In previous papers, we proposed the important Ztransformations and obtained general solutions to a large number of linear and quasi-linear partial differential equations for the first time. In this paper, we will use ...In previous papers, we proposed the important Ztransformations and obtained general solutions to a large number of linear and quasi-linear partial differential equations for the first time. In this paper, we will use the Z1transformation to get the general solutions of some nonlinear partial differential equations for the first time, and use the general solutions to obtain the exact solutions of some typical definite solution problems.展开更多
This paper considers the regularity of solutions to mixed boundary value problems in small-angle regions for elliptic equations. By constructing a specific barrier function, we proved that under the assumption of suff...This paper considers the regularity of solutions to mixed boundary value problems in small-angle regions for elliptic equations. By constructing a specific barrier function, we proved that under the assumption of sufficient regularity of boundary conditions and coefficients, as long as the angle is sufficiently small, the regularity of the solution to the mixed boundary value problem of the second-order elliptic equation can reach any order.展开更多
In this paper we consider the singularly perturbed boundary value problem for the fourth-order elliptic differential equation, establish the energy estimates of the solutionand its derivatives and construct the formal...In this paper we consider the singularly perturbed boundary value problem for the fourth-order elliptic differential equation, establish the energy estimates of the solutionand its derivatives and construct the formal asymptotic solution by Lyuternik- Vishik 's method. Finally, by means of the energy estimates we obtain the bound of the remainder of the asymptotic solution.展开更多
In this paper, we consider a singular perturbation elliptic-parabolic partial differential equation for periodic boundary value problem, and construct a difference scheme. Using the method of decomposing the singular ...In this paper, we consider a singular perturbation elliptic-parabolic partial differential equation for periodic boundary value problem, and construct a difference scheme. Using the method of decomposing the singular term from its solution and combining an asymptotic expansion of the equation, we prove that the scheme constructed by this paper converges uniformly to the solution of its original problem with O(r+h2).展开更多
In this paper, we consider the upwind difference scheme for singular perturbation problem (1.1). On a special discretization mesh, it is proved that the solution of the upwind difference scheme is first order converge...In this paper, we consider the upwind difference scheme for singular perturbation problem (1.1). On a special discretization mesh, it is proved that the solution of the upwind difference scheme is first order convergent, uniformly in the small parameter e , to the solution of problem (1.1). Numerical results are finally provided.展开更多
In the poper, the method of separating singularity is applied to study the uniformly difference scheme of a singular perturbation problem for a semilinear ordinary differential equation with mixed boundary value condi...In the poper, the method of separating singularity is applied to study the uniformly difference scheme of a singular perturbation problem for a semilinear ordinary differential equation with mixed boundary value condition. The uniform convergence on small parameter ε of order one for an IVin type difference scheme constructed is proved. At the end of the paper, a numerical example is given. The computing results coincide with the theoretical analysis.展开更多
The present paper deals with the mixed boundary value problem for elliptic equations with degenerate rank 0. We first give the formulation of the problem and estimates of solutions of the problem, and then prove the e...The present paper deals with the mixed boundary value problem for elliptic equations with degenerate rank 0. We first give the formulation of the problem and estimates of solutions of the problem, and then prove the existence of solutions of the above problem for elliptic equations by the above estimates and the method of parameter extension. We use the complex method, namely first discuss the corresponding problem for degenerate elliptic complex equations of first order, afterwards discuss the above problem for degenerate elliptic equations of second order展开更多
In this paper we study the forced oscillations of boundary value problems of a class of higher order functional partial differential equations.The principal tool is an everaging techniqe which enables one to establish...In this paper we study the forced oscillations of boundary value problems of a class of higher order functional partial differential equations.The principal tool is an everaging techniqe which enables one to establish oscillation in terms of related functional differential inequallities.展开更多
This paper studies the global existence of the classical solutions to the following problem:This problem describes the nonlinear vibrations of finite rods with nonlinear vis-coelasticity. Under certain conditions on ...This paper studies the global existence of the classical solutions to the following problem:This problem describes the nonlinear vibrations of finite rods with nonlinear vis-coelasticity. Under certain conditions on σand f , we obtained the unique existence of the global classical solution of this problem.展开更多
In this paper,the existence and uniqueness of iterative solutions to the boundary value problems for a class of first order impulsive integro-differential equations were studied. Under a new concept of upper and lower...In this paper,the existence and uniqueness of iterative solutions to the boundary value problems for a class of first order impulsive integro-differential equations were studied. Under a new concept of upper and lower solutions, a new monotone iterative technique on the boundary value problem of integro-differential equations was proposed. The existence and uniqueness of iterative solutions and the error estimation in certain interval were obtained.An example was also given to illustrate the results.展开更多
In this paper a singular perturbation of boundary value problem for elliptic partial differential equations of higher order is considered by using the differential inequalities. The uniformly valid asymptotic expansio...In this paper a singular perturbation of boundary value problem for elliptic partial differential equations of higher order is considered by using the differential inequalities. The uniformly valid asymptotic expansion in entire region is obtained.展开更多
We considerer partial differential equations of second order, for example the Klein-Gordon equation, the Poisson equation, on a region E = （a1, b1 ） × （a2, b2 ） x （a3, b3 ）. We will see that with a common p...We considerer partial differential equations of second order, for example the Klein-Gordon equation, the Poisson equation, on a region E = （a1, b1 ） × （a2, b2 ） x （a3, b3 ）. We will see that with a common procedure in all cases, we can write the equation in partial derivatives as an Fredholm integral equation of first kind and will solve this latter with the techniques of inverse problem moments. We will find an approximated solution and bounds for the error of the estimated solution using the techniques on problem of moments.展开更多
In this paper, we proposed new results in quadruple Laplace transform and proved some properties concerned with quadruple Laplace transform. We also developed some applications based on these results and solved homoge...In this paper, we proposed new results in quadruple Laplace transform and proved some properties concerned with quadruple Laplace transform. We also developed some applications based on these results and solved homogeneous as well as non-homogeneous partial differential equations involving four variables. The performance of quadruple Laplace transform is shown to be very encouraging by concrete examples. An elementary table of quadruple Laplace transform is also provided.展开更多
In this paper, we investigate the existence of solution for a class of impulse boundary value problem of nonlinear fractional functional differential equation of mixed type. We obtain the existence results of solution...In this paper, we investigate the existence of solution for a class of impulse boundary value problem of nonlinear fractional functional differential equation of mixed type. We obtain the existence results of solution by applying some well-known fixed point theorems. An example is given to illustrate the effectiveness of our result.展开更多
In this paper, by applying the Jacobi elliptic function expansion method, the periodic solutions for two coupled nonlinear partial differential equations are obtained.
In this paper,a class of singular perturbation of nonlocal boundary value problems for elliptic partial differential equations of higher order is considered by using the differential inequalities.The uniformly valid a...In this paper,a class of singular perturbation of nonlocal boundary value problems for elliptic partial differential equations of higher order is considered by using the differential inequalities.The uniformly valid asymptotic expansion of solution is obtained.展开更多
Accurately approximating higher order derivatives is an inherently difficult problem. It is shown that a random variable shape parameter strategy can improve the accuracy of approximating higher order derivatives with...Accurately approximating higher order derivatives is an inherently difficult problem. It is shown that a random variable shape parameter strategy can improve the accuracy of approximating higher order derivatives with Radial Basis Function methods. The method is used to solve fourth order boundary value problems. The use and location of ghost points are examined in order to enforce the extra boundary conditions that are necessary to make a fourth-order problem well posed. The use of ghost points versus solving an overdetermined linear system via least squares is studied. For a general fourth-order boundary value problem, the recommended approach is to either use one of two novel sets of ghost centers introduced here or else to use a least squares approach. When using either ghost centers or least squares, the random variable shape parameter strategy results in significantly better accuracy than when a constant shape parameter is used.展开更多
Solutions of quasilinear mixed boundary problems for the some parabolic an elliptic partial differential equations are interpreted as solutions of a kind of backward stochastic differential equations, which are associ...Solutions of quasilinear mixed boundary problems for the some parabolic an elliptic partial differential equations are interpreted as solutions of a kind of backward stochastic differential equations, which are associated with the classical Ito forward stochastic differential equations with reflecting boundary conditions.展开更多
We introduce the concept of <i>q</i>-calculus in quantum geometry. This involves the <i>q</i>-differential and <i>q</i>-integral operators. With these, we study the basic rules gove...We introduce the concept of <i>q</i>-calculus in quantum geometry. This involves the <i>q</i>-differential and <i>q</i>-integral operators. With these, we study the basic rules governing <i>q</i>-calculus as compared with the classical Newton-Leibnitz calculus, and obtain some important results. We introduce the reduced <i>q</i>-differential transform method (R<i>q</i>DTM) for solving partial <i>q</i>-differential equations. The solution is computed in the form of a convergent power series with easily computable coefficients. With the help of some test examples, we discover the effectiveness and performance of the proposed method and employing MathCAD 14 software for computation. It turns out that when <i>q</i> = 1, the solution coincides with that for the classical version of the given initial value problem. The results demonstrate that the R<i>q</i>DTM approach is quite efficient and convenient.展开更多
文摘In previous papers, we proposed the important Ztransformations and obtained general solutions to a large number of linear and quasi-linear partial differential equations for the first time. In this paper, we will use the Z1transformation to get the general solutions of some nonlinear partial differential equations for the first time, and use the general solutions to obtain the exact solutions of some typical definite solution problems.
文摘This paper considers the regularity of solutions to mixed boundary value problems in small-angle regions for elliptic equations. By constructing a specific barrier function, we proved that under the assumption of sufficient regularity of boundary conditions and coefficients, as long as the angle is sufficiently small, the regularity of the solution to the mixed boundary value problem of the second-order elliptic equation can reach any order.
文摘In this paper we consider the singularly perturbed boundary value problem for the fourth-order elliptic differential equation, establish the energy estimates of the solutionand its derivatives and construct the formal asymptotic solution by Lyuternik- Vishik 's method. Finally, by means of the energy estimates we obtain the bound of the remainder of the asymptotic solution.
基金This work is supported by the National Fujian Province Nature Science Research Funds
文摘In this paper, we consider a singular perturbation elliptic-parabolic partial differential equation for periodic boundary value problem, and construct a difference scheme. Using the method of decomposing the singular term from its solution and combining an asymptotic expansion of the equation, we prove that the scheme constructed by this paper converges uniformly to the solution of its original problem with O(r+h2).
文摘In this paper, we consider the upwind difference scheme for singular perturbation problem (1.1). On a special discretization mesh, it is proved that the solution of the upwind difference scheme is first order convergent, uniformly in the small parameter e , to the solution of problem (1.1). Numerical results are finally provided.
文摘In the poper, the method of separating singularity is applied to study the uniformly difference scheme of a singular perturbation problem for a semilinear ordinary differential equation with mixed boundary value condition. The uniform convergence on small parameter ε of order one for an IVin type difference scheme constructed is proved. At the end of the paper, a numerical example is given. The computing results coincide with the theoretical analysis.
文摘The present paper deals with the mixed boundary value problem for elliptic equations with degenerate rank 0. We first give the formulation of the problem and estimates of solutions of the problem, and then prove the existence of solutions of the above problem for elliptic equations by the above estimates and the method of parameter extension. We use the complex method, namely first discuss the corresponding problem for degenerate elliptic complex equations of first order, afterwards discuss the above problem for degenerate elliptic equations of second order
文摘In this paper we study the forced oscillations of boundary value problems of a class of higher order functional partial differential equations.The principal tool is an everaging techniqe which enables one to establish oscillation in terms of related functional differential inequallities.
文摘This paper studies the global existence of the classical solutions to the following problem:This problem describes the nonlinear vibrations of finite rods with nonlinear vis-coelasticity. Under certain conditions on σand f , we obtained the unique existence of the global classical solution of this problem.
基金National Natural Science Foundation of China(No.11271372)Hunan Provincial National Natural Science Foundation of China(No.12JJ2004)Central South University Graduate Innovation Project,China(No.2014zzts136)
文摘In this paper,the existence and uniqueness of iterative solutions to the boundary value problems for a class of first order impulsive integro-differential equations were studied. Under a new concept of upper and lower solutions, a new monotone iterative technique on the boundary value problem of integro-differential equations was proposed. The existence and uniqueness of iterative solutions and the error estimation in certain interval were obtained.An example was also given to illustrate the results.
文摘In this paper a singular perturbation of boundary value problem for elliptic partial differential equations of higher order is considered by using the differential inequalities. The uniformly valid asymptotic expansion in entire region is obtained.
文摘We considerer partial differential equations of second order, for example the Klein-Gordon equation, the Poisson equation, on a region E = （a1, b1 ） × （a2, b2 ） x （a3, b3 ）. We will see that with a common procedure in all cases, we can write the equation in partial derivatives as an Fredholm integral equation of first kind and will solve this latter with the techniques of inverse problem moments. We will find an approximated solution and bounds for the error of the estimated solution using the techniques on problem of moments.
文摘In this paper, we proposed new results in quadruple Laplace transform and proved some properties concerned with quadruple Laplace transform. We also developed some applications based on these results and solved homogeneous as well as non-homogeneous partial differential equations involving four variables. The performance of quadruple Laplace transform is shown to be very encouraging by concrete examples. An elementary table of quadruple Laplace transform is also provided.
基金Supported by the NNSF of China（ll071001） Supported by the NSF＂ of the Anhui Higher Education Institutions of China（KJ2013B276） Supporied by the Key Program of the Natural Science Foundation for the Excellent Youth Scholars of Anhui Higher Education Institutions of China （2013SQRL142ZD）
文摘In this paper, we investigate the existence of solution for a class of impulse boundary value problem of nonlinear fractional functional differential equation of mixed type. We obtain the existence results of solution by applying some well-known fixed point theorems. An example is given to illustrate the effectiveness of our result.
基金The project supported by National Natural Science Foundation of China under Grant Nos. 90511009 and 40305006 Cprrespondence author,
文摘In this paper, by applying the Jacobi elliptic function expansion method, the periodic solutions for two coupled nonlinear partial differential equations are obtained.
文摘In this paper,a class of singular perturbation of nonlocal boundary value problems for elliptic partial differential equations of higher order is considered by using the differential inequalities.The uniformly valid asymptotic expansion of solution is obtained.
文摘Accurately approximating higher order derivatives is an inherently difficult problem. It is shown that a random variable shape parameter strategy can improve the accuracy of approximating higher order derivatives with Radial Basis Function methods. The method is used to solve fourth order boundary value problems. The use and location of ghost points are examined in order to enforce the extra boundary conditions that are necessary to make a fourth-order problem well posed. The use of ghost points versus solving an overdetermined linear system via least squares is studied. For a general fourth-order boundary value problem, the recommended approach is to either use one of two novel sets of ghost centers introduced here or else to use a least squares approach. When using either ghost centers or least squares, the random variable shape parameter strategy results in significantly better accuracy than when a constant shape parameter is used.
文摘Solutions of quasilinear mixed boundary problems for the some parabolic an elliptic partial differential equations are interpreted as solutions of a kind of backward stochastic differential equations, which are associated with the classical Ito forward stochastic differential equations with reflecting boundary conditions.
文摘We introduce the concept of <i>q</i>-calculus in quantum geometry. This involves the <i>q</i>-differential and <i>q</i>-integral operators. With these, we study the basic rules governing <i>q</i>-calculus as compared with the classical Newton-Leibnitz calculus, and obtain some important results. We introduce the reduced <i>q</i>-differential transform method (R<i>q</i>DTM) for solving partial <i>q</i>-differential equations. The solution is computed in the form of a convergent power series with easily computable coefficients. With the help of some test examples, we discover the effectiveness and performance of the proposed method and employing MathCAD 14 software for computation. It turns out that when <i>q</i> = 1, the solution coincides with that for the classical version of the given initial value problem. The results demonstrate that the R<i>q</i>DTM approach is quite efficient and convenient.
