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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
文摘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.
文摘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.
文摘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.
文摘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.
文摘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.
文摘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.
基金Supported by NECC and NSF of Shandong Proyilice,China(Y2005A06).
文摘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.
基金Supported by Key Scientific Research Project of Colleges and Universities in Henan Province of China(Grant No.20B110012)。
文摘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.
文摘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).
基金supported by the National Natural Science Foundation of China (No. 10662003)the Doctoral Fund of Ministry of Education of China (No. 20040787013)
文摘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.
文摘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.
文摘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.
文摘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.
文摘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.
文摘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.
文摘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.
文摘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.
基金supported by the National Natural Science Foundation of China (No. 12001311)Science Foundation of China University of Petroleum,Beijing (No. 2462021YJRC025)the State Key Laboratory of Petroleum Resources and Prospecting,China University of Petroleum。
文摘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.
基金Research supported by China National Science Foundation
文摘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.