We consider the following (1 + 3)-dimensional P(1,4)-invariant partial differential equations (PDEs): the Eikonal equation, the Euler-Lagrange-Born-Infeld equation, the homogeneous Monge-Ampère equation, the inho...We consider the following (1 + 3)-dimensional P(1,4)-invariant partial differential equations (PDEs): the Eikonal equation, the Euler-Lagrange-Born-Infeld equation, the homogeneous Monge-Ampère equation, the inhomogeneous Monge-Ampère equation. The purpose of this paper is to construct and classify the common invariant solutions for those equations. For this aim, we have used the results concerning construction and classification of invariant solutions for the (1 + 3)-dimensional P(1,4)-invariant Eikonal equation, since this equation is the simplest among the equations under investigation. The direct checked allowed us to conclude that the majority of invariant solutions of the (1 + 3)-dimensional Eikonal equation, obtained on the base of low-dimensional (dimL ≤ 3) nonconjugate subalgebras of the Lie algebra of the Poincaré group P(1,4), satisfy all the equations under investigation. In this paper, we present obtained common invariant solutions of the equations under study as well as the classification of those invariant solutions.展开更多
In this short note we examine the connection between weakly isotone maps and common solutions for first order Cauchy problems in R^n and, as a rule, in Banach lattices.
Solving the famous Hermite, Legendre, Laguerre and Chebyshev equations requires different techniques of unique character for each equation. By reducing these differential equations of second order to a common solvable...Solving the famous Hermite, Legendre, Laguerre and Chebyshev equations requires different techniques of unique character for each equation. By reducing these differential equations of second order to a common solvable differential equation of first order, a simple common solution is provided to cover all the existing standard solutions of these named equations. It is easier than the method of generating functions and more powerful than the Probenius method of power series.展开更多
This paper extends the previous work on common positive definite solutions (CPDSs) to planar algebraic Riccati inequalities (ARIs) to those with arbitrary dimensions.The topological structure of the set of all positiv...This paper extends the previous work on common positive definite solutions (CPDSs) to planar algebraic Riccati inequalities (ARIs) to those with arbitrary dimensions.The topological structure of the set of all positive definite solutions of an ARI is investigated.This leads to a necessary and sufficient condition for the existence of CPDSs to a set of Riccati inequalities.It also reveals that the solution set of ARIs is a positive cube in Rn,which arouses a new method to search the CPDS.Some examples of three-dimensional ARIs are presented to show the effectiveness of the proposed methods.Unlike linear matrix inequality (LMI) method,the computing collapse will not occur with the increase of the number of Riccati inequalities due to the fact that our approach handles the ARIs one by one rather than simultaneously.展开更多
文摘We consider the following (1 + 3)-dimensional P(1,4)-invariant partial differential equations (PDEs): the Eikonal equation, the Euler-Lagrange-Born-Infeld equation, the homogeneous Monge-Ampère equation, the inhomogeneous Monge-Ampère equation. The purpose of this paper is to construct and classify the common invariant solutions for those equations. For this aim, we have used the results concerning construction and classification of invariant solutions for the (1 + 3)-dimensional P(1,4)-invariant Eikonal equation, since this equation is the simplest among the equations under investigation. The direct checked allowed us to conclude that the majority of invariant solutions of the (1 + 3)-dimensional Eikonal equation, obtained on the base of low-dimensional (dimL ≤ 3) nonconjugate subalgebras of the Lie algebra of the Poincaré group P(1,4), satisfy all the equations under investigation. In this paper, we present obtained common invariant solutions of the equations under study as well as the classification of those invariant solutions.
文摘In this short note we examine the connection between weakly isotone maps and common solutions for first order Cauchy problems in R^n and, as a rule, in Banach lattices.
文摘Solving the famous Hermite, Legendre, Laguerre and Chebyshev equations requires different techniques of unique character for each equation. By reducing these differential equations of second order to a common solvable differential equation of first order, a simple common solution is provided to cover all the existing standard solutions of these named equations. It is easier than the method of generating functions and more powerful than the Probenius method of power series.
基金supported by the Scientific Research Foundation for the Returned Overseas Chinese Scholars,State Education Ministryby the Aerospace Science Foundation of China (No. 2009ZH68022)the Program of 985 Innovation Engineering on Information at Xiamen University(2009-2011)
文摘This paper extends the previous work on common positive definite solutions (CPDSs) to planar algebraic Riccati inequalities (ARIs) to those with arbitrary dimensions.The topological structure of the set of all positive definite solutions of an ARI is investigated.This leads to a necessary and sufficient condition for the existence of CPDSs to a set of Riccati inequalities.It also reveals that the solution set of ARIs is a positive cube in Rn,which arouses a new method to search the CPDS.Some examples of three-dimensional ARIs are presented to show the effectiveness of the proposed methods.Unlike linear matrix inequality (LMI) method,the computing collapse will not occur with the increase of the number of Riccati inequalities due to the fact that our approach handles the ARIs one by one rather than simultaneously.
