In this paper we study a matrix equation AX+BX=C(I)over an arbitrary skew field,and give a consistency criterion of(I)and an explicit expression of general solutions of(I).A convenient,simple and practical method of s...In this paper we study a matrix equation AX+BX=C(I)over an arbitrary skew field,and give a consistency criterion of(I)and an explicit expression of general solutions of(I).A convenient,simple and practical method of solving(I)is also given.As a particular case,we also give a simple method of finding a system of fundamental solutions of a homogeneous system of right linear equations over a skew field.展开更多
We apply the local method of fundamental solutions(LMFS)to boundary value problems(BVPs)for the Laplace and homogeneous biharmonic equations in annuli.By appropriately choosing the collocation points,the LMFS discreti...We apply the local method of fundamental solutions(LMFS)to boundary value problems(BVPs)for the Laplace and homogeneous biharmonic equations in annuli.By appropriately choosing the collocation points,the LMFS discretization yields sparse block circulant system matrices.As a result,matrix decomposition algorithms(MDAs)and fast Fourier transforms(FFTs)can be used for the solution of the systems resulting in considerable savings in both computational time and storage requirements.The accuracy of the method and its ability to solve large scale problems are demonstrated by applying it to several numerical experiments.展开更多
It is known that the solution to a Cauchy problem of linear differential equations:x'(t)=A(t)x(t),with x(t0)=x0,can be presented by the matrix exponential as exp(∫_(t0)^(t)A(s)ds)x0,if the commutativity condition...It is known that the solution to a Cauchy problem of linear differential equations:x'(t)=A(t)x(t),with x(t0)=x0,can be presented by the matrix exponential as exp(∫_(t0)^(t)A(s)ds)x0,if the commutativity condition for the coefficient matrix A(t)holds:[∫_(t0)^(t)A(s)ds,A(t)]=0.A natural question is whether this is true without the commutativity condition.To give a definite answer to this question,we present two classes of illustrative examples of coefficient matrices,which satisfy the chain rule d/dt exp(∫_(t0)^(t)A(s)ds)=A(t)exp(∫_(t0)^(t)A(s)ds),but do not possess the commutativity condition.The presented matrices consist of finite-times continuously differentiable entries or smooth entries.展开更多
文摘In this paper we study a matrix equation AX+BX=C(I)over an arbitrary skew field,and give a consistency criterion of(I)and an explicit expression of general solutions of(I).A convenient,simple and practical method of solving(I)is also given.As a particular case,we also give a simple method of finding a system of fundamental solutions of a homogeneous system of right linear equations over a skew field.
文摘We apply the local method of fundamental solutions(LMFS)to boundary value problems(BVPs)for the Laplace and homogeneous biharmonic equations in annuli.By appropriately choosing the collocation points,the LMFS discretization yields sparse block circulant system matrices.As a result,matrix decomposition algorithms(MDAs)and fast Fourier transforms(FFTs)can be used for the solution of the systems resulting in considerable savings in both computational time and storage requirements.The accuracy of the method and its ability to solve large scale problems are demonstrated by applying it to several numerical experiments.
基金supported in part by the Established Researcher Grant and the CAS Faculty Development Grant of the University of South Florida,Chunhui Plan of the Ministry of Education of China,Wang Kuancheng Foundation,the National Natural Science Foundation of China(Grant Nos.10332030,10472091 and 10502042)the Doctorate Foundation of Northwestern Polytechnical University(Grant No.CX200616).
文摘It is known that the solution to a Cauchy problem of linear differential equations:x'(t)=A(t)x(t),with x(t0)=x0,can be presented by the matrix exponential as exp(∫_(t0)^(t)A(s)ds)x0,if the commutativity condition for the coefficient matrix A(t)holds:[∫_(t0)^(t)A(s)ds,A(t)]=0.A natural question is whether this is true without the commutativity condition.To give a definite answer to this question,we present two classes of illustrative examples of coefficient matrices,which satisfy the chain rule d/dt exp(∫_(t0)^(t)A(s)ds)=A(t)exp(∫_(t0)^(t)A(s)ds),but do not possess the commutativity condition.The presented matrices consist of finite-times continuously differentiable entries or smooth entries.