The singularly perturbed initial value problem for a nonlinear singular equation is considered. By using a simple and special method the asymptotic behavior of solution is studied.
We study the existence and the regularity of solutions for a class of nonlocal equations involving the fractional Laplacian operator with singular nonlinearity and Radon measure data.
Let?denote a smooth,bounded domain in R^(N)(N≥2).Suppose that g is a nondecreasing C^(1)positive function and assume that b(x)is continuous and nonnegative inΩ,and that it may be singular on■Ω.In this paper,we pro...Let?denote a smooth,bounded domain in R^(N)(N≥2).Suppose that g is a nondecreasing C^(1)positive function and assume that b(x)is continuous and nonnegative inΩ,and that it may be singular on■Ω.In this paper,we provide sufficient and necessary conditions on the existence of boundary blow-up solutions to the p-Laplacian problem△_(p)u=b(x)g(u)for x∈Ω,u(x)→+∞as dist(x,■Ω)→0.The estimates of such solutions are also investigated.Moreover,when b has strong singularity,the nonexistence of boundary blow-up(radial)solutions and infinitely many radial solutions are also considered.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
In this paper, we are concerned with a singular parabolic equation subject to Dirichlet boundary condition and initial condition. Under different assumptions on μ, v and ψ, some existence results are obtained by app...In this paper, we are concerned with a singular parabolic equation subject to Dirichlet boundary condition and initial condition. Under different assumptions on μ, v and ψ, some existence results are obtained by applying parabolic regularization method and the sub-super solutions method.展开更多
Some conclusions about the smooth function classes stability for the basic system of equations of atmospheric motion and instability for Navier-Stokes equation are summarized. On the basis of this, by taking the basic...Some conclusions about the smooth function classes stability for the basic system of equations of atmospheric motion and instability for Navier-Stokes equation are summarized. On the basis of this, by taking the basic system of equations of atmospheric motion via Boussinesq approximation as example to explain in detail that the instability about some simplified models of the basic system of equations for atmospheric motion is caused by the instability of Navier-Stokes equation, thereby, a principle to guarantee the stability of simplified equation is drawn in simplifying the basic system of equations.展开更多
A class of quasi-linear singularly perturbed boundary value problems with singular equation is considered. Under suitable conditions, the existence and asymptotic behavior of a solution for the boundary value problem ...A class of quasi-linear singularly perturbed boundary value problems with singular equation is considered. Under suitable conditions, the existence and asymptotic behavior of a solution for the boundary value problem are studied by using the theory of differential inequalities, and the uniformly valid asymptotic expansion of solution with boundary layer is obtained.展开更多
The singularly perturbed boundary value problem for a class of semilinearsingular equation is considered. Using a simple and special method the asym-ptotic behavior of solution is studied.
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 paper, the author discusses the multiple positive solutions for an infinite boundary value problem of first order impulsive singular integro-differential equations on the half line by means of the fixed point ...In this paper, the author discusses the multiple positive solutions for an infinite boundary value problem of first order impulsive singular integro-differential equations on the half line by means of the fixed point theorem of cone expansion and compression with norm type.展开更多
The solutions of the nonlinear singular integral equation , t 6 L, are considered, where L is a closed contour in the complex plane, b ≠ 0 is a constant and f(t) is a polynomial. It is an extension of the results obt...The solutions of the nonlinear singular integral equation , t 6 L, are considered, where L is a closed contour in the complex plane, b ≠ 0 is a constant and f(t) is a polynomial. It is an extension of the results obtained in [1] when f(t) is a constant. Certain special cases are illustrated.展开更多
For domains composed by balls in C^n, this paper studies the boundary behaviour of Cauchy type integrals with discrete holomorphic kernels and the corresponding linear singular integral equation on each piece of smoot...For domains composed by balls in C^n, this paper studies the boundary behaviour of Cauchy type integrals with discrete holomorphic kernels and the corresponding linear singular integral equation on each piece of smooth lower dimensional edges on the boundary of the domain.展开更多
In this article, we study the existence of multiple solutions for the singular semilinear elliptic equation involving critical Sobolev-Hardy exponents -△μ-μ｜x｜^2^-μ=α｜x｜^s^-｜μ｜^2＊（s）-2u＋βα（x）｜u｜^...In this article, we study the existence of multiple solutions for the singular semilinear elliptic equation involving critical Sobolev-Hardy exponents -△μ-μ｜x｜^2^-μ=α｜x｜^s^-｜μ｜^2＊（s）-2u＋βα（x）｜u｜^r-2u,x∈R^n. By means of the concentration-compactness principle and minimax methods, we obtain infinitely many solutions which tend to zero for suitable positive parameters α,β.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 article, a degenerate and singular diffusion problem is studied. The existence of solutions is established by parabolic regularization. Some properties of solutions, for instance, asymptotic behavior, are also...In this article, a degenerate and singular diffusion problem is studied. The existence of solutions is established by parabolic regularization. Some properties of solutions, for instance, asymptotic behavior, are also discussed.展开更多
In this paper,we study the initial-boundary value problem for a class of singular parabolic equations.Under some conditions,we obtain the existence and asymptotic behavior of solutions to the problem by parabolic regu...In this paper,we study the initial-boundary value problem for a class of singular parabolic equations.Under some conditions,we obtain the existence and asymptotic behavior of solutions to the problem by parabolic regularization method and the sub-super solutions method.As a byproduct,we prove the existence of solutions to some problems with gradient terms,which blow up on the boundary.展开更多
By using the concept of finite-part integral, a set of hypersingular integro-differential equations for multiple interracial cracks in a three-dimensional infinite bimaterial subjected to arbitrary loads is derived. I...By using the concept of finite-part integral, a set of hypersingular integro-differential equations for multiple interracial cracks in a three-dimensional infinite bimaterial subjected to arbitrary loads is derived. In the numerical analysis, unknown displacement discontinuities are approximated with the products of the fundamental density functions and power series. The fundamental functions are chosen to express a two-dimensional interface crack rigorously. As illustrative examples, the stress intensity factors for two rectangular interface cracks are calculated for various spacing, crack shape and elastic constants. It is shown that the stress intensity factors decrease with the crack spacing.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 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
基金Supported by Important Project of the National Natural Science Foundation of China( 90 2 1 1 0 0 4 ) andby the"Hundred Talents Project" of Chinese Academy of Science
文摘The singularly perturbed initial value problem for a nonlinear singular equation is considered. By using a simple and special method the asymptotic behavior of solution is studied.
文摘We study the existence and the regularity of solutions for a class of nonlocal equations involving the fractional Laplacian operator with singular nonlinearity and Radon measure data.
基金supported by the Beijing Natural Science Foundation(1212003)。
文摘Let?denote a smooth,bounded domain in R^(N)(N≥2).Suppose that g is a nondecreasing C^(1)positive function and assume that b(x)is continuous and nonnegative inΩ,and that it may be singular on■Ω.In this paper,we provide sufficient and necessary conditions on the existence of boundary blow-up solutions to the p-Laplacian problem△_(p)u=b(x)g(u)for x∈Ω,u(x)→+∞as dist(x,■Ω)→0.The estimates of such solutions are also investigated.Moreover,when b has strong singularity,the nonexistence of boundary blow-up(radial)solutions and infinitely many radial solutions are also considered.
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.
基金partially supported by NSFC(11201311,11201181,11301345)NSF of Guangdong Province(2015A030313623,2016A030313048)+1 种基金Department of Education of Guangdong Province(YQ2015077,YQ2015020)Shenzhen supporting scheme(ZYC201105130121A)
文摘In this paper, we are concerned with a singular parabolic equation subject to Dirichlet boundary condition and initial condition. Under different assumptions on μ, v and ψ, some existence results are obtained by applying parabolic regularization method and the sub-super solutions method.
基金Project supported by the National Natural Science Foundation of China (Nos.40175014, 90411006)the Science Foundation of Shanghai Municipal Commission of Science and Technology(No.02DJ14032)
文摘Some conclusions about the smooth function classes stability for the basic system of equations of atmospheric motion and instability for Navier-Stokes equation are summarized. On the basis of this, by taking the basic system of equations of atmospheric motion via Boussinesq approximation as example to explain in detail that the instability about some simplified models of the basic system of equations for atmospheric motion is caused by the instability of Navier-Stokes equation, thereby, a principle to guarantee the stability of simplified equation is drawn in simplifying the basic system of equations.
基金Supported by the National Natural Science Foundation of China (40876010)the Main Direction Program of the Knowledge Innovation Project of Chinese Academy of Sciences (KZCX2-YW-Q03-08)the R & D Special Fund for Public Welfare Industry (meteorology) (GYHY200806010)
文摘A class of quasi-linear singularly perturbed boundary value problems with singular equation is considered. Under suitable conditions, the existence and asymptotic behavior of a solution for the boundary value problem are studied by using the theory of differential inequalities, and the uniformly valid asymptotic expansion of solution with boundary layer is obtained.
基金Supported by important study project of the National Natural Science Foundation of China(No.90211004),and by the"Hundred Talents Project"of the Chinese Academy of Sciences.
文摘The singularly perturbed boundary value problem for a class of semilinearsingular equation is considered. Using a simple and special method the asym-ptotic behavior of solution is studied.
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 Nature Science Foundation of China (10671167)
文摘In this paper, the author discusses the multiple positive solutions for an infinite boundary value problem of first order impulsive singular integro-differential equations on the half line by means of the fixed point theorem of cone expansion and compression with norm type.
文摘The solutions of the nonlinear singular integral equation , t 6 L, are considered, where L is a closed contour in the complex plane, b ≠ 0 is a constant and f(t) is a polynomial. It is an extension of the results obtained in [1] when f(t) is a constant. Certain special cases are illustrated.
基金Project supported by the National Science Foundation of China (10271097)
文摘For domains composed by balls in C^n, this paper studies the boundary behaviour of Cauchy type integrals with discrete holomorphic kernels and the corresponding linear singular integral equation on each piece of smooth lower dimensional edges on the boundary of the domain.
文摘In this article, we study the existence of multiple solutions for the singular semilinear elliptic equation involving critical Sobolev-Hardy exponents -△μ-μ｜x｜^2^-μ=α｜x｜^s^-｜μ｜^2＊（s）-2u＋βα（x）｜u｜^r-2u,x∈R^n. By means of the concentration-compactness principle and minimax methods, we obtain infinitely many solutions which tend to zero for suitable positive parameters α,β.
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 National Natural Science Foundation of China(10171113 and 10471156)Natural Science Foundation of GuangDong 2004(4009793).
文摘In this article, a degenerate and singular diffusion problem is studied. The existence of solutions is established by parabolic regularization. Some properties of solutions, for instance, asymptotic behavior, are also discussed.
基金Supported by Natural Science Foundation of Youth and Tianyuan (11001177,11026156,10926141)Startup Program of Shenzhen University
文摘In this paper,we study the initial-boundary value problem for a class of singular parabolic equations.Under some conditions,we obtain the existence and asymptotic behavior of solutions to the problem by parabolic regularization method and the sub-super solutions method.As a byproduct,we prove the existence of solutions to some problems with gradient terms,which blow up on the boundary.
基金supported by the National Natural Science Foundation of China (No. 10872213)
文摘By using the concept of finite-part integral, a set of hypersingular integro-differential equations for multiple interracial cracks in a three-dimensional infinite bimaterial subjected to arbitrary loads is derived. In the numerical analysis, unknown displacement discontinuities are approximated with the products of the fundamental density functions and power series. The fundamental functions are chosen to express a two-dimensional interface crack rigorously. As illustrative examples, the stress intensity factors for two rectangular interface cracks are calculated for various spacing, crack shape and elastic constants. It is shown that the stress intensity factors decrease with the crack spacing.
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.
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.
