For analysis of displacement and stress, an elastic sloping pile embedded in a homogeneous isotropic elastic half space under arbitrary loads at the top can be decomposed into two plane systems, i.e., the inclined pla...For analysis of displacement and stress, an elastic sloping pile embedded in a homogeneous isotropic elastic half space under arbitrary loads at the top can be decomposed into two plane systems, i.e., the inclined plane xOz and its normal plane yOz . Let Mindlin's forces be the fundamental loads with unknown intensity function X(t),Y(t),Z(t) ,parallel to x,y,z_axis respectively, be distributed along the t axis of the pile in and concentrated forces Q x,Q y,Z ,couples M y,M x at top of the pile. Then, according to the boundary conditions of elastic pile, the problem is reduced to a set of Fredholm_Volterra type equations. Numerical solution is given and the accuracy of calculation can be checked by the reciprocal theorem of work.展开更多
The meshless method is a new numerical technique presented in recent years.It uses the moving least square(MLS)approximation as a shape function.The smoothness of the MLS approximation is determined by that of the bas...The meshless method is a new numerical technique presented in recent years.It uses the moving least square(MLS)approximation as a shape function.The smoothness of the MLS approximation is determined by that of the basic function and of the weight function,and is mainly determined by that of the weight function.Therefore,the weight function greatly affects the accuracy of results obtained.Different kinds of weight functions,such as the spline function, the Gauss function and so on,are proposed recently by many researchers.In the present work,the features of various weight functions are illustrated through solving elasto-static problems using the local boundary integral equation method.The effect of various weight functions on the accuracy, convergence and stability of results obtained is also discussed.Examples show that the weight function proposed by Zhou Weiyuan and Gauss and the quartic spline weight function are better than the others if parameters c and α in Gauss and exponential weight functions are in the range of reasonable values,respectively,and the higher the smoothness of the weight function,the better the features of the solutions.展开更多
This paper determines the exact error order on optimization of adaptive direct methods of approximate solution of the class of Fredholm integral equations of the second kind with kernel belonging to the anisotropic So...This paper determines the exact error order on optimization of adaptive direct methods of approximate solution of the class of Fredholm integral equations of the second kind with kernel belonging to the anisotropic Sobolev classes, and also gives an optimal algorithm.展开更多
In this paper, we suggest a method for solving Fredholm integral equation of the first kind based on wavelet basis. The continuous Legendre and Chebyshev wavelets of the first, second, third and fourth kind on [0,1] a...In this paper, we suggest a method for solving Fredholm integral equation of the first kind based on wavelet basis. The continuous Legendre and Chebyshev wavelets of the first, second, third and fourth kind on [0,1] are used and are utilized as a basis in Galerkin method to approximate the solution of integral equations. Then, in some examples the mentioned wavelets are compared with each other.展开更多
In this article, a general formula of the first integral method has been extended to celebrate the exact solution of nonlinear time-space differential equations of fractional orders. The proposed method is easy, direc...In this article, a general formula of the first integral method has been extended to celebrate the exact solution of nonlinear time-space differential equations of fractional orders. The proposed method is easy, direct and concise as compared with other existent methods.展开更多
The meshless local boundary integral equation method is a currently developed numerical method, which combines the advantageous features of Galerkin finite element method(GFEM), boundary element method(BEM) and elemen...The meshless local boundary integral equation method is a currently developed numerical method, which combines the advantageous features of Galerkin finite element method(GFEM), boundary element method(BEM) and element free Galerkin method(EFGM), and is a truly meshless method possessing wide prospects in engineering applications. The companion solution and all the other formulas required in the meshless local boundary integral equation for a thin plate were presented, in order to make this method apply to solve the thin plate problem.展开更多
Using the Picard iteration method and treating the involved integration by numerical quadrature formulas, we propose a numerical scheme for the second kind nonlinear Volterra integral equations. For enlarging the conv...Using the Picard iteration method and treating the involved integration by numerical quadrature formulas, we propose a numerical scheme for the second kind nonlinear Volterra integral equations. For enlarging the convergence region of the Picard iteration method, multistage algorithm is devised. We also introduce an algorithm for problems with some singularities at the limits of integration including fractional integral equations. Numerical tests verify the validity of the proposed schemes.展开更多
In this paper, we present a new method, a mixture of homotopy perturbation method and a new integral transform to solve some nonlinear partial differential equations. The proposed method introduces also He’s polynomi...In this paper, we present a new method, a mixture of homotopy perturbation method and a new integral transform to solve some nonlinear partial differential equations. The proposed method introduces also He’s polynomials [1]. The analytical results of examples are calculated in terms of convergent series with easily computed components [2].展开更多
In recent years, many methods have been used to find the exact solutions of nonlinear partial differential equations. One of them is called the first integral method, which is based on the ring theory of commutative a...In recent years, many methods have been used to find the exact solutions of nonlinear partial differential equations. One of them is called the first integral method, which is based on the ring theory of commutative algebra. In this paper, exact travelling wave solutions of the Non-Boussinesq wavepacket model and the (2 + 1)-dimensional Zoomeron equation are studied by using the first integral method. From the solving process and results, the first integral method has the characteristics of simplicity, directness and effectiveness about solving the exact travelling wave solutions of nonlinear partial differential equations. In other words, tedious calculations can be avoided by Maple software;the solutions of more accurate and richer travelling wave solutions are obtained. Therefore, this method is an effective method for solving exact solutions of nonlinear partial differential equations.展开更多
In this paper, Adomian decomposition method (ADM) is used to solve the Volterra-Fredholm integral equation. A number of examples have been presented to explain the numerical results, which is the comparison between th...In this paper, Adomian decomposition method (ADM) is used to solve the Volterra-Fredholm integral equation. A number of examples have been presented to explain the numerical results, which is the comparison between the exact solution and the numerical solution, and it is found through the tables and the amount of error between the exact solution and the numerical solution, it is very small and almost non-existent and is also illustrated through the graph how the exact solution of completely applies to the numerical solution This proves the accuracy of the method, which is the Adomian decomposition method (ADM) for solving the Volterra Fredholm integral equation using Maple 18. And that this method is characterized by ease, speed and great accuracy in obtaining numerical results.展开更多
The generalized Zakharov equation is a coupled equation which is a classic nonlinear mathematic model in plasma. A series of new exact explicit solutions of the system are obtained, by means of the first integral meth...The generalized Zakharov equation is a coupled equation which is a classic nonlinear mathematic model in plasma. A series of new exact explicit solutions of the system are obtained, by means of the first integral method, in the form of trigonometric and exponential functions. The results show the first integral method is an efficient way to solve the coupled nonlinear equations and get rich explicit analytical solutions.展开更多
Pocklington’s integral equation is presented for analysis of current distributions on wire antenna above ground. Sommerfeld type integrals, the kernel functions of the integral equation, can be approximately expresse...Pocklington’s integral equation is presented for analysis of current distributions on wire antenna above ground. Sommerfeld type integrals, the kernel functions of the integral equation, can be approximately expressed as the elementary functions using the Fresnel plane wave reflection coefficients method; and the Pocklington’s integral equation will be rearranged into a linear equation with solution easily obtained by using the method of moments, when the sinusoidal sub domain expansion is chosen to express the current distributions.展开更多
The aim of this paper is to investigate the numerical solution of the hypersingular integral equation reduced by the harmonic equation. First, we transform the hypersingular integral equation into 2π-periodic hypersi...The aim of this paper is to investigate the numerical solution of the hypersingular integral equation reduced by the harmonic equation. First, we transform the hypersingular integral equation into 2π-periodic hypersingular integral equation with the map x=cot（θ/2）. Second, we initiate the study of the multiscale Galerkin method for the 2π-periodic hypersingular integral equation. The trigonometric wavelets are used as trial functions. Consequently, the 2j＋1 × 2j＋1 stiffness matrix Kj can be partitioned j×j block matrices. Furthermore, these block matrices are zeros except main diagonal block matrices. These main diagonal block matrices are symmetrical and circulant matrices, and hence the solution of the associated linear algebraic system can be solved with the fast Fourier transform and the inverse fast Fourier transform instead of the inverse matrix. Finally, we provide several numerical examples to demonstrate our method has good accuracy even though the exact solutions are multi-peak and almost singular.展开更多
The boundary-volume integral equation numerical technique can be a powerful tool for piecewise heterogeneous media, but it is limited to small problems or low frequencies because of great computational cost. Therefore...The boundary-volume integral equation numerical technique can be a powerful tool for piecewise heterogeneous media, but it is limited to small problems or low frequencies because of great computational cost. Therefore, a restarted GMRES method is applied to solve large-scale boundary-volume scattering problems in this paper to overcome the computational barrier. The iterative method is firstly applied to responses of dimensionless frequency to a semicircular alluvial valley filled with sediments, compared with the standard Gaussian elimination method. Then the method is tested by a heterogeneous multilayered model to show its applicability. Numerical experiments indicate that the preconditioned GMRES method can significantly improve computational efficiency especially for large Earth models and high frequencies, but with a faster convergence for the left diagonal preconditioning.展开更多
When the source nodes are on the global boundary in the implementation of local boundary integral equation method （LBIEM）, singularities in the local boundary integrals need to be treated specially. In the current p...When the source nodes are on the global boundary in the implementation of local boundary integral equation method （LBIEM）, singularities in the local boundary integrals need to be treated specially. In the current paper, local integral equations are adopted for the nodes inside the domain and moving least square approximation （MLSA） for the nodes on the global boundary, thus singularities will not occur in the new al- gorithm. At the same time, approximation errors of boundary integrals are reduced significantly. As applications and numerical tests, Laplace equation and Helmholtz equation problems are considered and excellent numerical results are obtained. Furthermore, when solving the Helmholtz problems, the modified basis functions with wave solutions are adapted to replace the usually-used monomial basis functions. Numerical results show that this treatment is simple and effective and its application is promising in solutions for the wave propagation problem with high wave number.展开更多
From the dislocation type solution of the torsion of single crack, by using the concept of finite part integrals. The torsion problem of cylinder with a single crack was reduced into an integral equation with strong s...From the dislocation type solution of the torsion of single crack, by using the concept of finite part integrals. The torsion problem of cylinder with a single crack was reduced into an integral equation with strong singularity. The numerical method was also obtained and several numerical examples were calculated successfully at the end of this paper.,展开更多
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
From the mode-marching method,by doing some transforms and derivations,the scatteringand propagation integral equations for solving the problems of discontinuous dielectric structure are ob-tained.The necessary and su...From the mode-marching method,by doing some transforms and derivations,the scatteringand propagation integral equations for solving the problems of discontinuous dielectric structure are ob-tained.The necessary and sufficient condition for the existence of solution of propagating equations is giv-en.It is really the dispersion equation of the dielectric structure.As an example,a succinct solution ofone-step discontinuous dielectric structure is derived.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 the view of Reissner's and Kirchhoff's theories, respectively, we formulate the isotropicalized governing equations for the anisotropic plates, and give the proof of the equivalence relation between these t...In the view of Reissner's and Kirchhoff's theories, respectively, we formulate the isotropicalized governing equations for the anisotropic plates, and give the proof of the equivalence relation between these two plate-models for the simply-supported rectangular orthotropic plates. The well-known fundamental solutions of the isotrqpic plates are utlized for the spline integral equation analysis of anisotropic plates.Even with sparse meshes the satisfactory results can be obtained. The analysis of plates on two-parameter elastic foundation is so simple as the common case that only a few terms should be added to the formulas of fictitious loads.展开更多
文摘For analysis of displacement and stress, an elastic sloping pile embedded in a homogeneous isotropic elastic half space under arbitrary loads at the top can be decomposed into two plane systems, i.e., the inclined plane xOz and its normal plane yOz . Let Mindlin's forces be the fundamental loads with unknown intensity function X(t),Y(t),Z(t) ,parallel to x,y,z_axis respectively, be distributed along the t axis of the pile in and concentrated forces Q x,Q y,Z ,couples M y,M x at top of the pile. Then, according to the boundary conditions of elastic pile, the problem is reduced to a set of Fredholm_Volterra type equations. Numerical solution is given and the accuracy of calculation can be checked by the reciprocal theorem of work.
文摘The meshless method is a new numerical technique presented in recent years.It uses the moving least square(MLS)approximation as a shape function.The smoothness of the MLS approximation is determined by that of the basic function and of the weight function,and is mainly determined by that of the weight function.Therefore,the weight function greatly affects the accuracy of results obtained.Different kinds of weight functions,such as the spline function, the Gauss function and so on,are proposed recently by many researchers.In the present work,the features of various weight functions are illustrated through solving elasto-static problems using the local boundary integral equation method.The effect of various weight functions on the accuracy, convergence and stability of results obtained is also discussed.Examples show that the weight function proposed by Zhou Weiyuan and Gauss and the quartic spline weight function are better than the others if parameters c and α in Gauss and exponential weight functions are in the range of reasonable values,respectively,and the higher the smoothness of the weight function,the better the features of the solutions.
基金Project supported by the Natural Science Foundation of China(10371009)Research Fund for the Doctoral Program Higher Education
文摘This paper determines the exact error order on optimization of adaptive direct methods of approximate solution of the class of Fredholm integral equations of the second kind with kernel belonging to the anisotropic Sobolev classes, and also gives an optimal algorithm.
文摘In this paper, we suggest a method for solving Fredholm integral equation of the first kind based on wavelet basis. The continuous Legendre and Chebyshev wavelets of the first, second, third and fourth kind on [0,1] are used and are utilized as a basis in Galerkin method to approximate the solution of integral equations. Then, in some examples the mentioned wavelets are compared with each other.
文摘In this article, a general formula of the first integral method has been extended to celebrate the exact solution of nonlinear time-space differential equations of fractional orders. The proposed method is easy, direct and concise as compared with other existent methods.
文摘The meshless local boundary integral equation method is a currently developed numerical method, which combines the advantageous features of Galerkin finite element method(GFEM), boundary element method(BEM) and element free Galerkin method(EFGM), and is a truly meshless method possessing wide prospects in engineering applications. The companion solution and all the other formulas required in the meshless local boundary integral equation for a thin plate were presented, in order to make this method apply to solve the thin plate problem.
文摘Using the Picard iteration method and treating the involved integration by numerical quadrature formulas, we propose a numerical scheme for the second kind nonlinear Volterra integral equations. For enlarging the convergence region of the Picard iteration method, multistage algorithm is devised. We also introduce an algorithm for problems with some singularities at the limits of integration including fractional integral equations. Numerical tests verify the validity of the proposed schemes.
文摘In this paper, we present a new method, a mixture of homotopy perturbation method and a new integral transform to solve some nonlinear partial differential equations. The proposed method introduces also He’s polynomials [1]. The analytical results of examples are calculated in terms of convergent series with easily computed components [2].
文摘In recent years, many methods have been used to find the exact solutions of nonlinear partial differential equations. One of them is called the first integral method, which is based on the ring theory of commutative algebra. In this paper, exact travelling wave solutions of the Non-Boussinesq wavepacket model and the (2 + 1)-dimensional Zoomeron equation are studied by using the first integral method. From the solving process and results, the first integral method has the characteristics of simplicity, directness and effectiveness about solving the exact travelling wave solutions of nonlinear partial differential equations. In other words, tedious calculations can be avoided by Maple software;the solutions of more accurate and richer travelling wave solutions are obtained. Therefore, this method is an effective method for solving exact solutions of nonlinear partial differential equations.
文摘In this paper, Adomian decomposition method (ADM) is used to solve the Volterra-Fredholm integral equation. A number of examples have been presented to explain the numerical results, which is the comparison between the exact solution and the numerical solution, and it is found through the tables and the amount of error between the exact solution and the numerical solution, it is very small and almost non-existent and is also illustrated through the graph how the exact solution of completely applies to the numerical solution This proves the accuracy of the method, which is the Adomian decomposition method (ADM) for solving the Volterra Fredholm integral equation using Maple 18. And that this method is characterized by ease, speed and great accuracy in obtaining numerical results.
文摘The generalized Zakharov equation is a coupled equation which is a classic nonlinear mathematic model in plasma. A series of new exact explicit solutions of the system are obtained, by means of the first integral method, in the form of trigonometric and exponential functions. The results show the first integral method is an efficient way to solve the coupled nonlinear equations and get rich explicit analytical solutions.
文摘Pocklington’s integral equation is presented for analysis of current distributions on wire antenna above ground. Sommerfeld type integrals, the kernel functions of the integral equation, can be approximately expressed as the elementary functions using the Fresnel plane wave reflection coefficients method; and the Pocklington’s integral equation will be rearranged into a linear equation with solution easily obtained by using the method of moments, when the sinusoidal sub domain expansion is chosen to express the current distributions.
文摘The aim of this paper is to investigate the numerical solution of the hypersingular integral equation reduced by the harmonic equation. First, we transform the hypersingular integral equation into 2π-periodic hypersingular integral equation with the map x=cot（θ/2）. Second, we initiate the study of the multiscale Galerkin method for the 2π-periodic hypersingular integral equation. The trigonometric wavelets are used as trial functions. Consequently, the 2j＋1 × 2j＋1 stiffness matrix Kj can be partitioned j×j block matrices. Furthermore, these block matrices are zeros except main diagonal block matrices. These main diagonal block matrices are symmetrical and circulant matrices, and hence the solution of the associated linear algebraic system can be solved with the fast Fourier transform and the inverse fast Fourier transform instead of the inverse matrix. Finally, we provide several numerical examples to demonstrate our method has good accuracy even though the exact solutions are multi-peak and almost singular.
基金supported by the National Natural Science Foundation of China(Nos. 41130418 and 40925013)the National Basic Research Program(973 Program)(No.2009CB219403)
文摘The boundary-volume integral equation numerical technique can be a powerful tool for piecewise heterogeneous media, but it is limited to small problems or low frequencies because of great computational cost. Therefore, a restarted GMRES method is applied to solve large-scale boundary-volume scattering problems in this paper to overcome the computational barrier. The iterative method is firstly applied to responses of dimensionless frequency to a semicircular alluvial valley filled with sediments, compared with the standard Gaussian elimination method. Then the method is tested by a heterogeneous multilayered model to show its applicability. Numerical experiments indicate that the preconditioned GMRES method can significantly improve computational efficiency especially for large Earth models and high frequencies, but with a faster convergence for the left diagonal preconditioning.
文摘When the source nodes are on the global boundary in the implementation of local boundary integral equation method （LBIEM）, singularities in the local boundary integrals need to be treated specially. In the current paper, local integral equations are adopted for the nodes inside the domain and moving least square approximation （MLSA） for the nodes on the global boundary, thus singularities will not occur in the new al- gorithm. At the same time, approximation errors of boundary integrals are reduced significantly. As applications and numerical tests, Laplace equation and Helmholtz equation problems are considered and excellent numerical results are obtained. Furthermore, when solving the Helmholtz problems, the modified basis functions with wave solutions are adapted to replace the usually-used monomial basis functions. Numerical results show that this treatment is simple and effective and its application is promising in solutions for the wave propagation problem with high wave number.
基金Project supported by P.H.D.Foundation of the State Education Commission of China
文摘From the dislocation type solution of the torsion of single crack, by using the concept of finite part integrals. The torsion problem of cylinder with a single crack was reduced into an integral equation with strong singularity. The numerical method was also obtained and several numerical examples were calculated successfully at the end of this paper.,
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.
文摘From the mode-marching method,by doing some transforms and derivations,the scatteringand propagation integral equations for solving the problems of discontinuous dielectric structure are ob-tained.The necessary and sufficient condition for the existence of solution of propagating equations is giv-en.It is really the dispersion equation of the dielectric structure.As an example,a succinct solution ofone-step discontinuous dielectric structure is derived.
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 the view of Reissner's and Kirchhoff's theories, respectively, we formulate the isotropicalized governing equations for the anisotropic plates, and give the proof of the equivalence relation between these two plate-models for the simply-supported rectangular orthotropic plates. The well-known fundamental solutions of the isotrqpic plates are utlized for the spline integral equation analysis of anisotropic plates.Even with sparse meshes the satisfactory results can be obtained. The analysis of plates on two-parameter elastic foundation is so simple as the common case that only a few terms should be added to the formulas of fictitious loads.
