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1.
Existence of solutions for dual singular integral equations with convolution kernels in case of non-normal type
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Pingrun Li 《Journal of Applied Analysis & Computation》2020,10(6):2756-2766
This paper is devoted to the study of dual singular integral equations with convolution kernels in the case of non-normal type. Via using the Fourier transforms, we transform such equations into Riemann boundary value problems. To solve the equation, we establish the regularity theory of solvability. The general solutions and the solvable conditions of the equation are obtained. Especially, we investigate the asymptotic property of solutions at nodes. This paper will have a significant meaning for the study of improving and developing complex analysis, integral equations and Riemann boundary value problems. 相似文献
2.
Pingrun Li 《Mathematical Methods in the Applied Sciences》2019,42(8):2631-2645
In this paper, we first establish a locality theory for the Noethericity of generalized boundary value problems on the spaces . By means of this theory, of the classical boundary value theory, and of the theory of Fourier analysis, we discuss the necessary and sufficient conditions of the solvability and obtain the general solutions and the Noether conditions for one class of generalized boundary value problems. All cases as regards the index of the coefficients in the equations are considered in detail. Moreover, we apply our theoretical results to the solvability of singular integral equations with variable coefficients. Thus, this paper will be of great significance for the study of improving and developing complex analysis, integral equation, and boundary value theory. 相似文献
3.
SOLVABILITY FOR FRACTIONAL FUNCTIONAL DIFFERENTIAL EQUATION BOUNDARY VALUE PROBLEMS AT RESONANCE
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The paper deals a fractional functional boundary value problems with integral boundary conditions. Besed on the coincidence degree theory, some existence criteria of solutions at resonance are established. 相似文献
4.
L. A. Alexeyeva G. K. Zakir’yanova 《Computational Mathematics and Mathematical Physics》2011,51(7):1194-1207
The method of boundary integral equations is developed as applied to initial-boundary value problems for strictly hyperbolic
systems of second-order equations characteristic of anisotropic media dynamics. Based on the theory of distributions (generalized
functions), solutions are constructed in the space of generalized functions followed by passing to integral representations
and classical solutions. Solutions are considered in the class of singular functions with discontinuous derivatives, which
are typical of physical problems describing shock waves. The uniqueness of the solutions to the initial-boundary value problems
is proved under certain smoothness conditions imposed on the boundary functions. The Green’s matrix of the system and new
fundamental matrices based on it are used to derive integral analogues of the Gauss, Kirchhoff, and Green formulas for solutions
and solving singular boundary integral equations. 相似文献
5.
We consider several elliptic boundary value problems for which there is an overspecification of data on the boundary of the domain. After reformulating the problems in an equivalent integral form, we use the alternate integral formulation to deduce that if a solution exists, then the domain must be an N-ball. Various Green's functions and classical boundary value problems of second, fourth and higher order are included among the problems considered here. 相似文献
6.
David Natroshvili Shota Zazashvili 《Journal of Computational and Applied Mathematics》2010,234(6):1622-1630
The theory of elasticity of hemitropic materials has recently been the object of rigorous mathematical analysis. In particular, the potential method and the theory of pseudodifferential equations have been used in studying the solvability in various function spaces of the main boundary value and transmission problems, in smooth and in Lipschitz domains. The main features and results of this boundary integral equations approach are briefly reviewed here. 相似文献
7.
Merab Svanadze 《PAMM》2007,7(1):4060061-4060062
In this paper, the boundary value problems of steady oscillation (vibration) of the linear theory of thermoelasticity for binary mixtures are investigated by means of the boundary integral equation method (potential method). The uniqueness and existence theorems of solutions of the exterior boundary value problems by means potential method and multidimensional singular integral equations are proved. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
8.
Existence and uniqueness of solutions, as well as their explicit representations, are obtained for singular integral equations with weighted Carleman shift which cannot be reduced to binomial boundary value problems. 相似文献
9.
Solvability for Riemann-Stieltjes integral boundary value problems of Bagley-Torvik equations at resonance
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In this paper, we study the solvability for Riemann-Stieltjes integral boundary value problems of Bagley-Torvik equations with fractional derivative under resonant conditions. Firstly, the kernel function is presented through the Laplace transform and the properties of the kernel function are obtained. And then, some new results on the solvability for the boundary value problem are established by using Mawhin''s coincidence degree theory. Finally, two examples are presented to illustrate the applicability of our main results. 相似文献
10.
Kirill Serkh 《Applied and Computational Harmonic Analysis》2019,46(2):250-287
In this paper we investigate the solution of boundary value problems on polygonal domains for elliptic partial differential equations. Previously, we observed that when the boundary value problems are formulated as boundary integral equations of classical potential theory, the solutions are representable by series of elementary functions, to arbitrary order, for all but finitely many values of the angles. Here, we extend this observation to all values of the angles. We show that the solutions near corners are representable, to arbitrary order, by linear combinations of certain non-integer powers and non-integer powers multiplied by logarithms. 相似文献
11.
This paper examines the existence and uniqueness of solutions for the fractional boundary value problems with integral boundary conditions. Banach’s contraction mapping principle and Schaefer’s fixed point theorem have been used besides topological technique of approximate solutions. An example is propounded to uphold our results. 相似文献
12.
E. Lioubimova P. Schiavone 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2006,77(1):862-874
We consider fundamental (Dirichlet and Neumann-type) boundary value problems in a theory of generalized plane strain for the
steady-state vibrations of an infinite piezoelectric medium with transversely isotropic symmetry (6 mm). Using integral equation methods with the appropriate Sommerfeld-type radiation conditions, we prove existence and uniqueness
results for the corresponding exterior boundary value problems. Exact solutions are obtained in the form of integral potentials. 相似文献
13.
In this paper, a functional boundary value problem is studied. Based on Mawhin's coincidence degree theory, some existence theorems are obtained in the case of non-resonance and the cases of and at resonance. The results not only generalize and improve some known results of multi-point and integral boundary value problems, but also give some existence theorems for boundary value problems that all their boundary value conditions are relied on both x and x′. 相似文献
14.
E. Lioubimova P. Schiavone 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2006,57(5):862-874
We consider fundamental (Dirichlet and Neumann-type) boundary value problems in a theory of generalized plane strain for the
steady-state vibrations of an infinite piezoelectric medium with transversely isotropic symmetry (6 mm). Using integral equation methods with the appropriate Sommerfeld-type radiation conditions, we prove existence and uniqueness
results for the corresponding exterior boundary value problems. Exact solutions are obtained in the form of integral potentials.
(Received: September 27, 2005) 相似文献
15.
Ju. K. Podipenko 《Mathematical Methods in the Applied Sciences》1995,18(6):449-471
In the present paper we investigated the boundary value problems appearing in the study of diffraction of acoustical or electromagnetic waves on arbitrary bounded body contained within the wedge. The potential theory has been developed making it possible to reduce the boundary value problems to Fredholm integral equations on the body's boundary. We prove existence and uniqueness of solutions for these integral equations and the boundary value problems. For some other types of domains with infinite boundaries similar problems have been studied in [4, 6–8, 12–14, 19, 20, 22, 23, 25–28]. 相似文献
16.
17.
A. G. Petrov 《Computational Mathematics and Mathematical Physics》2008,48(8):1266-1283
Two-dimensional and axisymmetric boundary value problems for the Laplace equation in a domain bounded by a closed smooth contour are considered. The problems are reduced to integral equations with a periodic singular kernel, where the period is equal to the length of the contour. Taking into account the periodicity property, high-order accurate quadrature formulas are applied to the integral operator. As a result, the integral equations are reduced to a system of linear algebraic equations. This substantially simplifies the numerical schemes for solving boundary value problems and considerably improves the accuracy of approximation of the integral operator. The boundaries are specified by analytic functions, and the remainder of the quadrature formulas decreases faster than any power of the integration step size. The examples include the two-dimensional potential inviscid circulation flow past a single blade or a grid of blades; the axisymmetric flow past a torus; and free-surface flow problems, such as wave breakdown, standing waves, and the development of Rayleigh-Taylor instability. 相似文献
18.
Hari M. Srivastava Anvar Hasanov Tuhtasin Gulamjanovich Ergashev 《Mathematical Methods in the Applied Sciences》2020,43(10):6181-6199
Potentials play an important role in solving boundary value problems for elliptic equations. In the middle of the last century, a potential theory was constructed for a two-dimensional elliptic equation with one singular coefficient. In the study of potentials, the properties of the fundamental solutions of the given equation are essentially and fruitfully used. At the present time, fundamental solutions of a multidimensional elliptic equation with one degeneration line are already known. In this paper, we investigate the double- and simple-layer potentials for this kind of elliptic equations. Results from potential theory allow us to represent the solution of the boundary value problems in the form of an integral equation. By using some properties of the Gaussian hypergeometric function, we first prove limiting theorems and derive integral equations concerning the densities of the double- and simple-layer potentials. The obtained results are then applied in order to find an explicit solution of the Holmgren problem for the multidimensional singular elliptic equation in the half of the ball. 相似文献
19.
Adrianna Gillman 《Advances in Computational Mathematics》2017,43(2):351-364
This paper presents an integral formulation for Helmholtz problems with mixed boundary conditions. Unlike most integral equation techniques for mixed boundary value problems, the proposed method uses a global boundary charge density. As a result, Calderón identities can be utilized to avoid the use of hypersingular integral operators. Numerical results illustrate the performance of the proposed solution technique. 相似文献
20.
We consider initial boundary value problems for the homogeneous differential equation of hyperbolic or parabolic type in the unbounded two‐ or three‐dimensional spatial domain with the homogeneous initial conditions and with Dirichlet or Neumann boundary condition. The numerical solution is realized in two steps. At first using the Laguerre transformation or Rothe's method with respect to the time variable the non‐stationary problem is reduced to the sequence of boundary value problems for the non‐homogeneous Helmholtz equation. Further we construct the special integral representation for solutions and obtain the sequence of boundary integral equations (without volume integrals). For the full‐discretization of integral equations we propose some projection methods. 相似文献