In order to solve the problem of the variable coefficient ordinary differen-tial equation on the bounded domain,the Lagrange interpolation method is used to approximate the exact solution of the equation,and the error...In order to solve the problem of the variable coefficient ordinary differen-tial equation on the bounded domain,the Lagrange interpolation method is used to approximate the exact solution of the equation,and the error between the numerical solution and the exact solution is obtained,and then compared with the error formed by the difference method,it is concluded that the Lagrange interpolation method is more effective in solving the variable coefficient ordinary differential equation.展开更多
Based on a method of finite element model and combined with matrix theory, a method for solving differential equation with variable coefficients is proposed. With the method, it is easy to deal with the differential e...Based on a method of finite element model and combined with matrix theory, a method for solving differential equation with variable coefficients is proposed. With the method, it is easy to deal with the differential equations with variable coefficients. On most occasions and due to the nonuniformity nature, nonlinearity property can cause the equations of the kinds. Using the model, the satisfactory valuable results with only a few units can be obtained.展开更多
After reading the article "The Boundedness and Asymptotic Behavior of Solution of Differential System of Second-Order with Variable Coefficient" in "Applied Mathematics and Mechanics", Vol. 3, No. ...After reading the article "The Boundedness and Asymptotic Behavior of Solution of Differential System of Second-Order with Variable Coefficient" in "Applied Mathematics and Mechanics", Vol. 3, No. 4, 1982, we would like to put forward a few points to discuss with the author and the readers. Our opinions are presented as follows:展开更多
Many engineering problems can be reduced to the solution of a variable coefficient differential equation. In this paper, the exact analytic method is suggested to solve variable coefficient differential equations unde...Many engineering problems can be reduced to the solution of a variable coefficient differential equation. In this paper, the exact analytic method is suggested to solve variable coefficient differential equations under arbitrary boundary condition. By this method, the general computation formal is obtained. Its convergence in proved. We can get analytic expressions which converge to exact solution and its higher order derivatives uniformy Four numerical examples are given, which indicate that satisfactory results can he obtanedby this method.展开更多
How to solve the partial differential equation has been attached importance to by all kinds of fields. The exact solution to a class of partial differential equation with variable-coefficient is obtained in reproducin...How to solve the partial differential equation has been attached importance to by all kinds of fields. The exact solution to a class of partial differential equation with variable-coefficient is obtained in reproducing kernel space. For getting the approximate solution, give an iterative method, convergence of the iterative method is proved. The numerical example shows that our method is effective and good practicability.展开更多
A boundary integral method with radial basis function approximation is proposed for numerically solving an important class of boundary value problems governed by a system of thermoelastostatic equations with variable ...A boundary integral method with radial basis function approximation is proposed for numerically solving an important class of boundary value problems governed by a system of thermoelastostatic equations with variable coe?cients. The equations describe the thermoelastic behaviors of nonhomogeneous anisotropic materials with properties that vary smoothly from point to point in space. No restriction is imposed on the spatial variations of the thermoelastic coe?cients as long as all the requirements of the laws of physics are satis?ed. To check the validity and accuracy of the proposed numerical method, some speci?c test problems with known solutions are solved.展开更多
The radial basis functions(RBFs)play an important role in the numerical simulation processes of partial differential equations.Since the radial basis functions are meshless algorithms,its approximation is easy to impl...The radial basis functions(RBFs)play an important role in the numerical simulation processes of partial differential equations.Since the radial basis functions are meshless algorithms,its approximation is easy to implement and mathematically simple.In this paper,the commonly⁃used multiquadric RBF,conical RBF,and Gaussian RBF were applied to solve boundary value problems which are governed by partial differential equations with variable coefficients.Numerical results were provided to show the good performance of the three RBFs as numerical tools for a wide range of problems.It is shown that the conical RBF numerical results were more stable than the other two radial basis functions.From the comparison of three commonly⁃used RBFs,one may obtain the best numerical solutions for boundary value problems.展开更多
Consider the neutral differential equations with variable coefficients and delays [x(t)-p(t)x(t-r(t))]'+ Qj(t)x(t-σj(t))=0. (1)We establish sufficient conditions for the oscillation of equation (1). Our condition...Consider the neutral differential equations with variable coefficients and delays [x(t)-p(t)x(t-r(t))]'+ Qj(t)x(t-σj(t))=0. (1)We establish sufficient conditions for the oscillation of equation (1). Our condition is 'sharp' in the sense that when all the coefficients and delays of the equation are constants.Our conclusions improve and generalize some known results.展开更多
The paper presents the theoretical analysis of a variable stiffness beam. The bending stiffness EI varies continuously along the length of the beam. Dynamic equation yields differential equation with variable co- effi...The paper presents the theoretical analysis of a variable stiffness beam. The bending stiffness EI varies continuously along the length of the beam. Dynamic equation yields differential equation with variable co- efficients based on the model of the Euler-Bernoulli beam. Then differential equation with variable coefficients becomes that with constant coefficients by variable substitution. At last, the study obtains the solution of dy- namic equation. The cantilever beam is an object for analysis. When the flexural rigidity at free end is a constant and that at clamped end is varied, the dynamic characteristics are analyzed under several cases. The results dem- onstrate that the natural angular frequency reduces as the fiexural rigidity reduces. When the rigidity of clamped end is higher than that of free end, low-level mode contributes the larger displacement response to the total re- sponse. On the contrary, the contribution of low-level mode is lesser than that of hi^h-level mode.展开更多
An iterative process of positive solution for BVP w'+h(t)f(w)=0, w(0)=w(1)= 0 is established, where h(t) is allowed to changes sign on [0,1]. The process starts from a simple function.
This work aims to present nonlinear models that arise in ocean engineering.There are many models of ocean waves that are present in nature.In shallow water,the linearization of the equations requires critical conditio...This work aims to present nonlinear models that arise in ocean engineering.There are many models of ocean waves that are present in nature.In shallow water,the linearization of the equations requires critical conditions on wave capacity than it make in deep water,and the strong nonlinear belongings are spotted.We use Lie symmetry analysis to obtain different types of soliton solutions like one,two,and three-soliton solutions in a(2+1)dimensional variable-coefficient Bogoyavlensky Konopelchenko(VCBK)equation that describes the interaction of a Riemann wave reproducing along the y-axis and a long wave reproducing along the x-axis in engineering and science.We use the Lie symmetry analysis then the integrating factor method to obtain new solutions of the VCBK equation.To demonstrate the physical meaning of the solutions obtained by the presented techniques,the graphical performance has been demonstrated with some values.The presented equation has fewer dimensions and is reduced to ordinary differential equations using the Lie symmetry technique.展开更多
This work proposes a generalized boundary integral method for variable coefficients elliptic partial differential equations(PDEs),including both boundary value and interface problems.The method is kernel-free in the s...This work proposes a generalized boundary integral method for variable coefficients elliptic partial differential equations(PDEs),including both boundary value and interface problems.The method is kernel-free in the sense that there is no need to know analytical expressions for kernels of the boundary and volume integrals in the solution of boundary integral equations.Evaluation of a boundary or volume integral is replaced with interpolation of a Cartesian grid based solution,which satisfies an equivalent discrete interface problem,while the interface problem is solved by a fast solver in the Cartesian grid.The computational work involved with the generalized boundary integral method is essentially linearly proportional to the number of grid nodes in the domain.This paper gives implementation details for a secondorder version of the kernel-free boundary integral method in two space dimensions and presents numerical experiments to demonstrate the efficiency and accuracy of the method for both boundary value and interface problems.The interface problems demonstrated include those with piecewise constant and large-ratio coefficients and the heterogeneous interface problem,where the elliptic PDEs on two sides of the interface are of different types.展开更多
In the event of an instantaneous valve closure, the pressure transmitted to a surge tank induces the mass fluctuations that can cause high amplitude of water-level fluctuation in the surge tank for a reasonable cross-...In the event of an instantaneous valve closure, the pressure transmitted to a surge tank induces the mass fluctuations that can cause high amplitude of water-level fluctuation in the surge tank for a reasonable cross-sectional area. The height of the surge tank is then designed using this high water level mark generated by the completely closed penstock valve. Using a conical surge tank with a non-constant cross-sectional area can resolve the problems of space and height. When addressing issues in designing open surge tanks, key parameters are usually calculated by using complex equations, which may become cumbersome when multiple iterations are required. A more effective alternative in obtaining these values is the use of simple charts. Firstly, this paper presents and describes the equations used to design open conical surge tanks. Secondly, it introduces user-friendly charts that can be used in the design of cylindrical and conical open surge tanks. The contribution can be a benefit for practicing engineers in this field. A case study is also presented to illustrate the use of these design charts. The case study’s results show that key parameters obtained via successive approximation method required 26 iterations or complex calculations, whereas these values can be obtained by simple reading of the proposed chart. The use of charts to help surge tanks designing, in the case of preliminary designs, can save time and increase design efficiency, while reducing calculation errors.展开更多
文摘In order to solve the problem of the variable coefficient ordinary differen-tial equation on the bounded domain,the Lagrange interpolation method is used to approximate the exact solution of the equation,and the error between the numerical solution and the exact solution is obtained,and then compared with the error formed by the difference method,it is concluded that the Lagrange interpolation method is more effective in solving the variable coefficient ordinary differential equation.
文摘Based on a method of finite element model and combined with matrix theory, a method for solving differential equation with variable coefficients is proposed. With the method, it is easy to deal with the differential equations with variable coefficients. On most occasions and due to the nonuniformity nature, nonlinearity property can cause the equations of the kinds. Using the model, the satisfactory valuable results with only a few units can be obtained.
文摘After reading the article "The Boundedness and Asymptotic Behavior of Solution of Differential System of Second-Order with Variable Coefficient" in "Applied Mathematics and Mechanics", Vol. 3, No. 4, 1982, we would like to put forward a few points to discuss with the author and the readers. Our opinions are presented as follows:
文摘Many engineering problems can be reduced to the solution of a variable coefficient differential equation. In this paper, the exact analytic method is suggested to solve variable coefficient differential equations under arbitrary boundary condition. By this method, the general computation formal is obtained. Its convergence in proved. We can get analytic expressions which converge to exact solution and its higher order derivatives uniformy Four numerical examples are given, which indicate that satisfactory results can he obtanedby this method.
基金Project supported by the National Natural Science Foundation of China(No.10461005)
文摘How to solve the partial differential equation has been attached importance to by all kinds of fields. The exact solution to a class of partial differential equation with variable-coefficient is obtained in reproducing kernel space. For getting the approximate solution, give an iterative method, convergence of the iterative method is proved. The numerical example shows that our method is effective and good practicability.
文摘A boundary integral method with radial basis function approximation is proposed for numerically solving an important class of boundary value problems governed by a system of thermoelastostatic equations with variable coe?cients. The equations describe the thermoelastic behaviors of nonhomogeneous anisotropic materials with properties that vary smoothly from point to point in space. No restriction is imposed on the spatial variations of the thermoelastic coe?cients as long as all the requirements of the laws of physics are satis?ed. To check the validity and accuracy of the proposed numerical method, some speci?c test problems with known solutions are solved.
基金the Natural Science Foundation of Anhui Province(Grant No.1908085QA09)the University Natural Science Research Project of Anhui Province(KJ2019A0591).
文摘The radial basis functions(RBFs)play an important role in the numerical simulation processes of partial differential equations.Since the radial basis functions are meshless algorithms,its approximation is easy to implement and mathematically simple.In this paper,the commonly⁃used multiquadric RBF,conical RBF,and Gaussian RBF were applied to solve boundary value problems which are governed by partial differential equations with variable coefficients.Numerical results were provided to show the good performance of the three RBFs as numerical tools for a wide range of problems.It is shown that the conical RBF numerical results were more stable than the other two radial basis functions.From the comparison of three commonly⁃used RBFs,one may obtain the best numerical solutions for boundary value problems.
文摘Consider the neutral differential equations with variable coefficients and delays [x(t)-p(t)x(t-r(t))]'+ Qj(t)x(t-σj(t))=0. (1)We establish sufficient conditions for the oscillation of equation (1). Our condition is 'sharp' in the sense that when all the coefficients and delays of the equation are constants.Our conclusions improve and generalize some known results.
基金National Natural Science Foundation of China(No.51178175)
文摘The paper presents the theoretical analysis of a variable stiffness beam. The bending stiffness EI varies continuously along the length of the beam. Dynamic equation yields differential equation with variable co- efficients based on the model of the Euler-Bernoulli beam. Then differential equation with variable coefficients becomes that with constant coefficients by variable substitution. At last, the study obtains the solution of dy- namic equation. The cantilever beam is an object for analysis. When the flexural rigidity at free end is a constant and that at clamped end is varied, the dynamic characteristics are analyzed under several cases. The results dem- onstrate that the natural angular frequency reduces as the fiexural rigidity reduces. When the rigidity of clamped end is higher than that of free end, low-level mode contributes the larger displacement response to the total re- sponse. On the contrary, the contribution of low-level mode is lesser than that of hi^h-level mode.
文摘An iterative process of positive solution for BVP w'+h(t)f(w)=0, w(0)=w(1)= 0 is established, where h(t) is allowed to changes sign on [0,1]. The process starts from a simple function.
文摘This work aims to present nonlinear models that arise in ocean engineering.There are many models of ocean waves that are present in nature.In shallow water,the linearization of the equations requires critical conditions on wave capacity than it make in deep water,and the strong nonlinear belongings are spotted.We use Lie symmetry analysis to obtain different types of soliton solutions like one,two,and three-soliton solutions in a(2+1)dimensional variable-coefficient Bogoyavlensky Konopelchenko(VCBK)equation that describes the interaction of a Riemann wave reproducing along the y-axis and a long wave reproducing along the x-axis in engineering and science.We use the Lie symmetry analysis then the integrating factor method to obtain new solutions of the VCBK equation.To demonstrate the physical meaning of the solutions obtained by the presented techniques,the graphical performance has been demonstrated with some values.The presented equation has fewer dimensions and is reduced to ordinary differential equations using the Lie symmetry technique.
基金supported in part by the National Science Foundation of the USA under Grant DMS-0915023is supported by the National Natural Science Foundation of China under Grants DMS-11101278 and DMS-91130012+2 种基金supported by the Young Thousand Talents Program of Chinasupported in part by National Science Committee of Taiwan under Grant 99-2115-M-007-002-MY2supported in part by National Center for Theoretical Sciences of Taiwan,too.
文摘This work proposes a generalized boundary integral method for variable coefficients elliptic partial differential equations(PDEs),including both boundary value and interface problems.The method is kernel-free in the sense that there is no need to know analytical expressions for kernels of the boundary and volume integrals in the solution of boundary integral equations.Evaluation of a boundary or volume integral is replaced with interpolation of a Cartesian grid based solution,which satisfies an equivalent discrete interface problem,while the interface problem is solved by a fast solver in the Cartesian grid.The computational work involved with the generalized boundary integral method is essentially linearly proportional to the number of grid nodes in the domain.This paper gives implementation details for a secondorder version of the kernel-free boundary integral method in two space dimensions and presents numerical experiments to demonstrate the efficiency and accuracy of the method for both boundary value and interface problems.The interface problems demonstrated include those with piecewise constant and large-ratio coefficients and the heterogeneous interface problem,where the elliptic PDEs on two sides of the interface are of different types.
文摘In the event of an instantaneous valve closure, the pressure transmitted to a surge tank induces the mass fluctuations that can cause high amplitude of water-level fluctuation in the surge tank for a reasonable cross-sectional area. The height of the surge tank is then designed using this high water level mark generated by the completely closed penstock valve. Using a conical surge tank with a non-constant cross-sectional area can resolve the problems of space and height. When addressing issues in designing open surge tanks, key parameters are usually calculated by using complex equations, which may become cumbersome when multiple iterations are required. A more effective alternative in obtaining these values is the use of simple charts. Firstly, this paper presents and describes the equations used to design open conical surge tanks. Secondly, it introduces user-friendly charts that can be used in the design of cylindrical and conical open surge tanks. The contribution can be a benefit for practicing engineers in this field. A case study is also presented to illustrate the use of these design charts. The case study’s results show that key parameters obtained via successive approximation method required 26 iterations or complex calculations, whereas these values can be obtained by simple reading of the proposed chart. The use of charts to help surge tanks designing, in the case of preliminary designs, can save time and increase design efficiency, while reducing calculation errors.
