minkowski measurable with –Minkowskimeasure µ then there exists a positive constant Cn, dependingonlyon n and such that where  相似文献   

2.
Bivariational methods are presented for nonlinear integral equationsof Ham-merstein type = Kf(). With appropriate conditions onK and f, various upper and lower bounding functionals are derivedfor inner products and associated with the solution . Inthe latter case, suitable choices for g lead to point wise boundson both and its derivative. The methods are tested on a pendulumequation, and encouraging accuracy is obtained using simpletrial vectors.  相似文献   

3.
A method is developed for evaluating Fourier integrals of theform A() = 1–1f(x) efax dx, 0. The method consists of expanding the function f in a seriesof Chebyshev polynomials and expressing the integral A() asa series of the Bessel functionsJr+(), r= 0, 1, 2,.... A partialsum AN() of the series provides an approximant to A(). The principalfeature of the method is that one set of N+1 evaluations off(x) suffices for the calculation of AN() for all , and alsothe truncation error A()–AN() is essentially independentof . Numerical tests show that the method is accurate, economicaland reliable. An application to the inversion of Fourier andLaplace transforms is briefly described.  相似文献   

4.
A method, due to Fox, is used to derive asymptotic error formulaefor numerical procedures having the form (z+h, h).–(z,h)=(f,z,h).These procedures correspond to numerical quadrature for theintegrand (/h)(f,z,0)and compact expressions are given for determiningthe order of convergence as h 0, and the leading term in theerror. It is shown that a natural generalization of the Euler-Maclaurinexpansion is available. These results are applied to the particularcase where the Pt are polynomials in the differentiation operator.A related interpolation problem is also studied, and it is shownthat in certain cases higher order quadrature formulae are possiblewhen this interpolation problem is not poised.  相似文献   

5.
The decay of the eddy-currents that are induced in a thin, uniform,imperfectly-conducting sheet by switching off the source ofan external magnetic field is investigated. For the two-dimensionalproblem of an infinite strip the (non-dimensional) decay constantsn and eddy-current distributions in(x) are the eigenvalues andeigenfunctions of the integral equation with the constraint. For the circular disc the corresponding equation is where and K and E are complete elliptic integrals. For both problemsthe initial eddy-currents have inverse-square-root singularitiesat the edges but during their decay the eddy currents are finiteat the edges and the normal magnetic fields have logarithmicsingularities there. Numerical results are given for variousinitial-value problems. The eddy current problems are closely related to water-waveproblems in which there is a strip-shaped or circular aperturein a horizontal rigid dock. If n and n are the decay constantsand magnetic scalar potentials for the strip and n and n theangular frequencies and velocity potentials for the normal modesin the strip-shaped aperture, then n =n2 and n and n are thereal and imaginary parts respectively of a holomorphic function.The velocities in the normal modes are deduced from the solutionof the eddy-current problem and are found to agree with resultsgiven in Miles (1972). For circular geometries the eigenvaluesand eigenfunctions of the axisymmetric eddy-current problemare the same as those of the water-wave problem that has angularvariation ei; where (, , z) are cylindrical polar co-ordinateslocated at the centre of the basin.  相似文献   

6.
We are interested in the model plasma problem –u = u+in ,u = –d on , au+ dx=j where is a bounded domain in with boundary ; here, j isa given positive number, the function u and the positive number are the unknowns of the problem, and d is a real parameter.Using a variant of the implicit function theorem, we can provethe existence of a global solution branch parametrized by d.The method has the advantage that it can be used for analysingthe approximation of the above problem by a finite-element method.  相似文献   

7.
Two theorems related to equilibrium free-boundary problems arepresented. One arises as a time-independent solution to thephase-field equations. The other is the relevant time-independentproblem for the Stefan model, modified for the surface tensioneffect. It also serves as a preliminary result for the phase-fieldformulation. Under appropriate conditions, we prove that, givenan appropriate positive constant and a smooth function u: R;,where is an annular domain in R2, there exists a curve suchthat u(x)=—K(x) for all x , where K is the curvature.Using this result, we prove the existence of solutions to O=2+ ?(—3) + 2u that have a transition layer behaviour (from=—1 to =+1) for small and make the transition on thecurve . This proves there exist solutions to the phase fieldmodel that satisfy a Gibbs-Thompson relation.  相似文献   

8.
Experiments with a nonlinear electronic model show that certainsimple features of the solutions of where f(u) is an odd monotonic function of u for example u3,repeat in a regular pattern as either is decreased or U isincreased. For fixed U, the position of these features is periodicin 1/ and, when f(u) has the form u|u|k–1 a quantitativerelation between the period in 1/ and U can be found. The occurrenceof large-amplitude chaotic solutions is found to depend notonly on the nonlinearity of f(u) for large U but also on itsbehaviour near u = 0. For the Duffing equation, which can bereduced to the range of parameters accessible to experiment is 0<1 and0<F5000.  相似文献   

9.
For f: [0,) R, the JMN approximant of f(t) is where M and N are non-negative integers and ßk, i,Kiare defined constants. Under appropriate conditions on f andprovided Re(i) > 0 The approximants are the bases of recursions for numerical initial-valueproblems in linear differential-algebraic systems with constantcoefficients. The recursions are stable when N M N–2.Each step of a recursion involves mainly the solution of N/2uncoupled algebraic systems.  相似文献   

10.
On hearing the shape of a bounded domain with Robin boundary conditions   总被引:2,自引:0,他引:2  
The asymptotic expansions of the trace of the heat kernel (t)= [sum ]j=1 exp(-tj) for small positive t, where {j} j=1 arethe eigenvalues of the negative Laplacian -n = -[sum ]nk=1 (/xk)2in Rn (n = 2 or 3), are studied for a general multiply connectedbounded domain which is surrounded by simply connected boundeddomains i with smooth boundaries i (i = 1,...,m), where smoothfunctions Yi (i = 1,...,m) are assuming the Robin boundary conditions(ni + Yi) = 0 on i. Here /ni denote differentiations along theinward-pointing normals to i (i = 1,...,m). Some applicationsof an ideal gas enclosed in the multiply connected bounded containerwith Neumann or Robin boundary conditions are given.  相似文献   

11.
The autonomous differential equations for the temperature andreactant consumption in a first-order well-stirred exothermicreaction are considered. An examination of the phase-plane solutionsallows the qualitative behaviour of the Semenov number as afunction of maximum temperature rise * to be established. Inthe limit of infinite adiabatic temperature rise (B) and zeroactivation energy parameter ( = 0), the relationship between and stationary temperature s is known to be e1 = s. Criticalityarises at the maximum of (s) and leads to the critical Semenovvalues (s)cr = 1, cr = e–1. For sufficiently large B,it is shown that the (*) curve has a bifurcation at * = 1, withthe upper branch monotonically increasing and the lower branchmonotonically decreasing for * > 1. In the limit B thesebecome respectively the straight line = e–1, s 1 andthe unstable branch of = se–1, s 1 and the unstablebranch of = s e. Criticality for finite B is definedas occurring at the bifurcation, namely *cr = 1, with cr(B)the value of at this point. Values of these Semenoy numbersare obtainable from the numerical calculations of Boddingtonet al. [Proc. R. Soc. Lond. (1983), 390, 13–30]. The newcriterion is applied to an approximate phase-plane solution.The corresponding critical parameter is found to be cr = e–1[1+B(2–e–1)+O(B–1)].  相似文献   

12.
Liouville's non-linear partial differential equation is consideredfor an infinite rectangular strip domain with a slowly varyingboundary condition. The equation describes a layer of chemicallyreactive material under conditions where the resistance to surfaceheat transfer is negligible and the ambient temperature variesslowly along the surface. Symmetrical heating by a zero orderexothermic reaction is assumed. If is a small dimensionlesstemperature difference between regions where the surface temperatureis effectively constant, a perturbation series solution in may be determined provided the Frank-Kamenetskii parameter satisfies c(). It is shown that a plausible value for thecritical parameter is c() = c(0) e–e,where c(0) = 0.878.The corresponding critical temperature distribution is shownto have a dependence on different from that for subcriticalcases.  相似文献   

13.
An integral representation for the electrostatic capacity matrixC=[cij]i,j=1,2 of two conducting spheres of radii R1, and R2is obtained. A short-distance asymptotic expansion is then derivedand its approximation properties for fixed (surface) distancer between the spheres are investigated. An error function is defined for cij(r) and its nthorder asymptotic approximant it has the property following from the divergence of the expansion, and thereby shows thatthe optimal approximation of cij(r) is achieved by an approximantof finite order n = nij(r) depending possibly on r and the indicesi,j. The value gives the quality of approximation of cij by the asymptotic expansion for a givendistance r between the spheres. The point sets and are introduced in order to describe the distance ranges where cij can be approximatedwithin a given error >0 by an asymptotic approximant of given order n, or at least by theoptimal approximant, respectively. The optimal order nij(r)and the -approximation sets and D() are investigated numerically.  相似文献   

14.
An invariant of quasiprojective -varieties X with values ina commutative ring is motivic if (X) = (Y) + (X\ Y) for Y closedin X, and (X x Y) = (X)(Y). Examples include Euler characteristics and virtual Poincaré and Hodge polynomials. We firstdefine a unique extension ' of to finite type Artin -stacks, which is motivic and satisfies '([X/G]) = (X)/(G) when X is a -variety, G a special -groupacting on X, and [X/G] is the quotient stack. This only worksif (G) is invertible in for all special -groups G, which excludes = as (m) = 0. But we can extend the construction to get roundthis. Then we develop the theory of stack functions on Artin stacks.These are a universal generalization of constructible functionson Artin stacks. There are several versions of the construction:the basic one , and variants ‘twisted’ by motivic invariants. We associate a -vector space or a -module to each Artin stack , with functorial operations of multiplication, pullbacks * and pushforwards *under 1-morphisms ;, and so on. They will be important tools in the author's series on ‘Configurationsin abelian categories’.  相似文献   

15.
This paper presents a method of obtaining the complete asymptoticsolution of boundary value problems of the form for x [0,1] where b(x) is strictly positive andfor small and positive. Physically, the problem arises in determiningthe steady-state concentration of a substance in a chemicalflow reactor. A "two-variable" expansion procedure is used.  相似文献   

16.
An integral representation of the exact solution of the initialvalue problem for the hyperbolic equation of the form is derived. Here Ao, Av, B, and Care constant m x m matrices, u(t, X; ) is an m-component columnvector, and is a positive parameter. Various conditions areimposed on the coefficient matrices that permit the applicationof the method of stationary phase in several variables to theintegral representation of the exact solution. The leading termof the asymptotic expansion as of the exact solution is obtainedfor several types of initial data and source functions whichdepend on the parameter .  相似文献   

17.
The nonlinear nonlocal system of the equilibrium equations ofan elastic ring under the action of an external two-dimensionaluniformly subsonic potential barotropic steady-state gas flowis considered. The configurations of the elastic ring are identifiedby a pair of functions (, ). The simple curve represents theshape of the ring and the real-valued function identifies theorientation of the material sections of the ring. The pressurefield on the ring depends nonlocally on , and on two parametersU and P which represent the pressure and the velocity at infinity.The system is shown to be equivalent to a fixed-point problem,which is then treated with continuation methods. It is shownthat the solution branch ensuing from certain equilibrium states((0, 0), 0, P0) in the solution-parameter space of ((0, 0),0, P0) either approaches the boundary of the admissible ((,), U,p)'s in a well-defined sense, or is unbounded, or is homotopicallynontrivial in the sense that there exists a continuous map from the branch to a two-dimensional sphere which is not homotopicin the sphere to a constant, while restricted to the branchminus ((0, 0), 0, P0) is homotopic to a constant in the sphere.Furthermore, by fixing the pressure parameter at P0 and by consideringthe one-parameter problem in ((, ), U), the following holds.Every hyperplane in the solution-parameter space of the ((,), U)'s which contains the equilibrium state ((0, 0), 0) anddoes not include a welldetermined one-dimensional subspace intersectsthe solution branch above at a point different from ((0, 0),0).  相似文献   

18.
An asymptotic expansion is constructed for the double integral as +, where f(x, y) hasa curve of stationary points in D. The first two coefficientsof the expansion are explicitly calculated.  相似文献   

19.
The constitutive relations for the transport of heat, stress,electric charge, etc., in a continuum must be chosen so thatthe second law of thermodynamics is not violated; the constraintstake the form of inequalities, typically requiring the entropygenerated within a material element to be non-negative. Thepaper is concerned with this concept—its history, thephysical principles on which it depends, how to apply it whensecond-order or non-linear effects are important and how itis widely misused in modern continuum mechanics. The history is reduced to the contributions of five leadingthermodynamicists—Clausius, Maxwell, Gibbs, Boltzmannand Duhem. The object here was to try to discover which formof the inequality one should regard as being fundamental. Oneimportant conclusion is that entropy S must be defined simultaneouslywith the identification of the inequality, and that in generalthis cannot be done until the constitutive equations are known.The empirical element enters with the notion of irreversibility,which is given a precise meaning with the aid of the motionreversed parity (x), a variable x having = +1 or = –1if, when time and motions are reversed, x x or xx.The macroscopic parity of x, *(x), is obtained by first replacingx by the constitutive equation for x. The entropy production rate has both irreversible (f) and reversible(r) parts. It is shown that the reciprocal relations followfrom the requirement that the macroscopic parity of (i) mustbe +1. Continuum thermodynamics is based on various principles extractedfrom theory developed for uniform systems, the example chosento illustrate the ideas being the simple monatomic gas. Second-orderconstitutive relations are introduced, and the expressions forentropy and its production rate per unit volume, , obtained.It is shown that the stability condition 0 cannot, in general,be satisfied merely by imposing constraints on the constitutiverelations. To second-order = 1 + 2, where 1 is the usual bilinearform, and the terms in 2 have an additional derivative. Thesecond-order term 2 can have both signs, and is not dissipative.The relation between this fact and the frame-dependence of constitutiverelations is explained. The final section illustrates the errors frequently found inthe thermodynamic arguments appearing in books and papers onrational continuum mechanics. The principle of these is that 0 is interpreted as being a constraint on the constitutiverelations alone. Another is the idea that the balance equationscan be set aside as constraints by regarding them as mere definitionsof a heat source and a body force, an error based partly onthe misconception that constitutive relations should be frame-indifferent.Finally, an inequality due to Glansdorff & Prigogine isexamined and found to be in error.  相似文献   

20.
In the one-dimensional Stefan problem, the standard equilibriumcondition ; = 0 at the free boundary x = s(t) is here replacedby the kinetic law s'(t) = ß((s(t), t)), where ß:R R is continuous and increasing and ß(0) = 0. Thisrepresents supercooling and superheating effects. The standardStefan problem is then obtained in the limit as ß'(0) + A similar condition is considered for a radially symmetric system,taking also account of the surface tension effect. A kineticcondition is introduced also for phase transitions in binaryalloys, represented by means of the system of the Fourier'sand Fick's laws. In the case of several space dimensions, denoting by [0, 1]the concentration of the more energetic phase, the followinglaw is considered this is also extendedto binary systems. For all of the previous models of phase transitions, existenceresults are proved for the variational problems obtained bycoupling the free boundary condition with the energy conservationequation (and with the mass diffusion equation, for alloys).For heterogeneous systems, also a different model based on "non-equilibriumthermodynamics" is considered. This paper reviews the results of Visintin [IMA J. appl. Math.(1985) 34, 225–245] and announces those of Visintin (1985,to appear in Q appl. Math, and in Ann. Mat. pura appl.).  相似文献   

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