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1.
We investigate the branching of solutions of a Hammerstein-type nonlinear two-dimensional integral equation that arises in
the problems of mean-square approximation of a real finite nonnegative function of two variables by the modulus of double
Fourier integral, depending on two parameters [Mat. Metody Fiz.-Mekh. Polya,
51, No. 1, 53–64; No. 4, 80–85 (2008)]. We have derived analytical expressions for eigenfunctions of the corresponding linear
homogeneous integral equation, necessary for the construction of branched-off solutions, and obtained systems of transcendental
equations for finding the points of their branching. We also present, in the first approximation, the analytical representations
of complex solutions branched-off from the real solution for the two-dimensional case of branching. 相似文献
2.
We consider a nonlinear system of integral equations describing the structure of a plane shock wave. Based on physical reasoning, we propose an iterative method for constructing an approximate solution of this system. The problem reduces to studying decoupled scalar nonlinear and linear integral equations for the gas temperature, density, and velocity. We formulate a theorem on the existence of a positive bounded solution of a nonlinear equation of the Uryson type. We also prove theorems on the existence and uniqueness of bounded positive solutions for linear integral equations in the space L 1[?r, r] for all finite r < +∞. For a more general nonlinear integral equation, we prove a theorem on the existence of a positive solution and also find a lower bound and an integral upper bound for the constructed solution. 相似文献
3.
Zainab R. Al‐Yasiri Klaus Gürlebeck 《Mathematical Methods in the Applied Sciences》2016,39(14):4056-4068
The p‐Laplace equation is a nonlinear generalization of the Laplace equation. This generalization is often used as a model problem for special types of nonlinearities. The p‐Laplace equation can be seen as a bridge between very general nonlinear equations and the linear Laplace equation. The aim of this paper is to solve the p‐Laplace equation for 1 < p < 2 and to find strong solutions. The idea is to apply a hypercomplex integral operator and spatial function theoretic methods to transform the p‐Laplace equation into the p‐Dirac equation. This equation will be solved iteratively by using a fixed‐point theorem. Applying operator‐theoretical methods for the p‐Dirac equation and p‐Laplace equation, the existence and uniqueness of solutions in certain Sobolev spaces will be proved. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
4.
Abdelkader Boucherif Ali S. Al-Qahtani Bilal Chanane 《Numerical Functional Analysis & Optimization》2013,34(6):730-747
We discuss the propagation of heat along a homogeneous rod of length A under the influence of a nonlinear heat source and impulsive effects at fixed times. This problem is described by an initial-boundary value problem for a nonlinear parabolic partial differential equation subjected to impulsive effects at fixed times. Using Green's function, we convert the problem into a nonlinear integral equation. Sufficient conditions are provided that enable the application of fixed point theorems to prove existence and uniqueness of solutions. 相似文献
5.
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
6.
《偏微分方程通讯》2013,38(1-2):409-438
Abstract We study the asymptotic behavior of solutions of the Cauchy problem for a functional partial differential equation with a small parameter as the parameter tends to zero. We establish a convergence theorem in which the limit problem is identified with the Cauchy problem for a nonlinear parabolic partial differential equation. We also present comparison and existence results for the Cauchy problem for the functional partial differential equation and the limit problem. 相似文献
7.
Peter Somora 《Mathematica Slovaca》2008,58(4):439-454
A second order nonlinear differential equation with homogeneous Dirichlet boundary conditions is considered. An explicit expression
for the root functions for an autonomous nonlinear boundary value problem is obtained using the results of the paper [SOMORA,
P.: The lower bound of the number of solutions for the second order nonlinear boundary value problem via the root functions method, Math. Slovaca 57 (2007), 141–156]. Other assumptions are supposed to prove the monotonicity of root functions and to get the exact number
of solutions. The existence of infinitely many solutions of the boundary value problem with strong nonlinearity is obtained
by the root function method as well.
The paper was supported by the Grant VEGA No. 2/7140/27, Bratislava. 相似文献
8.
Anna Lisa Amadori 《Ricerche di matematica》2007,56(1):1-17
Abstract We study the obstacle problem for a class of nonlinear integro-partial differential equations of second order, possibly degenerate,
which includes the equation modeling American options in a jump-diffusion market with large investor. The viscosity solutions
setting reveals appropriate, because of a monotonicity property with respect to the integral term. The same property allows
to approximate the problem by penalization and to obtain the existence and uniqueness of solutions via a comparison principle.
We also give uniform estimates of the solutions of the penalized problems which allow to prove further regularity.
Keywords: Integro-differential equations, Obstacle problem, Viscosity solutions, American options
Mathematics Subject Classification (2000): 45K05, 35K85, 49L25, 91B24 相似文献
9.
《偏微分方程通讯》2013,38(1-2):349-379
Abstract In this article we study the asymptotic behavior of incompressible, ideal, time-dependent two dimensional flow in the exterior of a single smooth obstacle when the size of the obstacle becomes very small. Our main purpose is to identify the equation satisfied by the limit flow. We will see that the asymptotic behavior depends on γ, the circulation around the obstacle. For smooth flow around a single obstacle, γ is a conserved quantity which is determined by the initial data. We will show that if γ = 0, the limit flow satisfies the standard incompressible Euler equations in the full plane but, if γ≠ 0, the limit equation acquires an additional forcing term. We treat this problem by first constructing a sequence of approximate solutions to the incompressible 2D Euler equation in the full plane from the exact solutions obtained when solving the equation on the exterior of each obstacle and then passing to the limit on the weak formulation of the equation. We use an explicit treatment of the Green's function of the exterior domain based on conformal maps, a priori estimates obtained by carefully examining the limiting process and the Div-Curl Lemma, together with a standard weak convergence treatment of the nonlinearity for the passage to the limit. 相似文献
10.
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
11.
R. D. M. Garcia C. E. Siewert 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2007,11(5):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
12.
《复变函数与椭圆型方程》2012,57(6):537-552
A nonlinear integro-ordinary differential equation built up by a linear ordinary differential operator of n th order with constant coefficients and a quadratic integral term is dealt with. The integral term represents the so-called autocorrelation of the unknown function. Applying the Fourier cosine transformation, the integral-differential equation is reduced to a quadratic boundary value problem for the complex Fourier transform of the solution in the upper half-plane. This problem in turn is reduced to a linear boundary value problem which can be solved in closed form. There are infinitely many solutions of the integral-differential equation depending on the prescribed zeros of a function related to the complex Fourier transform. 相似文献
13.
Pui-Kei Wong 《Annali di Matematica Pura ed Applicata》1969,81(1):331-346
Summary In this paper we study the nonlinear fourth order differentiai equation uiv±F(u, v)u=0, where F is a pssitive monotone function of u. Asymptotic behaviour of certain solutions are treated in sections 2 and 4
while a two point boundary value problem is studied in section 3.
Entrata in Redazione il 26 agosto 1968. 相似文献
14.
Boundary value problems for the Poisson equation are considered in a multilevel thick junction consisting of a junction body
and a lot of alternating thin rectangles of two levels depending on their lengths. Rectangles of the first level have a finite
length, whereas rectangles of the second level have a length ε
α
, 0 < α < 1, where ε is the alternation period. On the boundary of thin rectangles, an inhomogeneous Neumann boundary condition involving additional
perturbation parameters is imposed. We prove convergence theorems for solutions and energy integrals. Regarding the convergence
of solutions of the original problem to solutions of the homogenized problem, we establish some (auxiliary) estimates necessary
for obtaining the convergence rate. Bibliography: 48 titles. Illustrations: 3 figures.
Dedicated to Nina Nikolaevna Uraltseva
Translated from Problemy Matematicheskogo Analiza, 40, May 2009, pp. 113–132. 相似文献
15.
《随机分析与应用》2013,31(5):1189-1205
Abstract In this paper, we establish the existence of solutions of a more general class of stochastic functional integral equations. The main tools here are the measure of noncompactness and the fixed point theorem of Darbo type. The results of this paper generalize the results of Rao–Tsokos [Rao, A.N.V.; Tsokos, C.P. A class of stochastic functional integral equations. Coll. Math. 1976, 35, 141–146.] and Szynal–Wedrychowicz [Szynal, D.; Wedrychowicz, S. On existence and an asymptotic behaviour of random solutions of a class of stochastic functional integral equations. Coll. Math. 1987, 51, 349–364.]. 相似文献
16.
Jana Björn 《Calculus of Variations and Partial Differential Equations》2009,35(4):481-496
We use variational methods to obtain a pointwise estimate near a boundary point for quasisubminimizers of the p-energy integral and other integral functionals in doubling metric measure spaces admitting a p-Poincaré inequality. It implies a Wiener type condition necessary for boundary regularity for p-harmonic functions on metric spaces, as well as for (quasi)minimizers of various integral functionals and solutions of nonlinear
elliptic equations on R
n . 相似文献
17.
Mikhail Kovalyov 《纯数学与应用数学通讯》1987,40(5):589-607
In this paper we consider the initial value problem for the nonlinear wave equation □u = F(u, u′) in Friedman-Robertson-Walker space-time, □ being the D'Alambertian in local coordinates of space-time. We obtain decay estimates and show that the equation has global solutions for small initial data. We do it by reducing the problem to an initial value problem for the wave equation over hyperbolic space. As byproduct we derive decay and global existence for solutions of the wave equation over the hyperbolic space with small initial data. The same technique with some auxiliary lemmas similar to the ones proved in [6], [7] can be used to generalize the result to the case when F depends also on second derivatives of u in a certain way. 相似文献
18.
Olivier Guibé 《Annali di Matematica Pura ed Applicata》2002,180(4):441-449
We give a partial uniqueness result concerning comparable renormalized solutions of the nonlinear elliptic problem -div(a(x,Du))=μ in Ω, u=0 on ∂Ω, where μ is a Radon measure with bounded variation on Ω.
Received: December 27, 2000 Published online: December 19, 2001 相似文献
19.
We prove a theorem on the existence of solutions of some nonlinear functional integral equations in the Banach algebra of continuous functions on the interval [0,a]. Then we consider a nonlinear integral equation of fractional order and give some sufficient conditions for existence of solutions of this equation. We use fixed point theorems associated with the measure of noncompactness as the main tool. Our existence results include several results obtained in previous studies. Finally we present some examples which show that our results are applicable. 相似文献
20.
Analysis of a Free Boundary Problem Modeling Tumor Growth 总被引:4,自引:0,他引:4
Shang Bin CUI 《数学学报(英文版)》2005,21(5):1071-1082
In this paper, we study a free boundary problem arising from the modeling of tumor growth. The problem comprises two unknown functions: R = R(t), the radius of the tumor, and u = u(r, t), the concentration of nutrient in the tumor. The function u satisfies a nonlinear reaction diffusion equation in the region 0 〈 r 〈 R(t), t 〉 0, and the function R satisfies a nonlinear integrodifferential equation containing u. Under some general conditions, we establish global existence of transient solutions, unique existence of a stationary solution, and convergence of transient solutions toward the stationary solution as t →∞. 相似文献