In this paper, we prove the global existence of generalized solutions to a two-dimensional Cauchy problem of a hyperbolic system by introducing a new definition of generalized solution. Moreover, the solution may invo...In this paper, we prove the global existence of generalized solutions to a two-dimensional Cauchy problem of a hyperbolic system by introducing a new definition of generalized solution. Moreover, the solution may involve delta-wave.展开更多
As a new structure of solid matter quasicrystal brings profound new ideas to the traditional condensed matter physics, its elastic equations are more complicated than that of traditional crystal. A contact problem of ...As a new structure of solid matter quasicrystal brings profound new ideas to the traditional condensed matter physics, its elastic equations are more complicated than that of traditional crystal. A contact problem of decagonal two? dimensional quasicrystal material under the action of a rigid flat die is solved satisfactorily by introducing displacement function and using Fourier analysis and dual integral equations theory, and the analytical expressions of stress and displacement fields of the contact problem are achieved. The results show that if the contact displacement is a constant in the contact zone, the vertical contact stress has order -1/2 singularity on the edge of contact zone, which provides the important mechanics parameter for contact deformation of the quasicrystal.展开更多
In this paper we survey the authors＇ and related work on two-dimensional Riemann problems for hyperbolic conservation laws, mainly those related to the compressible Euler equations in gas dynamics. It contains four s...In this paper we survey the authors＇ and related work on two-dimensional Riemann problems for hyperbolic conservation laws, mainly those related to the compressible Euler equations in gas dynamics. It contains four sections： 1. Historical review. 2. Scalar conservation laws. 3. Euler equations. 4. Simplified models.展开更多
This paper proposes an eigenfunction expansion method to solve twodimensional （2D） elasticity problems based on stress formulation. By introducing appropriate state functions, the fundamental system of partial diffe...This paper proposes an eigenfunction expansion method to solve twodimensional （2D） elasticity problems based on stress formulation. By introducing appropriate state functions, the fundamental system of partial differential equations of the above 2D problems is rewritten as an upper triangular differential system. For the associated operator matrix, the existence and the completeness of two normed orthogonal eigenfunction systems in some space are obtained, which belong to the two block operators arising in the operator matrix. Moreover, the general solution to the above 2D problem is given by the eigenfunction expansion method.展开更多
The Riemann problem for a two-dimensional 2 x 2 nonstrictly hyperbolic system of nonlinear conservation laws has been solved thoroughly for any given initial data which are constant in each quadrant. The non-classical...The Riemann problem for a two-dimensional 2 x 2 nonstrictly hyperbolic system of nonlinear conservation laws has been solved thoroughly for any given initial data which are constant in each quadrant. The non-classical shockwaves, which are labelled as delta-shock waves, appear in some solutions. The solutions have been obtained are not unique. Due to the specific property of the system considered, there are no rarefaction waves in solution. This paper is divided into three parts. The first part constructs Riemann solutions for initial data involving two contact discontinuities while the second considers the case for other initial data. The last part briefly discusses the non-uniqueness of the solutions.展开更多
Using a polarization method, the scattering problem for a two-dimensional inclusion embedded in infinite piezoelectric/piezomagnetic matrices is investigated. To achieve the purpose, the polarization method for a two-...Using a polarization method, the scattering problem for a two-dimensional inclusion embedded in infinite piezoelectric/piezomagnetic matrices is investigated. To achieve the purpose, the polarization method for a two-dimensional piezoelectric/piezomagnetic ＂comparison body＂ is formulated. For simple harmonic motion, kernel of the polarization method reduces to a 2-D time-harmonic Green＇s function, which is obtained using the Radon transform. The expression is further simplified under conditions of low frequency of the incident wave and small diameter of the inclusion. Some analytical expressions are obtained. The analytical solutions for generalized piezoelectric/piezomagnetic anisotropic composites are given followed by simplified results for piezoelectric composites. Based on the latter results, two numerical results are provided for an elliptical cylindrical inclusion in a PZT-5H-matrix, showing the effect of different factors including size, shape, material properties, and piezoelectricity on the scattering cross-section.展开更多
We are concerned with global solutions of multidimensional(M-D)Riemann problems for nonlinear hyperbolic systems of conservation laws,focusing on their global configurations and structures.We present some recent devel...We are concerned with global solutions of multidimensional(M-D)Riemann problems for nonlinear hyperbolic systems of conservation laws,focusing on their global configurations and structures.We present some recent developments in the rigorous analysis of two-dimensional(2-D)Riemann problems involving transonic shock waves through several prototypes of hyperbolic systems of conservation laws and discuss some further M-D Riemann problems and related problems for nonlinear partial differential equations.In particular,we present four different 2-D Riemann problems through these prototypes of hyperbolic systems and show how these Riemann problems can be reformulated/solved as free boundary problems with transonic shock waves as free boundaries for the corresponding nonlinear conservation laws of mixed elliptic-hyperbolic type and related nonlinear partial differential equations.展开更多
This paper studies the eigenfunction expansion method to solve the two dimensional （2D） elasticity problems based on the stress formulation. The fundamental system of partial differential equations of the 2D problem...This paper studies the eigenfunction expansion method to solve the two dimensional （2D） elasticity problems based on the stress formulation. The fundamental system of partial differential equations of the 2D problems is rewritten as an upper tri angular differential system based on the known results, and then the associated upper triangular operator matrix matrix is obtained. By further research, the two simpler com plete orthogonal systems of eigenfunctions in some space are obtained, which belong to the two block operators arising in the operator matrix. Then, a more simple and conve nient general solution to the 2D problem is given by the eigenfunction expansion method. Furthermore, the boundary conditions for the 2D problem, which can be solved by this method, are indicated. Finally, the validity of the obtained results is verified by a specific example.展开更多
The simplified linear model of Grad-Shafranov （GS） reconstruction can be reformulated into an inverse boundary value problem of Laplace＇s equation. Therefore, in this paper we focus on the method of solving the inv...The simplified linear model of Grad-Shafranov （GS） reconstruction can be reformulated into an inverse boundary value problem of Laplace＇s equation. Therefore, in this paper we focus on the method of solving the inverse boundary value problem of Laplace＇s equation. In the first place, the variational regularization method is used to deal with the ill- posedness of the Cauchy problem for Laplace＇s equation. Then, the ＇L-Curve＇ principle is suggested to be adopted in choosing the optimal regularization parameter. Finally, a numerical experiment is implemented with a section of Neumann and Dirichlet boundary conditions with observation errors. The results well converge to the exact solution of the problem, which proves the efficiency and robustness of the proposed method. When the order of observation error δ is 10-1, the order of the approximate result error can reach 10-3.展开更多
In this paper, we consider the Cauchy problem for the Laplace equation, which is severely ill-posed in the sense that the solution does not depend continuously on the data. A modified Tikhonov regularization method is...In this paper, we consider the Cauchy problem for the Laplace equation, which is severely ill-posed in the sense that the solution does not depend continuously on the data. A modified Tikhonov regularization method is proposed to solve this problem. An error estimate for the a priori parameter choice between the exact solution and its regularized approximation is obtained. Moreover, an a posteriori parameter choice rule is proposed and a stable error estimate is also obtained. Numerical examples illustrate the validity and effectiveness of this method.展开更多
This article is contributed to the Cauchy problem {δu/δt=△u＋K（｜x｜）u^p in R^n×（0,T）, u（x,0）=φ（x） in R^n;with initial function φ≡/0. The stability of positive radial steady state, which are positiv...This article is contributed to the Cauchy problem {δu/δt=△u＋K（｜x｜）u^p in R^n×（0,T）, u（x,0）=φ（x） in R^n;with initial function φ≡/0. The stability of positive radial steady state, which are positive solutions of △u ＋ K（｜x｜）u^p =0, is obtained when p is critical for general K（｜x｜）.展开更多
We consider the growth rate and quenching rate of the following problem with singular nonlinearityfor some positive constants b：, b2 （see Theorem 3.3 for the parametersfor some constantsHence, the solution （u, v） ...We consider the growth rate and quenching rate of the following problem with singular nonlinearityfor some positive constants b：, b2 （see Theorem 3.3 for the parametersfor some constantsHence, the solution （u, v） quenches at the originx = 0 at the same time ＇1＇ （see Theorem 4.3）. We also tind various other conditions tor the solution to quench in a finite time and obtain the corresponding decay rate of the solution near the quenching time.展开更多
In this paper, the existence and the uniqueness of the local generalized solution and the local classical solution of the Cauchy problem for the generalized BBM-Burgers equationare proved. The existence and the unique...In this paper, the existence and the uniqueness of the local generalized solution and the local classical solution of the Cauchy problem for the generalized BBM-Burgers equationare proved. The existence and the uniqueness of the global generalized solution and the global classical solution for the Cauchy problem of equation （1） are proved when n = 3, 2, 1. Moreover, the decay property of the solution is discussed.展开更多
This article considers Cauchy problem u(t) - (uv)(x) = 0, v(t) - u(x) = 0, u(x, 0) = u(0) (x) > 0, v(x, 0) = v(0)(x). A necessary and sufficient condition in guaranteeing that Cauchy problem admits a global C-1-sol...This article considers Cauchy problem u(t) - (uv)(x) = 0, v(t) - u(x) = 0, u(x, 0) = u(0) (x) > 0, v(x, 0) = v(0)(x). A necessary and sufficient condition in guaranteeing that Cauchy problem admits a global C-1-solution on t greater than or equal to 0 is obtained.展开更多
In the present paper,the local existence of classical solutions to the periodic boundary problem and the Cauchy problem of a quasilinear evolution equation are studied under the assumptions that do not require the mon...In the present paper,the local existence of classical solutions to the periodic boundary problem and the Cauchy problem of a quasilinear evolution equation are studied under the assumptions that do not require the monotonicity of σi(s) (i= 1,…, n). The nonexistence of global solutions to the initial-boundary value problem of the equation is also discussed, a blowup theorem is proved and a concrete example is given.展开更多
This paper is concerned with the Cauchy problem for the modified Helmholtz equation in an infinite strip domain 0<x≤1,y∈R.The Cauchy data at x = 0 is given and the solution is then sought for the interval 0<x...This paper is concerned with the Cauchy problem for the modified Helmholtz equation in an infinite strip domain 0<x≤1,y∈R.The Cauchy data at x = 0 is given and the solution is then sought for the interval 0<x≤1.This problem is highly ill-posed and the solution(if it exists) does not depend continuously on the given data. In this paper,we propose a fourth-order modified method to solve the Cauchy problem. Convergence estimates are presented under the suitable choices of regularization parameters and the a priori assumption on the bounds of the exact solution.Numerical implementation is considered and the numerical examples show that our proposed method is effective and stable.展开更多
A Cauchy problem for the semi-linear elliptic equation is investigated. We use a filtering function method to define a regularization solution for this ill-posed problem. The existence, uniqueness and stability of the...A Cauchy problem for the semi-linear elliptic equation is investigated. We use a filtering function method to define a regularization solution for this ill-posed problem. The existence, uniqueness and stability of the regularization solution are proven;a convergence estimate of H?lder type for the regularization method is obtained under the a-priori bound assumption for the exact solution. An iterative scheme is proposed to calculate the regularization solution;some numerical results show that this method works well.展开更多
In this paper,we consider a Cauchy problem of the time fractional diffusion equation(TFDE)in x∈[0,L].This problem is ubiquitous in science and engineering applications.The illposedness of the Cauchy problem is explai...In this paper,we consider a Cauchy problem of the time fractional diffusion equation(TFDE)in x∈[0,L].This problem is ubiquitous in science and engineering applications.The illposedness of the Cauchy problem is explained by its solution in frequency domain.Furthermore,the problem is formulated into a minimization problem with a modified Tikhonov regularization method.The gradient of the regularization functional based on an adjoint problem is deduced and the standard conjugate gradient method is presented for solving the minimization problem.The error estimates for the regularized solutions are obtained under Hp norm priori bound assumptions.Finally,numerical examples illustrate the effectiveness of the proposed method.展开更多
文摘In this paper, we prove the global existence of generalized solutions to a two-dimensional Cauchy problem of a hyperbolic system by introducing a new definition of generalized solution. Moreover, the solution may involve delta-wave.
文摘As a new structure of solid matter quasicrystal brings profound new ideas to the traditional condensed matter physics, its elastic equations are more complicated than that of traditional crystal. A contact problem of decagonal two? dimensional quasicrystal material under the action of a rigid flat die is solved satisfactorily by introducing displacement function and using Fourier analysis and dual integral equations theory, and the analytical expressions of stress and displacement fields of the contact problem are achieved. The results show that if the contact displacement is a constant in the contact zone, the vertical contact stress has order -1/2 singularity on the edge of contact zone, which provides the important mechanics parameter for contact deformation of the quasicrystal.
基金supported by 973 Key program and the Key Program from Beijing Educational Commission with No. KZ200910028002Program for New Century Excellent Talents in University (NCET)+4 种基金Funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality (PHR-IHLB)The research of Sheng partially supported by NSFC (10671120)Shanghai Leading Academic Discipline Project: J50101The research of Zhang partially supported by NSFC (10671120)The research of Zheng partially supported by NSF-DMS-0603859
文摘In this paper we survey the authors＇ and related work on two-dimensional Riemann problems for hyperbolic conservation laws, mainly those related to the compressible Euler equations in gas dynamics. It contains four sections： 1. Historical review. 2. Scalar conservation laws. 3. Euler equations. 4. Simplified models.
基金Project supported by the National Natural Science Foundation of China (No. 10962004)the Special-ized Research Fund for the Doctoral Program of Higher Education of China (No. 20070126002)+1 种基金the Chunhui Program of Ministry of Education of China (No. Z2009-1-01010)the Natural Science Foundation of Inner Mongolia (No. 2009BS0101)
文摘This paper proposes an eigenfunction expansion method to solve twodimensional （2D） elasticity problems based on stress formulation. By introducing appropriate state functions, the fundamental system of partial differential equations of the above 2D problems is rewritten as an upper triangular differential system. For the associated operator matrix, the existence and the completeness of two normed orthogonal eigenfunction systems in some space are obtained, which belong to the two block operators arising in the operator matrix. Moreover, the general solution to the above 2D problem is given by the eigenfunction expansion method.
文摘The Riemann problem for a two-dimensional 2 x 2 nonstrictly hyperbolic system of nonlinear conservation laws has been solved thoroughly for any given initial data which are constant in each quadrant. The non-classical shockwaves, which are labelled as delta-shock waves, appear in some solutions. The solutions have been obtained are not unique. Due to the specific property of the system considered, there are no rarefaction waves in solution. This paper is divided into three parts. The first part constructs Riemann solutions for initial data involving two contact discontinuities while the second considers the case for other initial data. The last part briefly discusses the non-uniqueness of the solutions.
基金supported by the National Natural Science Foundation of China (Nos. 10732100, 10572155)the Science and Technology Planning Project of Guangdong Province of China (No. 2006A11001002)the Ph. D. Programs Foundation of Ministry of Education of China (No. 2006300004111179)
文摘Using a polarization method, the scattering problem for a two-dimensional inclusion embedded in infinite piezoelectric/piezomagnetic matrices is investigated. To achieve the purpose, the polarization method for a two-dimensional piezoelectric/piezomagnetic ＂comparison body＂ is formulated. For simple harmonic motion, kernel of the polarization method reduces to a 2-D time-harmonic Green＇s function, which is obtained using the Radon transform. The expression is further simplified under conditions of low frequency of the incident wave and small diameter of the inclusion. Some analytical expressions are obtained. The analytical solutions for generalized piezoelectric/piezomagnetic anisotropic composites are given followed by simplified results for piezoelectric composites. Based on the latter results, two numerical results are provided for an elliptical cylindrical inclusion in a PZT-5H-matrix, showing the effect of different factors including size, shape, material properties, and piezoelectricity on the scattering cross-section.
基金The research of Gui-Qiang G.Chen was supported in part by the UK Engineering and Physical Sciences Research Council Awards EP/L015811/1,EP/V008854/1,EP/V051121/1the Royal Society-Wolfson Research Merit Award WM090014.
文摘We are concerned with global solutions of multidimensional(M-D)Riemann problems for nonlinear hyperbolic systems of conservation laws,focusing on their global configurations and structures.We present some recent developments in the rigorous analysis of two-dimensional(2-D)Riemann problems involving transonic shock waves through several prototypes of hyperbolic systems of conservation laws and discuss some further M-D Riemann problems and related problems for nonlinear partial differential equations.In particular,we present four different 2-D Riemann problems through these prototypes of hyperbolic systems and show how these Riemann problems can be reformulated/solved as free boundary problems with transonic shock waves as free boundaries for the corresponding nonlinear conservation laws of mixed elliptic-hyperbolic type and related nonlinear partial differential equations.
基金supported by the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20070126002)the National Natural Science Foundation of China (No. 10962004)
文摘This paper studies the eigenfunction expansion method to solve the two dimensional （2D） elasticity problems based on the stress formulation. The fundamental system of partial differential equations of the 2D problems is rewritten as an upper tri angular differential system based on the known results, and then the associated upper triangular operator matrix matrix is obtained. By further research, the two simpler com plete orthogonal systems of eigenfunctions in some space are obtained, which belong to the two block operators arising in the operator matrix. Then, a more simple and conve nient general solution to the 2D problem is given by the eigenfunction expansion method. Furthermore, the boundary conditions for the 2D problem, which can be solved by this method, are indicated. Finally, the validity of the obtained results is verified by a specific example.
基金Project supported by the National Natural Science Foundation of China(Grant No.41175025)
文摘The simplified linear model of Grad-Shafranov （GS） reconstruction can be reformulated into an inverse boundary value problem of Laplace＇s equation. Therefore, in this paper we focus on the method of solving the inverse boundary value problem of Laplace＇s equation. In the first place, the variational regularization method is used to deal with the ill- posedness of the Cauchy problem for Laplace＇s equation. Then, the ＇L-Curve＇ principle is suggested to be adopted in choosing the optimal regularization parameter. Finally, a numerical experiment is implemented with a section of Neumann and Dirichlet boundary conditions with observation errors. The results well converge to the exact solution of the problem, which proves the efficiency and robustness of the proposed method. When the order of observation error δ is 10-1, the order of the approximate result error can reach 10-3.
基金supported by the National Natural Science Foundation of China(1117113611261032)+2 种基金the Distinguished Young Scholars Fund of Lan Zhou University of Technology(Q201015)the basic scientific research business expenses of Gansu province collegethe Natural Science Foundation of Gansu province(1310RJYA021)
文摘In this paper, we consider the Cauchy problem for the Laplace equation, which is severely ill-posed in the sense that the solution does not depend continuously on the data. A modified Tikhonov regularization method is proposed to solve this problem. An error estimate for the a priori parameter choice between the exact solution and its regularized approximation is obtained. Moreover, an a posteriori parameter choice rule is proposed and a stable error estimate is also obtained. Numerical examples illustrate the validity and effectiveness of this method.
基金the National Natural Science Foundation of China(10471052,10631030)the PHD specialized grant of Ministry of Education of China(20060511001)
文摘This article is contributed to the Cauchy problem {δu/δt=△u＋K（｜x｜）u^p in R^n×（0,T）, u（x,0）=φ（x） in R^n;with initial function φ≡/0. The stability of positive radial steady state, which are positive solutions of △u ＋ K（｜x｜）u^p =0, is obtained when p is critical for general K（｜x｜）.
基金supported by NSFC(11201380)the Fundamental Research Funds for the Central Universities(XDJK2012B007)+1 种基金Doctor Fund of Southwest University(SWU111021)Educational Fund of Southwest University(2010JY053)
文摘We consider the growth rate and quenching rate of the following problem with singular nonlinearityfor some positive constants b：, b2 （see Theorem 3.3 for the parametersfor some constantsHence, the solution （u, v） quenches at the originx = 0 at the same time ＇1＇ （see Theorem 4.3）. We also tind various other conditions tor the solution to quench in a finite time and obtain the corresponding decay rate of the solution near the quenching time.
基金supported by the National Natural Science Foundation of China(11226175,11271336 and 11171311)Specialized Reseach Fund for the Docotoral Program of Higher Education(20124301120002)Foundation of He’nan Educational Committee(2009C110006)
文摘In this paper, the existence and the uniqueness of the local generalized solution and the local classical solution of the Cauchy problem for the generalized BBM-Burgers equationare proved. The existence and the uniqueness of the global generalized solution and the global classical solution for the Cauchy problem of equation （1） are proved when n = 3, 2, 1. Moreover, the decay property of the solution is discussed.
基金Project supported by the NSF of Fujian Province (A97020)
文摘This article considers Cauchy problem u(t) - (uv)(x) = 0, v(t) - u(x) = 0, u(x, 0) = u(0) (x) > 0, v(x, 0) = v(0)(x). A necessary and sufficient condition in guaranteeing that Cauchy problem admits a global C-1-solution on t greater than or equal to 0 is obtained.
基金Natural Science Foundation of Henan Province!(Grant No.98405070) National Natural Science Foundation of China (Grant No.19
文摘In the present paper,the local existence of classical solutions to the periodic boundary problem and the Cauchy problem of a quasilinear evolution equation are studied under the assumptions that do not require the monotonicity of σi(s) (i= 1,…, n). The nonexistence of global solutions to the initial-boundary value problem of the equation is also discussed, a blowup theorem is proved and a concrete example is given.
基金supported by the NSF of China(10571079,10671085)and the program of NCET
文摘This paper is concerned with the Cauchy problem for the modified Helmholtz equation in an infinite strip domain 0<x≤1,y∈R.The Cauchy data at x = 0 is given and the solution is then sought for the interval 0<x≤1.This problem is highly ill-posed and the solution(if it exists) does not depend continuously on the given data. In this paper,we propose a fourth-order modified method to solve the Cauchy problem. Convergence estimates are presented under the suitable choices of regularization parameters and the a priori assumption on the bounds of the exact solution.Numerical implementation is considered and the numerical examples show that our proposed method is effective and stable.
文摘A Cauchy problem for the semi-linear elliptic equation is investigated. We use a filtering function method to define a regularization solution for this ill-posed problem. The existence, uniqueness and stability of the regularization solution are proven;a convergence estimate of H?lder type for the regularization method is obtained under the a-priori bound assumption for the exact solution. An iterative scheme is proposed to calculate the regularization solution;some numerical results show that this method works well.
基金Supported by the National Natural Science Foundation of China(Grant No.11471253 and No.11571311)
文摘In this paper,we consider a Cauchy problem of the time fractional diffusion equation(TFDE)in x∈[0,L].This problem is ubiquitous in science and engineering applications.The illposedness of the Cauchy problem is explained by its solution in frequency domain.Furthermore,the problem is formulated into a minimization problem with a modified Tikhonov regularization method.The gradient of the regularization functional based on an adjoint problem is deduced and the standard conjugate gradient method is presented for solving the minimization problem.The error estimates for the regularized solutions are obtained under Hp norm priori bound assumptions.Finally,numerical examples illustrate the effectiveness of the proposed method.
