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1.
We continue the investigation of the nonlinear problem of mean-square approximation of a real finite nonnegative continuous
function of two variables by the modulus of a double Fourier integral depending on two parameters begun in the first part
of this work [J. Math. Sci., 160, No. 3, 343–356 (2009)]. Finding the solutions of this problem is reduced to the solution of a nonlinear two-dimensional
integral equation of the Hammerstein type. We construct and justify numerical algorithms for determination of branching lines
and branched solutions of this equation. Numerical examples are presented. 相似文献
2.
We study the nonlinear problem of mean-square approximation of a real finite nonnegative continuous function of two variables
by the modulus of a double Fourier integral depending on two parameters. The solution of this problem is reduced to the solution
of a nonlinear two-dimensional integral equation of the Hammerstein type. Numerical algorithms for determination of branching
lines and branched solutions of equation are constructed and substantiated. Some numerical examples are given.
Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 53–64, January–March, 2008. 相似文献
3.
M. Yu. Shvaiko 《Journal of Mathematical Sciences》2011,176(4):548-560
We have proposed a modification of the methods for solving the system of integral equations [M. Ya. Leonov and N. Yu. Shvaiko,
“Complex plane deformation,” Dokl. Akad. Nauk SSSR,
159, No. 2, 1007–1010 (1964); N. Yu. Shvaiko, “On the theory of slip with smooth and singular loading surfaces,” Mat. Metody Fiz.-Mekh. Polya,
48, No. 3, 129–137 (2005)]. These equations describe the development of plane plastic deformation for simple and complex loading
processes. A characteristic feature of these equations lies in the presence of unknown functions both under the integral sign
and in the integration limits. We have written analytical solutions for monotone deformation and in a small neighborhood of
an angular point of the loading trajectory. For arbitrary piecewise smooth trajectories, we have reduced this problem to the
Cauchy problem for a first-order differential equation with known initial conditions. The results obtained simplify significantly
the construction of constitutive equations [(s)\dot]mn ~ [(e)\dot]mn {\dot{\sigma }_{mn}} \sim {\dot{\varepsilon }_{mn}} and their use in applied problems of the theory of plasticity as compared with [N. Yu. Shvaiko, “On the theory of slip with smooth and singular loading surfaces,” Mat. Metody Fiz.-Mekh. Polya,
48, No. 3, 129–137 (2005); N. Yu. Shvaiko, Complex Loading and Problems of Stability [in Russian], Izd. DGU, Dnepropetrovsk (1989)]. 相似文献
4.
D. P. Novikov 《Theoretical and Mathematical Physics》2006,146(3):295-303
We show that under the Euler integral transformation with the kernel (x−z)−α, some solutions of the Fuchs equations (the original pair for the Painlevé VI equation) pass into solutions of a system of
the same form with the parameters changed according to the Okamoto transformation.
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 3, pp. 355–364, March, 2006. 相似文献
5.
The Volterra integral equation of the second order with a regular singularity is considered. Under the conditions that a kernel
K(x,t) is a real matrix function of order n×n with continuous partial derivatives up to order N+1 inclusively and K(0,0) has complex eigenvalues ν±i μ (ν>0), it is shown that if ν>2|‖K|‖
C
-N-1, then a given equation has two linearly independent solutions.
Voronezh Forest Industry Academy, Voronezh. Published in Ukrainskii Matematicheskii Zhurnal Vol. 49, No. 3, pp. 424–432, March,
1997. 相似文献
6.
Andreas Fischer 《Mathematical Programming》2002,94(1):91-124
An iterative framework for solving generalized equations with nonisolated solutions is presented. For generalized equations
with the structure , where is a multifunction and F is single-valued, the framework covers methods that, at each step, solve subproblems of the type . The multifunction approximates F around s. Besides a condition on the quality of this approximation, two other basic assumptions are employed to show Q-superlinear
or Q-quadratic convergence of the iterates to a solution. A key assumption is the upper Lipschitz-continuity of the solution
set map of the perturbed generalized equation . Moreover, the solvability of the subproblems is required. Conditions that ensure these assumptions are discussed in general
and by means of several applications. They include monotone mixed complementarity problems, Karush-Kuhn-Tucker systems arising
from nonlinear programs, and nonlinear equations. Particular results deal with error bounds and upper Lipschitz-continuity
properties for these problems.
Received: November 2001 / Accepted: November 2002 Published online: December 9, 2002
Key Words. generalized equation – nonisolated solutions – Newton's method – superlinear convergence – upper Lipschitz-continuity – mixed
complementarity problem – error bounds
Mathematics Subject Classification (1991): 90C30, 65K05, 90C31, 90C33 相似文献
7.
We obtain exact solutions U(x, y, z, t) of the three-dimensional sine-Gordon equation in a form that Lamb previously proposed
for integrating the two-dimensional sine-Gordon equation. The three-dimensional solutions depend on arbitrary functions F(α)
and ϕ(α,β), whose arguments are some functions α(x, y, z, t) and β(x, y, z, t). The ansatzes must satisfy certain equations. These
are an algebraic system of equations in the case of one ansatz. In the case of two ansatzes, the system of algebraic equations
is supplemented by first-order ordinary differential equations. The resulting solutions U(x, y, z, t) have an important property,
namely, the superposition principle holds for the function tan(U/4). The suggested approach can be used to solve the generalized sine-Gordon equation, which, in contrast to the classical equation,
additionally involves first-order partial derivatives with respect to the variables x, y, z, and t, and also to integrate
the sinh-Gordon equation. This approach admits a natural generalization to the case of integration of the abovementioned types
of equations in a space with any number of dimensions.
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 3, pp. 370–377, March, 2009. 相似文献
8.
Traveling waves in the complex Ginzburg-Landau equation 总被引:1,自引:0,他引:1
A. Doelman 《Journal of Nonlinear Science》1993,3(1):225-266
Summary In this paper we consider a modulation (or amplitude) equation that appears in the nonlinear stability analysis of reversible
or nearly reversible systems. This equation is the complex Ginzburg-Landau equation with coefficients with small imaginary
parts. We regard this equation as a perturbation of the real Ginzburg-Landau equation and study the persistence of the properties
of the stationary solutions of the real equation under this perturbation. First we show that it is necessary to consider a
two-parameter family of traveling solutions with wave speedυ and (temporal) frequencyθ; these solutions are the natural continuations of the stationary solutions of the real equation. We show that there exists
a two-parameter family of traveling quasiperiodic solutions that can be regarded as a direct continuation of the two-parameter
family of spatially quasi-periodic solutions of the integrable stationary real Ginzburg-Landau equation. We explicitly determine
a region in the (wave speedυ, frequencyθ)-parameter space in which the weakly complex Ginzburg-Landau equation has traveling quasi-periodic solutions. There are two
different one-parameter families of heteroclinic solutions in the weakly complex case. One of them consists of slowly varying
plane waves; the other is directly related to the analytical solutions due to Bekki & Nozaki [3]. These solutions correspond
to traveling localized structures that connect two different periodic patterns. The connections correspond to a one-parameter
family of heteroclinic cycles in an o.d.e. reduction. This family of cycles is obtained by determining the limit behaviour
of the traveling quasi-periodic solutions as the period of the amplitude goes to ∞. Therefore, the heteroclinic cycles merge
into the stationary homoclinic solution of the real Ginzburg-Landau equation in the limit in which the imaginary terms disappear. 相似文献
9.
P. O. Savenko 《Journal of Mathematical Sciences》2000,99(5):1557-1568
We investigate the branching of solutions of a nonlinear integral equation of the Hammerstein type which arises in the problem
of synthesis of a linear antenna with given directional amplitude diagram. Systems of transcendental equations for determination
of branching points of various types are deduced, and the number of branched solutions and their qualitative characteristics
are analyzed.
Pidtryhach Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences, L'viv. Translated from
Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 35–44, April–June, 1998. 相似文献
10.
V. I. Dmitriev P. S. Belkin N. A. Mershchikova 《Computational Mathematics and Modeling》2005,16(4):289-300
The article computes the electromagnetic field on the surface of a layered medium with a local nonhomogeneity. The problem
is transformed from three- to two-dimensional and the singulari-ties are investigated using the integral equation method.
The proposed algorithm efficiently sim-ulates two-dimensional H-polarization fields by solving a system of integral equations. The method is particularly effective for solving inverse problems.
__________
Translated from Prikladnaya Matematika i Informatika, No. 18, pp. 5–16, 2004. 相似文献
11.
V. I. Derguzov 《Journal of Mathematical Sciences》2000,98(6):679-683
An integral equation describing oscillations of a two-dimensional periodic lightguide is studied. The integral operator is
an analytic function of a spectral parameter. The homogenous integral equation has only the zero solution for all values of
the spectral parameter expect for some isolated values. Bibliography: 9 titles.
Translated fromProblemy Matematicheskogo Analiza, No. 18, 1998, pp. 78–84. 相似文献
12.
A. R. Bechelova 《Ukrainian Mathematical Journal》1998,50(7):1131-1134
For the diffusion equation of fractional order, we construct an approximation difference scheme of order 0(h
2 + τ). We present an algorithm for the solution of boundary-value problems for a generalized transfer equation of fractional
order.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 7, pp. 994–996, July, 1998. 相似文献
13.
We describe the basic propositions of the linear theory of viscoelasticity. We give transformation formulas for the resolvent
integral operators of viscoelasticity with an arbitrary analytic kernel of difference type. The method of computing the irrational
operator functions is illustrated by determining the real parameters of the two-dimensional stressed state of an orthotropic
plate. Three figures. Bibliography: 7 titles.
Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 86–96. 相似文献
14.
Alina Cristiana Gavriluţ 《Mathematica Slovaca》2010,60(3):289-318
In two earlier papers [GAVRILUŢ, A.: A Gould type integral with respect to a multisubmeasure, Math. Slovaca 58 (2008), 1–20] and [Gavriluţ, A.: On some properties of the Gould type integral with respect to a multisubmeasure, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 52 (2006), 177–194], we defined and studied a Gould type integral for a real valued, bounded function with respect to a multisubmeasure
having finite variation. In this paper, we introduce and study the properties of a Gould type integral in the general setting:
the function may be unbounded and the variation of the multisubmeasure may be infinite. 相似文献
15.
We consider an approximate method for the solution of the Cauchy problem for an operator differential equation. This method
is based on the expansion of an exponential in orthogonal Laguerre polynomials. We prove that the fact that an initial value
belongs to a certain space of smooth elements of the operator A is equivalent to the convergence of a certain weighted sum of integral residuals. As a corollary, we obtain direct and inverse
theorems of the theory of approximation in the mean.
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 4, pp. 557–563, April, 2008. 相似文献
16.
D. Yu. Ivanov 《Differential Equations》2010,46(8):1104-1113
We consider boundary value problems for the differential equations Δ2
u + B
u = 0 with operator coefficients B corresponding to initial-boundary value problems for the diffusion equation Δ3
u − pu = ∂
t
u (p > 0) on a right cylinder with inhomogeneous boundary conditions on the lateral surface of the cylinder with zero boundary
conditions on the bases of the cylinder and with zero initial condition. For their solution, we derive specific boundary integral
equations in which the space integration is performed only over the lateral surface of the cylinder and the kernels are expressed
via the fundamental solution of the two-dimensional heat equation and the Green function of corresponding one-dimensional
initial-boundary value problems of diffusion. We prove uniqueness theorems and obtain sufficient existence conditions for
such solutions in the class of functions with continuous L
2-norm. 相似文献
17.
We consider a graph, where the nodes have a pre-described degree distribution F, and where nodes are randomly connected in accordance to their degree. Based on a recent result (R. van der Hofstad, G. Hooghiemstra
and P. Van Mieghem, “Random graphs with finite variance degrees,” Random Structures and Algorithms, vol. 17(5) pp. 76–105, 2005), we improve the approximation of the mean distance between two randomly chosen nodes given
by M. E. J. Newman, S. H. Strogatz, and D. J. Watts, “Random graphs with arbitrary degree distribution and their application,”
Physical Review. E vol. 64, 026118, pp. 1–17, 2001. Our new expression for the mean distance involves the expectation of the logarithm of the
limit of a super-critical branching process. We compare simulations of the mean distance with the results of Newman et al.
and with our new approach.
AMS 2000 Subject Classification: 05C80, 60F05 相似文献
18.
R. D. M. Garcia C. E. Siewert 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2007,58(2):262-288
A Legendre expansion of the (matrix) scattering kernel relevant to the (vector- valued) linearized Boltzmann equation for
a binary mixture of rigid spheres is used to define twelve solutions that are linear in the spatial variables {x, y, z}. The twelve (asymptotic) solutions are expressed in terms of three vector-valued functions A
(1)(c), A(2)(c), and B(c). These functions are generalizations of the Chapman–Enskog functions used to define asymptotic solutions and viscosity and
heat conduction coefficients for the case of a single-species gas. To provide evidence that the three Chapman–Enskog vectors
exist as solutions of the defining linear integral equations, numerical results developed in terms of expansions based on
Hermite cubic splines and a collocation scheme are reported for two binary mixtures (Ne-Ar and He-Xe) with various molar concentrations. 相似文献
19.
V. Kanišauskas 《Lithuanian Mathematical Journal》1999,39(4):389-392
Asymptotic formulas of the Hellinger integral are used in the investigation of properties of optimal estimates and statistical
criteria. For a certain class of renewal processes, this formula was obtained by the author in [Lith. Math. J.,38(2), 131–143 (1998)]. In this paper, we obtain such a formula for all renewal processes whose intermediate renewal moments
have absolutely continuous distributions. We use the traditional representation of the Hellinger integral and the theory of
large deviations.
Šiauliai University, Višinskio 25, 5400 Šiauliai, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 4,
pp. 493–497, October–December, 1999.
Translated by V. Mackevičius 相似文献
20.
Olivier Teulié 《Monatshefte für Mathematik》2001,75(5):169-176
In this paper, we prove that for any real number ξ, which is not an algebraic number of degree , there exist infinitely many real algebraic units α of degree n + 1 such that . We also show how the flexibility of H. Davenport and W. M. Schmidt’s method allows to replace, with the same exponent of approximation, units of degree over Z (i.e. elements α with both α and integral over Z) by units of degree over a finite intersection . 相似文献