Under some conditions, one seows that the generalized solutions of the first boundary value problem for the equation [GRAPHICS] have the property of finite speed of propagation.
In this paper, the existence of a pair of ordered solutions for the following class of equations in ?(1)?was studied. A bounded (PS) (Palais-Smale) sequence was constructed and the related variational principle was us...In this paper, the existence of a pair of ordered solutions for the following class of equations in ?(1)?was studied. A bounded (PS) (Palais-Smale) sequence was constructed and the related variational principle was used to prove the existence of the positive solution. The existence of the ordered solutions is finally found.展开更多
Group classification of quasilinear third-order evolution equations is given by using the classical infinitesimal Lie method, the technique of equivalence transformations, and the theory of classification of abstract ...Group classification of quasilinear third-order evolution equations is given by using the classical infinitesimal Lie method, the technique of equivalence transformations, and the theory of classification of abstract low-dimensional Lie algebras. We show that there are three equations admitting simple Lie algebras of dimension three. All non-equivalent equations admitting simple Lie algebras are nothing but these three. Furthermore, we also show that there exist two, five, twenty-nine and twenty-six non- equivalent third-order nonlinear evolution equations admitting one-, two-, three-, and four-dimensional solvable Lie algebras, respectively.展开更多
Systems of quasilinear first order PDE are studied in the framework of contact manifold. All of the local stable geometric solutions of such systems are classified by using versal deformation and the classification of...Systems of quasilinear first order PDE are studied in the framework of contact manifold. All of the local stable geometric solutions of such systems are classified by using versal deformation and the classification of stable map germs of type ∑1 in singularity theory.展开更多
By means of the existence and uniqueness of semi-global C1 solution to the mixed initial-boundary value problem with general nonlinear boundary conditions for first order quasilinear hyperbolic systems with zero eigen...By means of the existence and uniqueness of semi-global C1 solution to the mixed initial-boundary value problem with general nonlinear boundary conditions for first order quasilinear hyperbolic systems with zero eigenvalues,the local exact boundary controllability for higher order quasilinear hyperbolic equations is established.展开更多
Optimal boundary control of semilinear parabolic equations requires efficient solution methods in applications. Solution methods bypass the nonlinearity in different approaches. One approach can be quasilinearization ...Optimal boundary control of semilinear parabolic equations requires efficient solution methods in applications. Solution methods bypass the nonlinearity in different approaches. One approach can be quasilinearization (QL) but its applicability is locally in time. Nonetheless, consecutive applications of it can form a new method which is applicable globally in time. Dividing the control problem equivalently into many finite consecutive control subproblems they can be solved consecutively by a QL method. The proposed QL method for each subproblem constructs an infinite sequence of linear-quadratic optimal boundary control problems. These problems have solutions which converge to any optimal solutions of the subproblem. This implies the uniqueness of optimal solution to the subproblem. Merging solutions to the subproblems the solution of original control problem is obtained and its uniqueness is concluded. This uniqueness result is new. The proposed consecutive quasilinearization method is numerically stable with convergence order at least linear. Its consecutive feature prevents large scale computations and increases machine applicability. Its applicability for globalization of locally convergent methods makes it attractive for designing fast hybrid solution methods with global convergence.展开更多
A family of high-order accuracy explict difference schemes for solving 3-dimension parabolic P. D. E. is constructed. The stability condition is r = Deltat/Deltax(2) Deltat/Deltay(2) = Deltat/Deltaz(2) < 1/2 ,and t...A family of high-order accuracy explict difference schemes for solving 3-dimension parabolic P. D. E. is constructed. The stability condition is r = Deltat/Deltax(2) Deltat/Deltay(2) = Deltat/Deltaz(2) < 1/2 ,and the truncation error is 0(<Delta>t(2) + Deltax(4)).展开更多
Based on the theory of semi-global C 1 solution and the local exact boundary controllability for first-order quasilinear hyperbolic systems,the local exact boundary controllability for a kind of second-order quasiline...Based on the theory of semi-global C 1 solution and the local exact boundary controllability for first-order quasilinear hyperbolic systems,the local exact boundary controllability for a kind of second-order quasilinear hyperbolic systems is obtained by a constructive method.展开更多
Based on the theory of semi-global classical solutions to quasilinear hyperbolic systems, the local exact boundary observability for a kind of second-order quasilinear hyperbolic systems is obtained by a constructive ...Based on the theory of semi-global classical solutions to quasilinear hyperbolic systems, the local exact boundary observability for a kind of second-order quasilinear hyperbolic systems is obtained by a constructive method.展开更多
Given a set of independent vector fields on a smooth manifold, we discuss how to find a function whose zero-level set is invariant under the flows of the vector fields. As an application, we study the solvability of o...Given a set of independent vector fields on a smooth manifold, we discuss how to find a function whose zero-level set is invariant under the flows of the vector fields. As an application, we study the solvability of overdetermined partial differential equations: Given a system of quasi-linear PDEs of first order for one unknown function we find a necessary and sufficient condition for the existence of solutions in terms of the second jet of the coefficients. This generalizes to certain quasi-linear systems of first order for several unknown functions.展开更多
We study overdetermined systems of first order partial differential equations with singular solutions.The main result gives a characterization of such systems and asserts that the singular solution is equal to the con...We study overdetermined systems of first order partial differential equations with singular solutions.The main result gives a characterization of such systems and asserts that the singular solution is equal to the contact singular set.展开更多
For first-order quasilinear hyperbolic systems with zero eigenvalues, the author establishes the local exact controllability in a shorter time-period by means of internal controls acting on suitable domains. In partic...For first-order quasilinear hyperbolic systems with zero eigenvalues, the author establishes the local exact controllability in a shorter time-period by means of internal controls acting on suitable domains. In particular, under certain special but reasonable hypotheses, the local exact controllability can be realized only by internal controls, and the control time can be arbitrarily small.展开更多
文摘Under some conditions, one seows that the generalized solutions of the first boundary value problem for the equation [GRAPHICS] have the property of finite speed of propagation.
文摘In this paper, the existence of a pair of ordered solutions for the following class of equations in ?(1)?was studied. A bounded (PS) (Palais-Smale) sequence was constructed and the related variational principle was used to prove the existence of the positive solution. The existence of the ordered solutions is finally found.
基金supported by the National Key Basic Research Project of China (973 Program)(No. 2004CB318000)
文摘Group classification of quasilinear third-order evolution equations is given by using the classical infinitesimal Lie method, the technique of equivalence transformations, and the theory of classification of abstract low-dimensional Lie algebras. We show that there are three equations admitting simple Lie algebras of dimension three. All non-equivalent equations admitting simple Lie algebras are nothing but these three. Furthermore, we also show that there exist two, five, twenty-nine and twenty-six non- equivalent third-order nonlinear evolution equations admitting one-, two-, three-, and four-dimensional solvable Lie algebras, respectively.
基金This work was supported by the National Natural Science Foundation of China (Grant No. 19971035).
文摘Systems of quasilinear first order PDE are studied in the framework of contact manifold. All of the local stable geometric solutions of such systems are classified by using versal deformation and the classification of stable map germs of type ∑1 in singularity theory.
文摘By means of the existence and uniqueness of semi-global C1 solution to the mixed initial-boundary value problem with general nonlinear boundary conditions for first order quasilinear hyperbolic systems with zero eigenvalues,the local exact boundary controllability for higher order quasilinear hyperbolic equations is established.
文摘Optimal boundary control of semilinear parabolic equations requires efficient solution methods in applications. Solution methods bypass the nonlinearity in different approaches. One approach can be quasilinearization (QL) but its applicability is locally in time. Nonetheless, consecutive applications of it can form a new method which is applicable globally in time. Dividing the control problem equivalently into many finite consecutive control subproblems they can be solved consecutively by a QL method. The proposed QL method for each subproblem constructs an infinite sequence of linear-quadratic optimal boundary control problems. These problems have solutions which converge to any optimal solutions of the subproblem. This implies the uniqueness of optimal solution to the subproblem. Merging solutions to the subproblems the solution of original control problem is obtained and its uniqueness is concluded. This uniqueness result is new. The proposed consecutive quasilinearization method is numerically stable with convergence order at least linear. Its consecutive feature prevents large scale computations and increases machine applicability. Its applicability for globalization of locally convergent methods makes it attractive for designing fast hybrid solution methods with global convergence.
文摘A family of high-order accuracy explict difference schemes for solving 3-dimension parabolic P. D. E. is constructed. The stability condition is r = Deltat/Deltax(2) Deltat/Deltay(2) = Deltat/Deltaz(2) < 1/2 ,and the truncation error is 0(<Delta>t(2) + Deltax(4)).
基金supported by the Excellent Doctoral Research Foundation for Key Subject of Fudan University (No.EHH1411208)
文摘Based on the theory of semi-global C 1 solution and the local exact boundary controllability for first-order quasilinear hyperbolic systems,the local exact boundary controllability for a kind of second-order quasilinear hyperbolic systems is obtained by a constructive method.
基金supported by the National Natural Science Foundation of China(No.11526050)
文摘Based on the theory of semi-global classical solutions to quasilinear hyperbolic systems, the local exact boundary observability for a kind of second-order quasilinear hyperbolic systems is obtained by a constructive method.
基金supported by National Research Foundation of Republic of Korea(Grant Nos.2011-0008976 and 2010-0011841)
文摘Given a set of independent vector fields on a smooth manifold, we discuss how to find a function whose zero-level set is invariant under the flows of the vector fields. As an application, we study the solvability of overdetermined partial differential equations: Given a system of quasi-linear PDEs of first order for one unknown function we find a necessary and sufficient condition for the existence of solutions in terms of the second jet of the coefficients. This generalizes to certain quasi-linear systems of first order for several unknown functions.
文摘We study overdetermined systems of first order partial differential equations with singular solutions.The main result gives a characterization of such systems and asserts that the singular solution is equal to the contact singular set.
文摘For first-order quasilinear hyperbolic systems with zero eigenvalues, the author establishes the local exact controllability in a shorter time-period by means of internal controls acting on suitable domains. In particular, under certain special but reasonable hypotheses, the local exact controllability can be realized only by internal controls, and the control time can be arbitrarily small.
