In this paper, the maximal and minimal ranks of the solution to a system of matrix equations over H, the real quaternion algebra, were derived. A previous known result could be regarded as a special case of the new re...In this paper, the maximal and minimal ranks of the solution to a system of matrix equations over H, the real quaternion algebra, were derived. A previous known result could be regarded as a special case of the new result.展开更多
Necessary and sufficient conditions are derived for some matrix equations that have a common least-squares solution.A general expression is provided when certain resolvable conditions are satisfied.This research exten...Necessary and sufficient conditions are derived for some matrix equations that have a common least-squares solution.A general expression is provided when certain resolvable conditions are satisfied.This research extends existing work in the literature.展开更多
A new expression is established for the common solution to six classical linear quaternion matrix equations A 1 X = C 1 , X B 1 = C 3 , A 2 X = C 2 , X B 2 = C 4 , A 3 X B 3 = C 5 , A 4 X B 4 = C 6 which was investiga...A new expression is established for the common solution to six classical linear quaternion matrix equations A 1 X = C 1 , X B 1 = C 3 , A 2 X = C 2 , X B 2 = C 4 , A 3 X B 3 = C 5 , A 4 X B 4 = C 6 which was investigated recently by Wang, Chang and Ning (Q. Wang, H. Chang, Q. Ning, The common solution to six quaternion matrix equations with applications, Appl. Math. Comput. 195: 721-732 (2008)). Formulas are derived for the maximal and minimal ranks of the common solution to this system. Moreover, corresponding results on some special cases are presented. As an application, a necessary and sufficient condition is presented for the invariance of the rank of the general solution to this system. Some known results can be regarded as the special cases of the results in this paper.展开更多
In this paper, we give the expression of the least square solution of the linear quaternion matrix equation AXB = C subject to a consistent system of quaternion matrix equations D1X = F1, XE2 =F2, and derive the maxim...In this paper, we give the expression of the least square solution of the linear quaternion matrix equation AXB = C subject to a consistent system of quaternion matrix equations D1X = F1, XE2 =F2, and derive the maximal and minimal ranks and the leastnorm of the above mentioned solution. The finding of this paper extends some known results in the literature.展开更多
Suppose that A1X = C1, XB2 = C2, A3XB3= 63 is a consistent system of matrix equations and partition its solution X into a 2× 2 block form. In this paper, we give formulas for the maximal and minimal ranks of the ...Suppose that A1X = C1, XB2 = C2, A3XB3= 63 is a consistent system of matrix equations and partition its solution X into a 2× 2 block form. In this paper, we give formulas for the maximal and minimal ranks of the submatrices in a solution X to the system. We also investigate the uniqueness and the independence of submatrices in a solution X. As applications, we give some properties of submatrices in generalized inverses of matrices. These extend some known results in the literature.展开更多
Let A be an arbitrary square matrix,then equation AXA=XAX with unknown X is called Yang-Baxter matrix equation.It is difficult to find all the solutions of this equation for any given A.In this paper,the relations bet...Let A be an arbitrary square matrix,then equation AXA=XAX with unknown X is called Yang-Baxter matrix equation.It is difficult to find all the solutions of this equation for any given A.In this paper,the relations between the matrices A and B are established via solving the associated rank optimization problem,where B=AXA=XAX,and some analytical formulas are derived,which may be useful for finding all the solutions and analyzing the structures of the solutions of the above Yang-Baxter matrix equation.展开更多
In this paper we investigate the system of linear matrix equations A1X = C1, YB2 = C2, A3XB3 = C3, A4YB4 = C4, BX + YC = A. We present some necessary and sufficient conditions for the existence of a solution to this ...In this paper we investigate the system of linear matrix equations A1X = C1, YB2 = C2, A3XB3 = C3, A4YB4 = C4, BX + YC = A. We present some necessary and sufficient conditions for the existence of a solution to this system and give an expression of the general solution to the system when the solvability conditions are satisfied.展开更多
We in this paper derive necessary and sufficient conditions for the system of the periodic discrete-time coupled Sylvester matrix equations AkXk + YkBk = Mk, CkXk+l + YkDk ---- Nk (k = 1, 2) over the quaternion a...We in this paper derive necessary and sufficient conditions for the system of the periodic discrete-time coupled Sylvester matrix equations AkXk + YkBk = Mk, CkXk+l + YkDk ---- Nk (k = 1, 2) over the quaternion algebra to be consistent in terms of ranks and generalized inverses of the coefficient matrices. We also give an expression of the general solution to the system when it is solvable. The findings of this paper generalize some known results in the literature.展开更多
We derive the solvability conditions and an expression of the general solution to the system of matrix equations A 1X=C1 , A2Y=C2 , YB2=D2 , Y=Y*, A3Z=C3 , ZB3=D3 , Z=Z*, B4X+(B4X)+C4YC4*+D4ZD4*=A4 . Moreover, we inve...We derive the solvability conditions and an expression of the general solution to the system of matrix equations A 1X=C1 , A2Y=C2 , YB2=D2 , Y=Y*, A3Z=C3 , ZB3=D3 , Z=Z*, B4X+(B4X)+C4YC4*+D4ZD4*=A4 . Moreover, we investigate the maximal and minimal ranks and inertias of Y and Z in the above system of matrix equations. As a special case of the results, we solve the problem proposed in Farid, Moslehian, Wang and Wu's recent paper (Farid F O, Moslehian M S, Wang Q W, et al. On the Hermitian solutions to a system of adjointable operator equations. Linear Algebra Appl, 2012, 437: 1854-1891).展开更多
In this paper,a high accurate numerical approach is investigated for solving the time-fractional linear and nonlinear Korteweg-de Vries(KdV)equations.These equations are the most appropriate and desirable definition f...In this paper,a high accurate numerical approach is investigated for solving the time-fractional linear and nonlinear Korteweg-de Vries(KdV)equations.These equations are the most appropriate and desirable definition for physical modeling.The spectral collocation method and the operational matrix of fractional derivatives are used together with the help of the Gauss-quadrature formula in order to reduce such problem into a problem consists of solving a system of algebraic equations which greatly simplifying the problem.Our approach is based on the shifted Jacobi polynomials and the fractional derivative is described in the sense of Caputo.In addition,the presented approach is applied also to solve the timefractional modified KdV equation.For testing the accuracy,validity and applicability of the developed numerical approach,we apply it to provide high accurate approximate solutions for four test problems.展开更多
In this paper, a new numerical method for solving fractional differential equations(FDEs) is presented. The method is based upon the fractional Taylor basis approximations. The operational matrix of the fractional int...In this paper, a new numerical method for solving fractional differential equations(FDEs) is presented. The method is based upon the fractional Taylor basis approximations. The operational matrix of the fractional integration for the fractional Taylor basis is introduced. This matrix is then utilized to reduce the solution of the fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of this technique.展开更多
An Euler wavelets method is proposed to solve a class of nonlinear variable order fractional differential equations in this paper.The properties of Euler wavelets and their operational matrix together with a family of...An Euler wavelets method is proposed to solve a class of nonlinear variable order fractional differential equations in this paper.The properties of Euler wavelets and their operational matrix together with a family of piecewise functions are first presented.Then they are utilized to reduce the problem to the solution of a nonlinear system of algebraic equations.And the convergence of the Euler wavelets basis is given.The method is computationally attractive and some numerical examples are provided to illustrate its high accuracy.展开更多
In this paper, operational matrices of Bernstein polynomials (BPs) are presented for solving the non-linear fractional Logistic differential equation (FLDE). The fractional derivative is described in the Riemann-Liouv...In this paper, operational matrices of Bernstein polynomials (BPs) are presented for solving the non-linear fractional Logistic differential equation (FLDE). The fractional derivative is described in the Riemann-Liouville sense. The operational matrices for the fractional integration in the Riemann-Liouville sense and the product are used to reduce FLDE to the solution of non-linear system of algebraic equations using Newton iteration method. Numerical results are introduced to satisfy the accuracy and the applicability of the proposed method.展开更多
The basic aim of this paper is to introduce and describe an efficient numerical scheme based on spectral approach coupled with Chebyshev wavelets for the approximate solutions of Klein-Gordon and Sine-Gordon equations...The basic aim of this paper is to introduce and describe an efficient numerical scheme based on spectral approach coupled with Chebyshev wavelets for the approximate solutions of Klein-Gordon and Sine-Gordon equations. The main characteristic is that, it converts the given problem into a system of algebraic equations that can be solved easily with any of the usual methods. To show the accuracy and the efficiency of the method, several benchmark problems are implemented and the comparisons are given with other methods existing in the recent literature. The results of numerical tests confirm that the proposed method is superior to other existing ones and is highly accurate展开更多
The performance of analytical derivative and sparse matrix techniques applied to a traditional dense sequential quadratic programming (SQP) is studied, and the strategy utilizing those techniques is also presented.Com...The performance of analytical derivative and sparse matrix techniques applied to a traditional dense sequential quadratic programming (SQP) is studied, and the strategy utilizing those techniques is also presented.Computational results on two typical chemical optimization problems demonstrate significant enhancement in efficiency, which shows this strategy is promising and suitable for large-scale process optimization problems.展开更多
We study nonhomogeneous systems of linear conformable fractional differential equations with pure delay.By using new conformable delayed matrix functions and the method of variation,we obtain a representation of their...We study nonhomogeneous systems of linear conformable fractional differential equations with pure delay.By using new conformable delayed matrix functions and the method of variation,we obtain a representation of their solutions.As an application,we derive a finite time stability result using the representation of solutions and a norm estimation of the conformable delayedmatrix functions.The obtained results are new,and they extend and improve some existing ones.Finally,an example is presented to illustrate the validity of our theoretical results.展开更多
基金Project supported by the National Natural Science Foundation of China (Grant No.60672160)
文摘In this paper, the maximal and minimal ranks of the solution to a system of matrix equations over H, the real quaternion algebra, were derived. A previous known result could be regarded as a special case of the new result.
文摘Necessary and sufficient conditions are derived for some matrix equations that have a common least-squares solution.A general expression is provided when certain resolvable conditions are satisfied.This research extends existing work in the literature.
基金Supported by the National Natural Science Foundation of Shanghai (No. 11ZR1412500)the Ph.D. Programs Foundation of Ministry of Education of China (No. 20093108110001)Shanghai Leading Academic Discipline Project (No. J50101)
文摘A new expression is established for the common solution to six classical linear quaternion matrix equations A 1 X = C 1 , X B 1 = C 3 , A 2 X = C 2 , X B 2 = C 4 , A 3 X B 3 = C 5 , A 4 X B 4 = C 6 which was investigated recently by Wang, Chang and Ning (Q. Wang, H. Chang, Q. Ning, The common solution to six quaternion matrix equations with applications, Appl. Math. Comput. 195: 721-732 (2008)). Formulas are derived for the maximal and minimal ranks of the common solution to this system. Moreover, corresponding results on some special cases are presented. As an application, a necessary and sufficient condition is presented for the invariance of the rank of the general solution to this system. Some known results can be regarded as the special cases of the results in this paper.
文摘In this paper, we give the expression of the least square solution of the linear quaternion matrix equation AXB = C subject to a consistent system of quaternion matrix equations D1X = F1, XE2 =F2, and derive the maximal and minimal ranks and the leastnorm of the above mentioned solution. The finding of this paper extends some known results in the literature.
文摘Suppose that A1X = C1, XB2 = C2, A3XB3= 63 is a consistent system of matrix equations and partition its solution X into a 2× 2 block form. In this paper, we give formulas for the maximal and minimal ranks of the submatrices in a solution X to the system. We also investigate the uniqueness and the independence of submatrices in a solution X. As applications, we give some properties of submatrices in generalized inverses of matrices. These extend some known results in the literature.
基金Supported by the National Natural Science Foundation of China(11961057)the Science and Technology Project of Gansu Province(21JR1RE287 and 2021B-221)+2 种基金the Fuxi Scientific Research Innovation Team of Tianshui Normal University(FXD2020-03)the Science Foundation(CXT2019-36 and CXJ2020-11)the Education and Teaching Reform Project of Tianshui Normal University(JY202004 and JY203008)。
文摘Let A be an arbitrary square matrix,then equation AXA=XAX with unknown X is called Yang-Baxter matrix equation.It is difficult to find all the solutions of this equation for any given A.In this paper,the relations between the matrices A and B are established via solving the associated rank optimization problem,where B=AXA=XAX,and some analytical formulas are derived,which may be useful for finding all the solutions and analyzing the structures of the solutions of the above Yang-Baxter matrix equation.
基金This research was supported by the grants from the National Natural Science Foundation of China (11571220, 11171205).
文摘In this paper we investigate the system of linear matrix equations A1X = C1, YB2 = C2, A3XB3 = C3, A4YB4 = C4, BX + YC = A. We present some necessary and sufficient conditions for the existence of a solution to this system and give an expression of the general solution to the system when the solvability conditions are satisfied.
基金This research was supported by the grant from the National Natural Science Foundation of China (11571220).
文摘We in this paper derive necessary and sufficient conditions for the system of the periodic discrete-time coupled Sylvester matrix equations AkXk + YkBk = Mk, CkXk+l + YkDk ---- Nk (k = 1, 2) over the quaternion algebra to be consistent in terms of ranks and generalized inverses of the coefficient matrices. We also give an expression of the general solution to the system when it is solvable. The findings of this paper generalize some known results in the literature.
基金National Natural Science Foundation of China (Grant No. 11171205)Natural Science Foundation of Shanghai (Grant No. 11ZR1412500)+2 种基金the Ph.D. Programs Foundation of Ministry of Education of China (Grant No. 20093108110001)Shanghai Leading Academic Discipline Project (Grant No. J50101)Innovation Foundation of Shanghai University (Grant No. SHUCX120109)
文摘We derive the solvability conditions and an expression of the general solution to the system of matrix equations A 1X=C1 , A2Y=C2 , YB2=D2 , Y=Y*, A3Z=C3 , ZB3=D3 , Z=Z*, B4X+(B4X)+C4YC4*+D4ZD4*=A4 . Moreover, we investigate the maximal and minimal ranks and inertias of Y and Z in the above system of matrix equations. As a special case of the results, we solve the problem proposed in Farid, Moslehian, Wang and Wu's recent paper (Farid F O, Moslehian M S, Wang Q W, et al. On the Hermitian solutions to a system of adjointable operator equations. Linear Algebra Appl, 2012, 437: 1854-1891).
文摘In this paper,a high accurate numerical approach is investigated for solving the time-fractional linear and nonlinear Korteweg-de Vries(KdV)equations.These equations are the most appropriate and desirable definition for physical modeling.The spectral collocation method and the operational matrix of fractional derivatives are used together with the help of the Gauss-quadrature formula in order to reduce such problem into a problem consists of solving a system of algebraic equations which greatly simplifying the problem.Our approach is based on the shifted Jacobi polynomials and the fractional derivative is described in the sense of Caputo.In addition,the presented approach is applied also to solve the timefractional modified KdV equation.For testing the accuracy,validity and applicability of the developed numerical approach,we apply it to provide high accurate approximate solutions for four test problems.
文摘In this paper, a new numerical method for solving fractional differential equations(FDEs) is presented. The method is based upon the fractional Taylor basis approximations. The operational matrix of the fractional integration for the fractional Taylor basis is introduced. This matrix is then utilized to reduce the solution of the fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of this technique.
基金The authors are grateful to the editor,the associate editor and the anonymous reviewers for their constructive and helpful comments.This work was supported by the Zhejiang Provincial Natural Science Foundation of China(No.LY18A010026),the National Natural Science Foundation of China(No.11701304,11526117)Zhejiang Provincial Natural Science Foundation of China(No.LQ16A010006)+1 种基金the Natural Science Foundation of Ningbo City,China(No.2017A610143)the Natural Science Foundation of Ningbo City,China(2018A610195).
文摘An Euler wavelets method is proposed to solve a class of nonlinear variable order fractional differential equations in this paper.The properties of Euler wavelets and their operational matrix together with a family of piecewise functions are first presented.Then they are utilized to reduce the problem to the solution of a nonlinear system of algebraic equations.And the convergence of the Euler wavelets basis is given.The method is computationally attractive and some numerical examples are provided to illustrate its high accuracy.
文摘In this paper, operational matrices of Bernstein polynomials (BPs) are presented for solving the non-linear fractional Logistic differential equation (FLDE). The fractional derivative is described in the Riemann-Liouville sense. The operational matrices for the fractional integration in the Riemann-Liouville sense and the product are used to reduce FLDE to the solution of non-linear system of algebraic equations using Newton iteration method. Numerical results are introduced to satisfy the accuracy and the applicability of the proposed method.
文摘The basic aim of this paper is to introduce and describe an efficient numerical scheme based on spectral approach coupled with Chebyshev wavelets for the approximate solutions of Klein-Gordon and Sine-Gordon equations. The main characteristic is that, it converts the given problem into a system of algebraic equations that can be solved easily with any of the usual methods. To show the accuracy and the efficiency of the method, several benchmark problems are implemented and the comparisons are given with other methods existing in the recent literature. The results of numerical tests confirm that the proposed method is superior to other existing ones and is highly accurate
基金Supported by the National Natural Science Foundation of China(No.29906010).
文摘The performance of analytical derivative and sparse matrix techniques applied to a traditional dense sequential quadratic programming (SQP) is studied, and the strategy utilizing those techniques is also presented.Computational results on two typical chemical optimization problems demonstrate significant enhancement in efficiency, which shows this strategy is promising and suitable for large-scale process optimization problems.
文摘We study nonhomogeneous systems of linear conformable fractional differential equations with pure delay.By using new conformable delayed matrix functions and the method of variation,we obtain a representation of their solutions.As an application,we derive a finite time stability result using the representation of solutions and a norm estimation of the conformable delayedmatrix functions.The obtained results are new,and they extend and improve some existing ones.Finally,an example is presented to illustrate the validity of our theoretical results.