Because exact analytic solution is not available, we use double expansion and boundary collocation to construct an approximate solution for a class of two-dimensional dual integral equations in mathematical physics. T...Because exact analytic solution is not available, we use double expansion and boundary collocation to construct an approximate solution for a class of two-dimensional dual integral equations in mathematical physics. The integral equations by this procedure are reduced to infinite algebraic equations. The accuracy of the solution lies in the boundary collocation technique. The application of which for some complicated initialboundary value problems in solid mechanics indicates the method is powerful.展开更多
In this paper,the approximate solutions for two different type of two-dimensional nonlinear integral equations:two-dimensional nonlinear Volterra-Fredholm integral equations and the nonlinear mixed Volterra-Fredholm i...In this paper,the approximate solutions for two different type of two-dimensional nonlinear integral equations:two-dimensional nonlinear Volterra-Fredholm integral equations and the nonlinear mixed Volterra-Fredholm integral equations are obtained using the Laguerre wavelet method.To do this,these two-dimensional nonlinear integral equations are transformed into a system of nonlinear algebraic equations in matrix form.By solving these systems,unknown coefficients are obtained.Also,some theorems are proved for convergence analysis.Some numerical examples are presented and results are compared with the analytical solution to demonstrate the validity and applicability of the proposed method.展开更多
A nonlinear problem of mean-square approximation of a real nonnegative continuous function with respect to two variables by the modulus of double Fourier integral dependent on two real parameters with use of the smoot...A nonlinear problem of mean-square approximation of a real nonnegative continuous function with respect to two variables by the modulus of double Fourier integral dependent on two real parameters with use of the smoothing functional is studied. Finding the optimal solutions of this problem is reduced to solution of the Hammerstein type two-dimensional nonlinear integral equation. The numerical algorithms to find the branching lines and branching-off solutions of this equation are constructed and justified. Numerical examples are presented.展开更多
In the investigation on fracture mechanics,the potential function was introduced, and the moving differential equation was constructed. By making Laplace and Fourier transformation as well as sine and cosine transform...In the investigation on fracture mechanics,the potential function was introduced, and the moving differential equation was constructed. By making Laplace and Fourier transformation as well as sine and cosine transformation to moving differential equations and various responses, the dual equation which is constructed from boundary conditions lastly was solved. This method of investigating dynamic crack has become a more systematic one that is used widely. Some problems are encountered when the dynamic crack is studied. After the large investigation on the problems, it is discovered that during the process of mathematic derivation, the method is short of precision, and the derived results in this method are accidental and have no credibility.A model for example is taken to explain the problems existing in initial deriving process of the integral_transformation method of dynamic crack.展开更多
A new type of dual boundary integral equations(DBIE)is presented first,through which,a smaller system of equations needs to be solved in fracture analysis.Then a non-conforming crack tip element in two-dimensional pro...A new type of dual boundary integral equations(DBIE)is presented first,through which,a smaller system of equations needs to be solved in fracture analysis.Then a non-conforming crack tip element in two-dimensional problems is proposed.The exact formula for the hypersingular integral over the non-con- forming crack tip element is given next.By virtue of Green's-function-library strategy,a series of stress in- tensity factors(SIF)of different crack orientations,locations and/or sizes in a complicated structure can be obtained easily and efficiently.Finally,several examples of fracture analysis in two dimensions are given to demonstrate the accuracy and efficiency of the method proposed.展开更多
Crack problems are often reduced to dual integral equations,which can be solved by expanding the displacement integral equation as a series in the form of Chebyshev-like or Jacobi polynomials.Schmidt’s multiplying-fa...Crack problems are often reduced to dual integral equations,which can be solved by expanding the displacement integral equation as a series in the form of Chebyshev-like or Jacobi polynomials.Schmidt’s multiplying-factor integration method has been one of the most favorable techniques for determining the expansion coefficients by constructing a well-posed system of linear algebraic equations.However,Schmidt’s method is less efficient for numerical computation because the matrix elements of the linear equations are evaluated from dual integrals.In this study,we propose a modified method to construct linear equations to efficiently determine the expansion coefficients.The modified technique is developed upon the application of certain multiplying factors to the traction integral equation and then integrating the resulting equation over“source”regions.Such manipulations simplify the matrix elements as single integrals.By carrying out numerical examples,we demonstrate that the technique is not only accurate but also very efficient.In particular,the method only needs approximately 1/5 of the computation time of Schmidt’s method.Therefore,this method can be used to replace Schmidt’s method and is expected to be very useful in solving crack problems.展开更多
基金Project supported by the National Natural Science Foundation of China(No.K19672007)
文摘Because exact analytic solution is not available, we use double expansion and boundary collocation to construct an approximate solution for a class of two-dimensional dual integral equations in mathematical physics. The integral equations by this procedure are reduced to infinite algebraic equations. The accuracy of the solution lies in the boundary collocation technique. The application of which for some complicated initialboundary value problems in solid mechanics indicates the method is powerful.
文摘In this paper,the approximate solutions for two different type of two-dimensional nonlinear integral equations:two-dimensional nonlinear Volterra-Fredholm integral equations and the nonlinear mixed Volterra-Fredholm integral equations are obtained using the Laguerre wavelet method.To do this,these two-dimensional nonlinear integral equations are transformed into a system of nonlinear algebraic equations in matrix form.By solving these systems,unknown coefficients are obtained.Also,some theorems are proved for convergence analysis.Some numerical examples are presented and results are compared with the analytical solution to demonstrate the validity and applicability of the proposed method.
文摘A nonlinear problem of mean-square approximation of a real nonnegative continuous function with respect to two variables by the modulus of double Fourier integral dependent on two real parameters with use of the smoothing functional is studied. Finding the optimal solutions of this problem is reduced to solution of the Hammerstein type two-dimensional nonlinear integral equation. The numerical algorithms to find the branching lines and branching-off solutions of this equation are constructed and justified. Numerical examples are presented.
文摘In the investigation on fracture mechanics,the potential function was introduced, and the moving differential equation was constructed. By making Laplace and Fourier transformation as well as sine and cosine transformation to moving differential equations and various responses, the dual equation which is constructed from boundary conditions lastly was solved. This method of investigating dynamic crack has become a more systematic one that is used widely. Some problems are encountered when the dynamic crack is studied. After the large investigation on the problems, it is discovered that during the process of mathematic derivation, the method is short of precision, and the derived results in this method are accidental and have no credibility.A model for example is taken to explain the problems existing in initial deriving process of the integral_transformation method of dynamic crack.
基金the Aeronautical Science Foundation of China (No.99C53026).
文摘A new type of dual boundary integral equations(DBIE)is presented first,through which,a smaller system of equations needs to be solved in fracture analysis.Then a non-conforming crack tip element in two-dimensional problems is proposed.The exact formula for the hypersingular integral over the non-con- forming crack tip element is given next.By virtue of Green's-function-library strategy,a series of stress in- tensity factors(SIF)of different crack orientations,locations and/or sizes in a complicated structure can be obtained easily and efficiently.Finally,several examples of fracture analysis in two dimensions are given to demonstrate the accuracy and efficiency of the method proposed.
基金This work was supported by the National Natural Science Foundation of China(Grant Nos.11802074,42074057,11972132,and 11734017)China National Postdoctoral Program for Innovative Talents(Grant No.BX201700066)+1 种基金China Postdoctoral Science Foundation(Grant No.2018M630345)the Fundamental Research Funds for the Central Universities(Grant No.HIT.NSRIF.2020016)。
文摘Crack problems are often reduced to dual integral equations,which can be solved by expanding the displacement integral equation as a series in the form of Chebyshev-like or Jacobi polynomials.Schmidt’s multiplying-factor integration method has been one of the most favorable techniques for determining the expansion coefficients by constructing a well-posed system of linear algebraic equations.However,Schmidt’s method is less efficient for numerical computation because the matrix elements of the linear equations are evaluated from dual integrals.In this study,we propose a modified method to construct linear equations to efficiently determine the expansion coefficients.The modified technique is developed upon the application of certain multiplying factors to the traction integral equation and then integrating the resulting equation over“source”regions.Such manipulations simplify the matrix elements as single integrals.By carrying out numerical examples,we demonstrate that the technique is not only accurate but also very efficient.In particular,the method only needs approximately 1/5 of the computation time of Schmidt’s method.Therefore,this method can be used to replace Schmidt’s method and is expected to be very useful in solving crack problems.
