In this article, by introducing characteristic singular integral operator and associate singular integral equations (SIEs), the authors discuss the direct method of solution for a class of singular integral equation...In this article, by introducing characteristic singular integral operator and associate singular integral equations (SIEs), the authors discuss the direct method of solution for a class of singular integral equations with certain analytic inputs. They obtain both the conditions of solvability and the solutions in closed form. It is noteworthy that the method is different from the classical one that is due to Lu.展开更多
In this article,the authors discuss a kind of modified singular integral equations on a disjoint union of closed contours or a disjoint union of open arcs.The authors introduce some singular integral operators associa...In this article,the authors discuss a kind of modified singular integral equations on a disjoint union of closed contours or a disjoint union of open arcs.The authors introduce some singular integral operators associated with this kind of singular integral equations,and obtain some useful properties for them.An operatorial approach is also given together with some illustrated examples.展开更多
By means of the method of solid angle coefficients and the permutation formula on the building domain of complex biballs,direct solutions of some singular integral equations with variable coefficients are discussed an...By means of the method of solid angle coefficients and the permutation formula on the building domain of complex biballs,direct solutions of some singular integral equations with variable coefficients are discussed and the explicit formulas for these solutions are obtained.展开更多
The singular integral equations with Cauchy kernels have studied in L_p(Γ),p>1,in detail.Here Γ stands for the set of a finite number of closed Liapunoff's curves which have no common points and T denotes the...The singular integral equations with Cauchy kernels have studied in L_p(Γ),p>1,in detail.Here Γ stands for the set of a finite number of closed Liapunoff's curves which have no common points and T denotes the completely continuous operator in the space sunder consideration. In this paper, we consider the equations mentioned above in Orlicz spaces L_M(Γ). It is proved that the Nether theorem and the index formula are hold true in the case of reflexive Orlicz spaces.展开更多
In this paper, the difficulties on calculation in solving singular integral equations are overcome when the restriction of curve of integration to be a closed contour is cancelled. When the curve is an open arc and th...In this paper, the difficulties on calculation in solving singular integral equations are overcome when the restriction of curve of integration to be a closed contour is cancelled. When the curve is an open arc and the solutions for singular integral equations possess singularities of higher order, the solution and the solvable condition for characteristic equations as well as the generalized Noether theorem for complete equations are given.展开更多
In this article, periodic Riemann boundary value problem with period 2aπ along closed smooth contours is discussed, and then singular integral equation with kernel csc t-t0/a along closed smooth contours restricted i...In this article, periodic Riemann boundary value problem with period 2aπ along closed smooth contours is discussed, and then singular integral equation with kernel csc t-t0/a along closed smooth contours restricted in the strip 0 〈 Rez 〈 aπ is discussed. Finally, the solutions with singularities of order one for the above two problems are discussed.展开更多
We transform the singular integral equations with solutions simultaneously having singularities of higher order at infinite point and at several finite points on the real axis into ones along a closed contour with sol...We transform the singular integral equations with solutions simultaneously having singularities of higher order at infinite point and at several finite points on the real axis into ones along a closed contour with solutions having singularities of higher order, and for the former obtain the extended Neother theorem of complete equation as well as the solutions and the solvable conditions of characteristic equation from the latter. The conclusions drawn by this article contain special cases discussed before.展开更多
In this paper, a class of singular integral equations with complex translations is discussed. By using the Plemelj projection method the authors reduce them to the boundary value problem of analytic functions in A(+)(...In this paper, a class of singular integral equations with complex translations is discussed. By using the Plemelj projection method the authors reduce them to the boundary value problem of analytic functions in A(+)(H) over cap(0) with upper translation and the boundary value problem of analytic functions in A-(H) over cap(0) with lower translation, which are solved here.展开更多
In this paper,some kinds of singular integral equations of convolution type with reflection and translation shift are discussed and they are turned into Riemann boundary value problems with both discontinuous coeffici...In this paper,some kinds of singular integral equations of convolution type with reflection and translation shift are discussed and they are turned into Riemann boundary value problems with both discontinuous coefficients and reflection by using the Fourier transform.In spite of the classical method for solution,we are to give another method,therefore the general solution and condition of solvability are obtained in class{0}.展开更多
The numerical solutions to the singular integral equations obtained by the fracture mechanical analyses of a cracked wedge under three different conditions are considered. The three considered conditions are:(i) a rad...The numerical solutions to the singular integral equations obtained by the fracture mechanical analyses of a cracked wedge under three different conditions are considered. The three considered conditions are:(i) a radial crack on a wedge with a nonfinite radius under the traction-traction boundary condition,(ii) a radial crack on a wedge with a finite radius under the traction-traction boundary condition, and(iii) a radial crack on a finite radius wedge under the traction-displacement boundary condition. According to the boundary conditions, the extracted singular integral equations have different forms. Numerical methods are used to solve the obtained coupled singular integral equations, where the Gauss-Legendre and the Gauss-Chebyshev polynomials are used to approximate the responses of the singular integral equations. The results are presented in figures and compared with those obtained by the analytical response. The results show that the obtained Gauss-Chebyshev polynomial response is closer to the analytical response.展开更多
Using the method of the boundary integral equation, a set of singular integral equations of the hear transfer problems and the thermo-elastic problems of a crack embedded in a two-dimensional finite body is derived, a...Using the method of the boundary integral equation, a set of singular integral equations of the hear transfer problems and the thermo-elastic problems of a crack embedded in a two-dimensional finite body is derived, and then,its numerical method is proposed by the numerical method of the singular integral equations combined with boundary element method. Moreover, the singular nature of temperature gradient field near the crack front is proved by the main-part analysis method of the singular integral equation, and the singular temperature gradients are exactly obtained. Finally, several typical examples calculated.展开更多
In this paper, it is considered for some two-dimensional singular integral equations of the hypercomplex functions in the Douglis sense. In some special cases, the Fredholm' conditions and index formula of such eq...In this paper, it is considered for some two-dimensional singular integral equations of the hypercomplex functions in the Douglis sense. In some special cases, the Fredholm' conditions and index formula of such equations are obtained.展开更多
We have discussed and solved the boundary value problem with period 2aπ and the singular integral equation with kernel csc t-tv/a in solution having singularities of high order, where the smooth contour of integratio...We have discussed and solved the boundary value problem with period 2aπ and the singular integral equation with kernel csc t-tv/a in solution having singularities of high order, where the smooth contour of integration is in the strip 0<Rez<aπ.展开更多
This paper develops fast multiscale collocation methods for a class of Fredholm integral equations of the second kind with singular kernels. A truncation strategy for the coefficient matrix of the corresponding discre...This paper develops fast multiscale collocation methods for a class of Fredholm integral equations of the second kind with singular kernels. A truncation strategy for the coefficient matrix of the corresponding discrete system is proposed, which forms a basis for fast algorithms. The convergence, stability and computational complexity of these algorithms are analyzed.展开更多
Daubechies interval cally weakly singular Fredholm kind. Utilizing the orthogonality equation is reduced into a linear wavelet is used to solve nurneriintegral equations of the second of the wavelet basis, the integra...Daubechies interval cally weakly singular Fredholm kind. Utilizing the orthogonality equation is reduced into a linear wavelet is used to solve nurneriintegral equations of the second of the wavelet basis, the integral system of equations. The vanishing moments of the wavelet make the wavelet coefficient matrices sparse, while the continuity of the derivative functions of basis overcomes naturally the singular problem of the integral solution. The uniform convergence of the approximate solution by the wavelet method is proved and the error bound is given. Finally, numerical example is presented to show the application of the wavelet method.展开更多
We mainly consider a class of two dimensional normal type singular integral equations, the solutions as well as the conditions of solvability of which are obtained .The methods used consist of transferring them to sev...We mainly consider a class of two dimensional normal type singular integral equations, the solutions as well as the conditions of solvability of which are obtained .The methods used consist of transferring them to several one dimensional Riemann boundary value problems, and then solving the latter.展开更多
Under the appropriate hypotheses subject to the unknown function and the free term, by means of our Lemma, we prove the rationality of order at x = ∞ on two sides for the characteristic singular integral equations wi...Under the appropriate hypotheses subject to the unknown function and the free term, by means of our Lemma, we prove the rationality of order at x = ∞ on two sides for the characteristic singular integral equations with solutions having singularities of higher order on the real axis X. We transform the equations into solving equivalent Riemann boundary value problems with solutions having singularities of higher order and with additional conditions on X. The solutions and the solvable conditions for the former are obtained from the latter.展开更多
In this paper, two kinds of convolution integral equations with both Cauchy kernel and reflection are discussed. Using Fourier transformation theory they can be transformed into Riemann boundary value problems with bo...In this paper, two kinds of convolution integral equations with both Cauchy kernel and reflection are discussed. Using Fourier transformation theory they can be transformed into Riemann boundary value problems with both discontinuous coefficients and reflection. And we give a new method, by which the general solution and the condition of solvability are obtained respectively.展开更多
In this paper, one class of nonlinear singular integral equation is discussed through Lagrange interpolation method. We research the connections between numerical solutions of the equations and chaos in the process of...In this paper, one class of nonlinear singular integral equation is discussed through Lagrange interpolation method. We research the connections between numerical solutions of the equations and chaos in the process of solving by iterative method.展开更多
In this paper, we derive a simple and efficient matrix formulation using Laguerre polynomials to solve the singular integral equation with degenerate kernel. This method is based on replacement of the unknown function...In this paper, we derive a simple and efficient matrix formulation using Laguerre polynomials to solve the singular integral equation with degenerate kernel. This method is based on replacement of the unknown function by truncated series of well known Laguerre expansion of functions. This leads to a system of algebraic equations with Laguerre coefficients. Thus, by solving the matrix equation, the coefficients are obtained. Some numerical examples are included to demonstrate the validity and applicability of the proposed method.展开更多
基金Project was supported by RFDP of Higher Education and NNSF of China, SF of Wuhan University
文摘In this article, by introducing characteristic singular integral operator and associate singular integral equations (SIEs), the authors discuss the direct method of solution for a class of singular integral equations with certain analytic inputs. They obtain both the conditions of solvability and the solutions in closed form. It is noteworthy that the method is different from the classical one that is due to Lu.
基金NNSF of China(10471107)RFDP of Higher Education(20060486001)
文摘In this article,the authors discuss a kind of modified singular integral equations on a disjoint union of closed contours or a disjoint union of open arcs.The authors introduce some singular integral operators associated with this kind of singular integral equations,and obtain some useful properties for them.An operatorial approach is also given together with some illustrated examples.
基金Supported by the NNSF of china(11171298)SuppoSed by the Natural Science Foundation of Zhejiang Province(Y6110425,Y604563)
文摘By means of the method of solid angle coefficients and the permutation formula on the building domain of complex biballs,direct solutions of some singular integral equations with variable coefficients are discussed and the explicit formulas for these solutions are obtained.
文摘The singular integral equations with Cauchy kernels have studied in L_p(Γ),p>1,in detail.Here Γ stands for the set of a finite number of closed Liapunoff's curves which have no common points and T denotes the completely continuous operator in the space sunder consideration. In this paper, we consider the equations mentioned above in Orlicz spaces L_M(Γ). It is proved that the Nether theorem and the index formula are hold true in the case of reflexive Orlicz spaces.
基金Foundation item is supported by the NNSF of China(19971064)
文摘In this paper, the difficulties on calculation in solving singular integral equations are overcome when the restriction of curve of integration to be a closed contour is cancelled. When the curve is an open arc and the solutions for singular integral equations possess singularities of higher order, the solution and the solvable condition for characteristic equations as well as the generalized Noether theorem for complete equations are given.
基金Supported by the National Natural Science Foundation of China (19971064)
文摘In this article, periodic Riemann boundary value problem with period 2aπ along closed smooth contours is discussed, and then singular integral equation with kernel csc t-t0/a along closed smooth contours restricted in the strip 0 〈 Rez 〈 aπ is discussed. Finally, the solutions with singularities of order one for the above two problems are discussed.
基金Supported by the NNSF of China (10471107)RFDP of Higher Education of China (20060486001)
文摘We transform the singular integral equations with solutions simultaneously having singularities of higher order at infinite point and at several finite points on the real axis into ones along a closed contour with solutions having singularities of higher order, and for the former obtain the extended Neother theorem of complete equation as well as the solutions and the solvable conditions of characteristic equation from the latter. The conclusions drawn by this article contain special cases discussed before.
文摘In this paper, a class of singular integral equations with complex translations is discussed. By using the Plemelj projection method the authors reduce them to the boundary value problem of analytic functions in A(+)(H) over cap(0) with upper translation and the boundary value problem of analytic functions in A-(H) over cap(0) with lower translation, which are solved here.
基金Supported by the Qufu Normal University Youth Fund(XJ201218)
文摘In this paper,some kinds of singular integral equations of convolution type with reflection and translation shift are discussed and they are turned into Riemann boundary value problems with both discontinuous coefficients and reflection by using the Fourier transform.In spite of the classical method for solution,we are to give another method,therefore the general solution and condition of solvability are obtained in class{0}.
文摘The numerical solutions to the singular integral equations obtained by the fracture mechanical analyses of a cracked wedge under three different conditions are considered. The three considered conditions are:(i) a radial crack on a wedge with a nonfinite radius under the traction-traction boundary condition,(ii) a radial crack on a wedge with a finite radius under the traction-traction boundary condition, and(iii) a radial crack on a finite radius wedge under the traction-displacement boundary condition. According to the boundary conditions, the extracted singular integral equations have different forms. Numerical methods are used to solve the obtained coupled singular integral equations, where the Gauss-Legendre and the Gauss-Chebyshev polynomials are used to approximate the responses of the singular integral equations. The results are presented in figures and compared with those obtained by the analytical response. The results show that the obtained Gauss-Chebyshev polynomial response is closer to the analytical response.
文摘Using the method of the boundary integral equation, a set of singular integral equations of the hear transfer problems and the thermo-elastic problems of a crack embedded in a two-dimensional finite body is derived, and then,its numerical method is proposed by the numerical method of the singular integral equations combined with boundary element method. Moreover, the singular nature of temperature gradient field near the crack front is proved by the main-part analysis method of the singular integral equation, and the singular temperature gradients are exactly obtained. Finally, several typical examples calculated.
文摘In this paper, it is considered for some two-dimensional singular integral equations of the hypercomplex functions in the Douglis sense. In some special cases, the Fredholm' conditions and index formula of such equations are obtained.
基金Supported by the National Natural Science Foundation of China(19971064 10161009)
文摘We have discussed and solved the boundary value problem with period 2aπ and the singular integral equation with kernel csc t-tv/a in solution having singularities of high order, where the smooth contour of integration is in the strip 0<Rez<aπ.
基金The NSF (10371137 and 10201034) of China,the Foundation (20030558008) of Doctoral Program of National Higher Education,Guangdong Provincial Natural Science Foundation (1011170) of China and the Foundation of Zhongshan University Advanced Research Center.
文摘This paper develops fast multiscale collocation methods for a class of Fredholm integral equations of the second kind with singular kernels. A truncation strategy for the coefficient matrix of the corresponding discrete system is proposed, which forms a basis for fast algorithms. The convergence, stability and computational complexity of these algorithms are analyzed.
基金Supported by the National Natural Science Foundation of China (60572048)the Natural Science Foundation of Guangdong Province(054006621)
文摘Daubechies interval cally weakly singular Fredholm kind. Utilizing the orthogonality equation is reduced into a linear wavelet is used to solve nurneriintegral equations of the second of the wavelet basis, the integral system of equations. The vanishing moments of the wavelet make the wavelet coefficient matrices sparse, while the continuity of the derivative functions of basis overcomes naturally the singular problem of the integral solution. The uniform convergence of the approximate solution by the wavelet method is proved and the error bound is given. Finally, numerical example is presented to show the application of the wavelet method.
文摘We mainly consider a class of two dimensional normal type singular integral equations, the solutions as well as the conditions of solvability of which are obtained .The methods used consist of transferring them to several one dimensional Riemann boundary value problems, and then solving the latter.
基金Supported by the National Natural Science Foundation of China (10471107)the Specialized Research Fund for the Doctoral Program of Higher Education of China (20060486001)
文摘Under the appropriate hypotheses subject to the unknown function and the free term, by means of our Lemma, we prove the rationality of order at x = ∞ on two sides for the characteristic singular integral equations with solutions having singularities of higher order on the real axis X. We transform the equations into solving equivalent Riemann boundary value problems with solutions having singularities of higher order and with additional conditions on X. The solutions and the solvable conditions for the former are obtained from the latter.
文摘In this paper, two kinds of convolution integral equations with both Cauchy kernel and reflection are discussed. Using Fourier transformation theory they can be transformed into Riemann boundary value problems with both discontinuous coefficients and reflection. And we give a new method, by which the general solution and the condition of solvability are obtained respectively.
文摘In this paper, one class of nonlinear singular integral equation is discussed through Lagrange interpolation method. We research the connections between numerical solutions of the equations and chaos in the process of solving by iterative method.
文摘In this paper, we derive a simple and efficient matrix formulation using Laguerre polynomials to solve the singular integral equation with degenerate kernel. This method is based on replacement of the unknown function by truncated series of well known Laguerre expansion of functions. This leads to a system of algebraic equations with Laguerre coefficients. Thus, by solving the matrix equation, the coefficients are obtained. Some numerical examples are included to demonstrate the validity and applicability of the proposed method.