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Efficient Solutions of Coupled Matrix and Matrix Differential Equations
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作者 Zeyad Al-Zhour 《Intelligent Control and Automation》 2012年第2期176-187,共12页
In Kronecker products works, matrices are sometimes regarded as vectors and vectors are sometimes made in to matrices. To be precise about these reshaping we use the vector and diagonal extraction operators. In the pr... In Kronecker products works, matrices are sometimes regarded as vectors and vectors are sometimes made in to matrices. To be precise about these reshaping we use the vector and diagonal extraction operators. In the present paper, the results are organized in the following ways. First, we formulate the coupled matrix linear least-squares problem and present the efficient solutions of this problem that arises in multistatic antenna array processing problem. Second, we extend the use of connection between the Hadamard (Kronecker) product and diagonal extraction (vector) operator in order to construct a computationally-efficient solution of non-homogeneous coupled matrix differential equations that useful in various applications. Finally, the analysis indicates that the Kronecker (Khatri-Rao) structure method can achieve good efficient while the Hadamard structure method achieve more efficient when the unknown matrices are diagonal. 展开更多
关键词 matrix Products LEAST-SQUARES Problem coupled matrix and matrix differential equations DIAGONAL Extraction OPERATOR
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HERMITE MATRIX POLYNOMIALS AND SECOND ORDER MATRIX DIFFERENTIAL EQUATIONS 被引量:6
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作者 L.Jódar R.Company 《Analysis in Theory and Applications》 1996年第2期20-30,共11页
In this paper we introduce the class of Hermite's matrix polynomials which appear as finite series solutions of second order matrix differential equations Y'-xAY'+BY=0.An explicit expression for the Hermit... In this paper we introduce the class of Hermite's matrix polynomials which appear as finite series solutions of second order matrix differential equations Y'-xAY'+BY=0.An explicit expression for the Hermite matrix polynomials,the orthogonality property and a Rodrigues' formula are given. 展开更多
关键词 exp HERMITE matrix POLYNOMIALS and SECOND ORDER matrix differential equations
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Hermite Matrix Polynomial Collocation Method for Linear Complex Differential Equations and Some Comparisons 被引量:1
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作者 Mina Bagherpoorfard Fahime Akhavan Ghassabzade 《Journal of Applied Mathematics and Physics》 2013年第5期58-64,共7页
In this paper, we introduce a Hermite operational matrix collocation method for solving higher-order linear complex differential equations in rectangular or elliptic domains. We show that based on a linear algebra the... In this paper, we introduce a Hermite operational matrix collocation method for solving higher-order linear complex differential equations in rectangular or elliptic domains. We show that based on a linear algebra theorem, the use of different polynomials such as Hermite, Bessel and Taylor in polynomial collocation methods for solving differential equations leads to an equal solution, and the difference in the numerical results arises from the difference in the coefficient matrix of final linear systems of equations. Some numerical examples will also be given. 展开更多
关键词 APPROXIMATE Solution COLLOCATION Methods Complex differential equations HERMITE POLYNOMIALS Operational matrix
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ON DETERMINATION OF THE IMPULSIVE SOLUTIONS AND IMPULSIVE PROPERTIES TO LINEAR NON-HOMOGENEOUS MATRIX DIFFERENTIAL EQUATIONS
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作者 Tan Liansheng 《Acta Mathematica Scientia》 SCIE CSCD 2000年第2期261-272,共12页
This note contains three main results.Firstly,a complete solution of the Linear Non-Homogeneous Matrix Differential Equations(LNHMDEs)is presented that takes into account both the non-zero initial conditions of the ps... This note contains three main results.Firstly,a complete solution of the Linear Non-Homogeneous Matrix Differential Equations(LNHMDEs)is presented that takes into account both the non-zero initial conditions of the pseudo state and the nonzero initial conditions of the input.Secondly,in order to characterise the dynamics of the LNHMDEs correctly,some important concepts such as the state,slow state(smooth state)and fast state(impulsive state)are generalized to the LNHMDE case and the solution of the LNHMDEs is separated into the smooth(slow)response and the fast(implusive)response.As a third result,a new characterization of the impulsive free initial conditions of the LNHMDEs is given. 展开更多
关键词 linear matrix differential equation impulsive and smooth solution impulsive free initial conditions {1}-inverse
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The Exact Solutions of Such Coupled Linear Matrix Fractional Differential Equations of Diagonal Unknown Matrices by Using Hadamard Product
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作者 Zayed Al-Zuhiri Zeyad Al-Zhour Khaled Jaber 《Journal of Applied Mathematics and Physics》 2016年第2期432-442,共11页
In this paper, we present the general exact solutions of such coupled system of matrix fractional differential equations for diagonal unknown matrices in Caputo sense by using vector extraction operators and Hadamard ... In this paper, we present the general exact solutions of such coupled system of matrix fractional differential equations for diagonal unknown matrices in Caputo sense by using vector extraction operators and Hadamard product. Some illustrated examples are also given to show our new approach. 展开更多
关键词 Fractional Operators matrix Fractional differential equations Hadamard Product Vector Extraction Operator
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An Extended Numerical Method by Stancu Polynomials for Solution of Integro-Differential Equations Arising in Oscillating Magnetic Fields
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作者 Neşe İşler Acar 《Advances in Pure Mathematics》 2024年第10期785-796,共12页
In this study, the Bernstein collocation method has been expanded to Stancu collocation method for numerical solution of the charged particle motion for certain configurations of oscillating magnetic fields modelled b... In this study, the Bernstein collocation method has been expanded to Stancu collocation method for numerical solution of the charged particle motion for certain configurations of oscillating magnetic fields modelled by a class of linear integro-differential equations. As the method has been improved, the Stancu polynomials that are generalization of the Bernstein polynomials have been used. The method has been tested on a physical problem how the method can be applied. Moreover, numerical results of the method have been compared with the numerical results of the other methods to indicate the efficiency of the method. 展开更多
关键词 Stancu Polynomials Collocation Method Integro-differential equations Linear Equation Systems matrix equations
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NEW OSCILLATION CRITERIA FOR SECOND-ORDER NONLINEAR MATRIX DIFFERENTIAL EQUATIONS 被引量:2
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作者 Xu Yancong Meng Fanwei 《Applied Mathematics(A Journal of Chinese Universities)》 SCIE CSCD 2006年第3期313-319,共7页
Some new oscillation criteria are established for the second-order matrix differential system (r(t)Z'(t))' + p(t)Z'(t) + Q(t)F(Z'(t))G(Z(t)) = 0, t ≥ to 〉 0, are different from most known ... Some new oscillation criteria are established for the second-order matrix differential system (r(t)Z'(t))' + p(t)Z'(t) + Q(t)F(Z'(t))G(Z(t)) = 0, t ≥ to 〉 0, are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of [to,∞), rather than on the whole half-line. The results weaken the condition of Q(t) and generalize some well-known results of Wong (1999) to nonlinear matrix differential equation. 展开更多
关键词 matrix differential equation OSCILLATION Riccati technique.
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AOR Iterative Method for Coupled Lyapunov Matrix Equations 被引量:3
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作者 ZHANG Shi-jun WANG Shi-heng WANG Ke 《Chinese Quarterly Journal of Mathematics》 2021年第2期141-148,共8页
An AOR(Accelerated Over-Relaxation)iterative method is suggested by introducing one more parameter than SOR(Successive Over-Relaxation)method for solving coupled Lyapunov matrix equations(CLMEs)that come from continuo... An AOR(Accelerated Over-Relaxation)iterative method is suggested by introducing one more parameter than SOR(Successive Over-Relaxation)method for solving coupled Lyapunov matrix equations(CLMEs)that come from continuous-time Markovian jump linear systems.The proposed algorithm improves the convergence rate,which can be seen from the given illustrative examples.The comprehensive theoretical analysis of convergence and optimal parameter needs further investigation. 展开更多
关键词 coupled Lyapunov matrix equations AOR iterative method SOR iterative method Markovian jump systems
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Numerical Solution of Nonlinear Integro-Differential Equations with Initial Conditions by Bernstein Operational Matrix of Derivative
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作者 Behrooz Basirat Mohammad Amin Shahdadi 《International Journal of Modern Nonlinear Theory and Application》 2013年第2期141-149,共9页
In this paper, we present a practical matrix method for solving nonlinear Volterra-Fredholm integro-differential equations under initial conditions in terms of Bernstein polynomials on the interval [0,1]. The nonlinea... In this paper, we present a practical matrix method for solving nonlinear Volterra-Fredholm integro-differential equations under initial conditions in terms of Bernstein polynomials on the interval [0,1]. The nonlinear part is approximated in the form of matrices’ equations by operational matrices of Bernstein polynomials, and the differential part is approximated in the form of matrices’ equations by derivative operational matrix of Bernstein polynomials. Finally, the main equation is transformed into a nonlinear equations system, and the unknown of the main equation is then approximated. We also give some numerical examples to show the applicability of the operational matrices for solving nonlinear Volterra-Fredholm integro-differential equations (NVFIDEs). 展开更多
关键词 BERNSTEIN POLYNOMIAL OPERATIONAL matrix Integro-differential equations
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Coupled Cross-correlation Neural Network Algorithm for Principal Singular Triplet Extraction of a Cross-covariance Matrix 被引量:2
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作者 Xiaowei Feng Xiangyu Kong Hongguang Ma 《IEEE/CAA Journal of Automatica Sinica》 SCIE EI 2016年第2期149-156,共8页
This paper proposes a novel coupled neural network learning algorithm to extract the principal singular triplet (PST) of a cross-correlation matrix between two high-dimensional data streams. We firstly introduce a nov... This paper proposes a novel coupled neural network learning algorithm to extract the principal singular triplet (PST) of a cross-correlation matrix between two high-dimensional data streams. We firstly introduce a novel information criterion (NIC), in which the stationary points are singular triplet of the crosscorrelation matrix. Then, based on Newton's method, we obtain a coupled system of ordinary differential equations (ODEs) from the NIC. The ODEs have the same equilibria as the gradient of NIC, however, only the first PST of the system is stable (which is also the desired solution), and all others are (unstable) saddle points. Based on the system, we finally obtain a fast and stable algorithm for PST extraction. The proposed algorithm can solve the speed-stability problem that plagues most noncoupled learning rules. Moreover, the proposed algorithm can also be used to extract multiple PSTs effectively by using sequential method. © 2014 Chinese Association of Automation. 展开更多
关键词 Clustering algorithms Covariance matrix Data mining differential equations EXTRACTION Learning algorithms Negative impedance converters Newton Raphson method Ordinary differential equations Singular value decomposition
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New matrix method for analyzing vibration and damping effect of sandwich circular cylindrical shell with viscoelastic core 被引量:1
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作者 向宇 黄玉盈 +2 位作者 陆静 袁丽芸 邹时智 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2008年第12期1587-1600,共14页
Based on the linear theories of thin cylindrical shells and viscoelastic materials, a governing equation describing vibration of a sandwich circular cylindrical shell with a viscoelastic core under harmonic excitation... Based on the linear theories of thin cylindrical shells and viscoelastic materials, a governing equation describing vibration of a sandwich circular cylindrical shell with a viscoelastic core under harmonic excitation is derived. The equation can be written as a matrix differential equation of the first order, and is obtained by considering the energy dissipation due to the shear deformation of the viscoelastic core layer and the interaction between all layers. A new matrix method for solving the governing equation is then presented With an extended homogeneous capacity precision integration approach. Having obtained these, vibration characteristics and damping effect of the sandwich cylindrical shell can be studied. The method differs from a recently published work as the state vector in the governing equation is composed of displacements and internal forces of the sandwich shell rather than displacements and their derivatives. So the present method can be applied to solve dynamic problems of the kind of sandwich shells with various boundary conditions and partially constrained layer damping. Numerical examples show that the proposed approach is effective and reliable compared with the existing methods. 展开更多
关键词 constrained layer damping matrix differential equation of first order circular cylindrical shell high precision integration approach
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Near Term Hybrid Quantum Computing Solution to the Matrix Riccati Equations 被引量:1
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作者 Augusto Gonzalez Bonorino Malick Ndiaye Casimer DeCusatis 《Journal of Quantum Computing》 2022年第3期135-146,共12页
The well-known Riccati differential equations play a key role in many fields,including problems in protein folding,control and stabilization,stochastic control,and cybersecurity(risk analysis and malware propaga-tion)... The well-known Riccati differential equations play a key role in many fields,including problems in protein folding,control and stabilization,stochastic control,and cybersecurity(risk analysis and malware propaga-tion).Quantum computer algorithms have the potential to implement faster approximate solutions to the Riccati equations compared with strictly classical algorithms.While systems with many qubits are still under development,there is significant interest in developing algorithms for near-term quantum computers to determine their accuracy and limitations.In this paper,we propose a hybrid quantum-classical algorithm,the Matrix Riccati Solver(MRS).This approach uses a transformation of variables to turn a set of nonlinear differential equation into a set of approximate linear differential equations(i.e.,second order non-constant coefficients)which can in turn be solved using a version of the Harrow-Hassidim-Lloyd(HHL)quantum algorithm for the case of Hermitian matrices.We implement this approach using the Qiskit language and compute near-term results using a 4 qubit IBM Q System quantum computer.Comparisons with classical results and areas for future research are discussed. 展开更多
关键词 Quantum computing matrix ricatti equations differential equations qiskit hybrid algorithm HHL algorithm
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Inverse Differential Operators in Time and Space
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作者 Edwin Eugene Klingman 《Journal of Applied Mathematics and Physics》 2023年第12期3789-3799,共11页
When one function is defined as a differential operation on another function, it’s often desirable to invert the definition, to effectively “undo” the differentiation. A Green’s function approach is often used to ... When one function is defined as a differential operation on another function, it’s often desirable to invert the definition, to effectively “undo” the differentiation. A Green’s function approach is often used to accomplish this, but variations on this theme exist, and we examine a few such variations. The mathematical analysis of  is sought in the form  if such an inverse operator exists, but physics is defined by both mathematical formula and ontological formalism, as I show for an example based on the Dirac equation. Finally, I contrast these “standard” approaches with a novel exact inverse operator for field equations. 展开更多
关键词 matrix-Inverse differential Operators Green’s Functions Dirac’s Equation Wave equations Inverse Curl Operator
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On <i>p</i>and <i>q</i>-Horn’s Matrix Function of Two Complex Variables
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作者 Ayman Shehata 《Applied Mathematics》 2011年第12期1437-1442,共6页
The main aim of this paper is to define and study of a new Horn’s matrix function, say, the p and q-Horn’s matrix function of two complex variables. The radius of regularity on this function is given when the positi... The main aim of this paper is to define and study of a new Horn’s matrix function, say, the p and q-Horn’s matrix function of two complex variables. The radius of regularity on this function is given when the positive integers p and q are greater than one, an integral representation of pHq 2 is obtained, recurrence relations are established. Finally, we obtain a higher order partial differential equation satisfied by the p and q-Horn’s matrix function. 展开更多
关键词 Hypergeometric matrix FUNCTIONS p and q-Horn’s matrix Function Contiguous Relations matrix FUNCTIONS matrix differential Equation differential Operator
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Legendre Approximation for Solving Linear HPDEs and Comparison with Taylor and Bernoulli Matrix Methods
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作者 Emran Tohidi 《Applied Mathematics》 2012年第5期410-416,共7页
The aim of this study is to give a Legendre polynomial approximation for the solution of the second order linear hyper-bolic partial differential equations (HPDEs) with two variables and constant coefficients. For thi... The aim of this study is to give a Legendre polynomial approximation for the solution of the second order linear hyper-bolic partial differential equations (HPDEs) with two variables and constant coefficients. For this purpose, Legendre matrix method for the approximate solution of the considered HPDEs with specified associated conditions in terms of Legendre polynomials at any point is introduced. The method is based on taking truncated Legendre series of the functions in the equation and then substituting their matrix forms into the given equation. Thereby the basic equation reduces to a matrix equation, which corresponds to a system of linear algebraic equations with unknown Legendre coefficients. The result matrix equation can be solved and the unknown Legendre coefficients can be found approximately. Moreover, the approximated solutions of the proposed method are compared with the Taylor [1] and Bernoulli [2] matrix methods. All of computations are performed on a PC using several programs written in MATLAB 7.12.0. 展开更多
关键词 LEGENDRE Operational matrix of DIFFERENTIATION HYPERBOLIC Partial differential equations LEGENDRE POLYNOMIAL Solutions Double LEGENDRE Series
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Certain <i>pl</i>(<i>m</i>,<i>n</i>)-Kummer Matrix Function of Two Complex Variables under Differential Operator
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作者 Ayman Shehata 《Applied Mathematics》 2013年第1期91-96,共6页
The main aim of this paper is to define and study of a new matrix functions, say, the pl(m,n)-Kummer matrix function of two complex variables. The radius of regularity, recurrence relation and several new results on t... The main aim of this paper is to define and study of a new matrix functions, say, the pl(m,n)-Kummer matrix function of two complex variables. The radius of regularity, recurrence relation and several new results on this function are established when the positive integers p is greater than one. Finally, we obtain a higher order partial differential equation satisfied by the pl(m,n)-Kummer matrix function and some special properties. 展开更多
关键词 HYPERGEOMETRIC matrix FUNCTION pl(m n)-Kummer matrix FUNCTION matrix differential Equation
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Fitting evolutionary process of matrix protein 2 family from influenza A virus using analytical solution of differential equation
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作者 Shao-Min Yan Zhen-Chong Li Guang Wu 《Journal of Biomedical Science and Engineering》 2009年第8期587-593,共7页
The evolution of protein family is a process along the time course, thus any mathematical methods that can describe a process over time could be possible to describe an evolutionary process. In our previously concept-... The evolution of protein family is a process along the time course, thus any mathematical methods that can describe a process over time could be possible to describe an evolutionary process. In our previously concept-initiated study, we attempted to use the differential equation to describe the evolution of hemagglutinins from influenza A viruses, and to discuss various issues related to the building of differential equation. In this study, we attempted not only to use the differential equation to describe the evolution of matrix protein 2 family from influenza A virus, but also to use the analytical solution to fit its evolutionary process. The results showed that the fitting was possible and workable. The fitted model parameters provided a way to further determine the evolutionary dynamics and kinetics, a way to more precisely predict the time of occurrence of mutation, and a way to figure out the interaction between protein family and its environment. 展开更多
关键词 AMINO-ACID Pair PREDICTABILITY differential Equation Evolution FITTING INFLUENZA A Virus matrix Protein 2
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On Horn Matrix Function <i>H<sub>2</sub>(A,A′,B,B′;C;z,w)</i>of Two Complex Variables under Differential Operator
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作者 Mohamed Saleh Metwally Mahmoud Tawfik Mohamed Ayman Shehata 《Advances in Linear Algebra & Matrix Theory》 2018年第2期96-110,共15页
The aim of this paper deals with the study of the Horn matrix function of two complex variables. The convergent properties, an integral representation of H2(A,A′,B,B′;C;z,w) is obtained and recurrence matrix relatio... The aim of this paper deals with the study of the Horn matrix function of two complex variables. The convergent properties, an integral representation of H2(A,A′,B,B′;C;z,w) is obtained and recurrence matrix relations are given. Some result when operating on Horn matrix function with the differential operator D and a solution of certain partial differential equations are established. The Hadamard product of two Horn’s matrix functions is studied, certain results as, the domain of regularity, contiguous functional relations and operating with the differential operator D and D2 are established. 展开更多
关键词 HYPERGEOMETRIC matrix FUNCTION HORN matrix FUNCTION Integral Form Recurrence matrix Relation matrix differential Equation differential Operator Hadamard Product
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A Minimum Residual Based Gradient Iterative Method for a Class of Matrix Equations
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作者 Qing-qing Zheng 《Acta Mathematicae Applicatae Sinica》 SCIE CSCD 2024年第1期17-34,共18页
In this paper, we present a minimum residual based gradient iterative method for solving a class of matrix equations including Sylvester matrix equations and general coupled matrix equations. The iterative method uses... In this paper, we present a minimum residual based gradient iterative method for solving a class of matrix equations including Sylvester matrix equations and general coupled matrix equations. The iterative method uses a negative gradient as steepest direction and seeks for an optimal step size to minimize the residual norm of next iterate. It is shown that the iterative sequence converges unconditionally to the exact solution for any initial guess and that the norm of the residual matrix and error matrix decrease monotonically. Numerical tests are presented to show the efficiency of the proposed method and confirm the theoretical results. 展开更多
关键词 Sylvester matrix equation coupled matrix equation minimum residual gradient descent convergence analysis
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Roughness of Exponential Estimates for Linear Functional Differential Equations 被引量:9
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作者 Wang Zhen (Anhui University) 《大学数学》 1994年第S1期146-150,共5页
Assume that the fundamental solution matrix U (t, s ) of x’(t)=L(t, x,) satisfies |U(t,s)|≤Ke-e(t-s) for t≥s.If|(t,φ)|≤δ|φ(0)|with δ【a/K, then the fundamental solution matrix of the perturbed equation x’(t)=... Assume that the fundamental solution matrix U (t, s ) of x’(t)=L(t, x,) satisfies |U(t,s)|≤Ke-e(t-s) for t≥s.If|(t,φ)|≤δ|φ(0)|with δ【a/K, then the fundamental solution matrix of the perturbed equation x’(t)=L(t,x,)+(t ,x,) also possesses similar exponential estimate. For α=0, a similar result is given. 展开更多
关键词 Linear functional differential equation FUNDAMENTAL solution matrix
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