In the paper, by using the methods of compensated compactness and energy estimate, the convergence of class of fourth and sixth orders singular perturbed, partial differential equations is obtained, and furthermore, t...In the paper, by using the methods of compensated compactness and energy estimate, the convergence of class of fourth and sixth orders singular perturbed, partial differential equations is obtained, and furthermore, the regularity of solutions are improved.展开更多
Linear Least Squares（LLS） problems are particularly difficult to solve because they are frequently ill-conditioned, and involve large quantities of data. Ill-conditioned LLS problems are commonly seen in mathematics...Linear Least Squares（LLS） problems are particularly difficult to solve because they are frequently ill-conditioned, and involve large quantities of data. Ill-conditioned LLS problems are commonly seen in mathematics and geosciences, where regularization algorithms are employed to seek optimal solutions. For many problems, even with the use of regularization algorithms it may be impossible to obtain an accurate solution. Riley and Golub suggested an iterative scheme for solving LLS problems. For the early iteration algorithm, it is difficult to improve the well-conditioned perturbed matrix and accelerate the convergence at the same time. Aiming at this problem, self-adaptive iteration algorithm（SAIA） is proposed in this paper for solving severe ill-conditioned LLS problems. The algorithm is different from other popular algorithms proposed in recent references. It avoids matrix inverse by using Cholesky decomposition, and tunes the perturbation parameter according to the rate of residual error decline in the iterative process. Example shows that the algorithm can greatly reduce iteration times, accelerate the convergence,and also greatly enhance the computation accuracy.展开更多
For the initial boundary value problem about a type of parabolicMonge Ampe re equation of the form (IBVP):{-D tu+( det D^(2)_(x)u) 1/n =f(x,t),(x,t)∈Q= Ω ×(0,T],u(x,t)=(x,t)(x,t)∈ pQ},where Ω is a ...For the initial boundary value problem about a type of parabolicMonge Ampe re equation of the form (IBVP):{-D tu+( det D^(2)_(x)u) 1/n =f(x,t),(x,t)∈Q= Ω ×(0,T],u(x,t)=(x,t)(x,t)∈ pQ},where Ω is a bounded convex domain in R n ,the result in by Ivochkina and Ladyzheskaya is improved in the sense that, under assumptions that the data of the problem possess lower regularity and satisfy lower order compatibility conditions than those in , the existence of classical solution to (IBVP) is still established (see Theorem 1.1 below). This can not be realized by only using the method in . The main additional effort the authors have done is a kind of nonlinear perturbation.展开更多
The Cauchy problem of the generalized Korteweg-de Vries-Benjamin-Ono equation is considered, and low regularity and limit behavior of the solutions are obtained. For k = 1, local well- posedness is obtained for data i...The Cauchy problem of the generalized Korteweg-de Vries-Benjamin-Ono equation is considered, and low regularity and limit behavior of the solutions are obtained. For k = 1, local well- posedness is obtained for data in H^s（R）（s 〉 -3/4）. For k = 2, local result for data in H^S（R）（s 〉1/4） is obtained. For k = 3, local result for data in H^S（R）（s 〉 -1/6） is obtained. Moreover, the solutions of generalized Korteweg-de Vries-Benjamin-Ono equation converge to the solutions of KdV equation if the term of Benjamin-Ono equation tends to zero.展开更多
In this paper, the integral problem for linear and nonlinear wave equations is studied. The equation involves abstract operator A in Hilbert space H. Here, assuming enough smoothness on the initial data and the operat...In this paper, the integral problem for linear and nonlinear wave equations is studied. The equation involves abstract operator A in Hilbert space H. Here, assuming enough smoothness on the initial data and the operators, the existence, uniqueness, regularity properties of solutions are established. By choosing the space H and A, the regularity properties of solutions of a wide class of wave equations in the field of physics are obtained.展开更多
文摘In the paper, by using the methods of compensated compactness and energy estimate, the convergence of class of fourth and sixth orders singular perturbed, partial differential equations is obtained, and furthermore, the regularity of solutions are improved.
基金supported by Open Fund of Engineering Laboratory of Spatial Information Technology of Highway Geological Disaster Early Warning in Hunan Province(Changsha University of Science&Technology,kfj150602)Hunan Province Science and Technology Program Funded Projects,China(2015NK3035)+1 种基金the Land and Resources Department Scientific Research Project of Hunan Province,China(2013-27)the Education Department Scientific Research Project of Hunan Province,China(13C1011)
文摘Linear Least Squares（LLS） problems are particularly difficult to solve because they are frequently ill-conditioned, and involve large quantities of data. Ill-conditioned LLS problems are commonly seen in mathematics and geosciences, where regularization algorithms are employed to seek optimal solutions. For many problems, even with the use of regularization algorithms it may be impossible to obtain an accurate solution. Riley and Golub suggested an iterative scheme for solving LLS problems. For the early iteration algorithm, it is difficult to improve the well-conditioned perturbed matrix and accelerate the convergence at the same time. Aiming at this problem, self-adaptive iteration algorithm（SAIA） is proposed in this paper for solving severe ill-conditioned LLS problems. The algorithm is different from other popular algorithms proposed in recent references. It avoids matrix inverse by using Cholesky decomposition, and tunes the perturbation parameter according to the rate of residual error decline in the iterative process. Example shows that the algorithm can greatly reduce iteration times, accelerate the convergence,and also greatly enhance the computation accuracy.
文摘For the initial boundary value problem about a type of parabolicMonge Ampe re equation of the form (IBVP):{-D tu+( det D^(2)_(x)u) 1/n =f(x,t),(x,t)∈Q= Ω ×(0,T],u(x,t)=(x,t)(x,t)∈ pQ},where Ω is a bounded convex domain in R n ,the result in by Ivochkina and Ladyzheskaya is improved in the sense that, under assumptions that the data of the problem possess lower regularity and satisfy lower order compatibility conditions than those in , the existence of classical solution to (IBVP) is still established (see Theorem 1.1 below). This can not be realized by only using the method in . The main additional effort the authors have done is a kind of nonlinear perturbation.
文摘The Cauchy problem of the generalized Korteweg-de Vries-Benjamin-Ono equation is considered, and low regularity and limit behavior of the solutions are obtained. For k = 1, local well- posedness is obtained for data in H^s（R）（s 〉 -3/4）. For k = 2, local result for data in H^S（R）（s 〉1/4） is obtained. For k = 3, local result for data in H^S（R）（s 〉 -1/6） is obtained. Moreover, the solutions of generalized Korteweg-de Vries-Benjamin-Ono equation converge to the solutions of KdV equation if the term of Benjamin-Ono equation tends to zero.
基金Supported by the Natural Science Foundation of Hebei University,the NationalNatural Science Foundation of China(10471039),the Zhejiang Provincial Natural Science Foundationof China(M103087)
文摘In this paper, the integral problem for linear and nonlinear wave equations is studied. The equation involves abstract operator A in Hilbert space H. Here, assuming enough smoothness on the initial data and the operators, the existence, uniqueness, regularity properties of solutions are established. By choosing the space H and A, the regularity properties of solutions of a wide class of wave equations in the field of physics are obtained.
