This paper is devoted to the study of second order nonlinear difference equations. A Nonlocal Perturbation of a Dirichlet Boundary Value Problem is considered. An exhaustive study of the related Green's function to t...This paper is devoted to the study of second order nonlinear difference equations. A Nonlocal Perturbation of a Dirichlet Boundary Value Problem is considered. An exhaustive study of the related Green's function to the linear part is done. The exact expression of the function is given, moreover the range of parameter for which it has constant sign is obtained. Using this, some existence results for the nonlinear problem are deduced from monotone iterative techniques, the classical Krasnoselski fixed point theorem or by application of recent fixed point theorems that combine both theories.展开更多
In this paper, by virtue of Leray-Lions theorem, we are mainly concerned with the existence of weak solutions to a Dirichlet boundary value problem with the p-Laplacian operator.
We study a functional modelling the progressive lens design,which is a combination of Willmore functional and total Gauss curvature.First,we prove the existence for the minimizers of this class of functionals among th...We study a functional modelling the progressive lens design,which is a combination of Willmore functional and total Gauss curvature.First,we prove the existence for the minimizers of this class of functionals among the class of revolution surfaces rotated by the curves y=f(x)about the x-axis.Then,choosing such a minimiser as background surfaces to approximate the functional by a quadratic functional,we prove the existence and uniqueness of the solution to the Euler-Lagrange equation for the quadratic functionals.Our results not only provide a strictly mathematical proof for numerical methods,but also give a more reasonable and more extensive choice for the background surfaces.展开更多
In this paper,based on physics-informed neural networks(PINNs),a good deep learning neural network framework that can be used to effectively solve the nonlinear evolution partial differential equations(PDEs)and other ...In this paper,based on physics-informed neural networks(PINNs),a good deep learning neural network framework that can be used to effectively solve the nonlinear evolution partial differential equations(PDEs)and other types of nonlinear physical models,we study the nonlinear Schrodinger equation(NLSE)with the generalized PT-symmetric Scarf-Ⅱpotential,which is an important physical model in many fields of nonlinear physics.Firstly,we choose three different initial values and the same Dinchlet boundaiy conditions to solve the NLSE with the generalized PT-symmetric Scarf-Ⅱpotential via the PINN deep learning method,and the obtained results are compared with ttose denved by the toditional numencal methods.Then,we mvestigate effect of two factors(optimization steps and activation functions)on the performance of the PINN deep learning method in the NLSE with the generalized PT-symmetric Scarf-Ⅱpotential.Ultimately,the data-driven coefficient discovery of the generalized PT-symmetric Scarf-Ⅱpotential or the dispersion and nonlinear items of the NLSE with the generalized PT-symmetric Scarf-Ⅱpotential can be approximately ascertained by using the PINN deep learning method.Our results may be meaningful for further investigation of the nonlinear Schrodmger equation with the generalized PT-symmetric Scarf-Ⅱpotential in the deep learning.展开更多
基金partially supported by Ministerio de Educación y Ciencia,Spain,and FEDER,Projects MTM2013-43014-P and MTM 2016-75140-P
文摘This paper is devoted to the study of second order nonlinear difference equations. A Nonlocal Perturbation of a Dirichlet Boundary Value Problem is considered. An exhaustive study of the related Green's function to the linear part is done. The exact expression of the function is given, moreover the range of parameter for which it has constant sign is obtained. Using this, some existence results for the nonlinear problem are deduced from monotone iterative techniques, the classical Krasnoselski fixed point theorem or by application of recent fixed point theorems that combine both theories.
基金supported by the NNSF of China(10971046)Shandong Provincial Natural Science Foundation(ZR2012AQ007)The Project of Shandong Province High Education Science and Tech-nology Program(J09LA55)
文摘In this paper, by virtue of Leray-Lions theorem, we are mainly concerned with the existence of weak solutions to a Dirichlet boundary value problem with the p-Laplacian operator.
基金This work was supported by the National Natural Science Foundation of China(Grant No.11771237).
文摘We study a functional modelling the progressive lens design,which is a combination of Willmore functional and total Gauss curvature.First,we prove the existence for the minimizers of this class of functionals among the class of revolution surfaces rotated by the curves y=f(x)about the x-axis.Then,choosing such a minimiser as background surfaces to approximate the functional by a quadratic functional,we prove the existence and uniqueness of the solution to the Euler-Lagrange equation for the quadratic functionals.Our results not only provide a strictly mathematical proof for numerical methods,but also give a more reasonable and more extensive choice for the background surfaces.
基金supported by the National Natural Science Foundation of China under Grant Nos.11775121,11435005the K.C.Wong Magna Fund of Ningbo University。
文摘In this paper,based on physics-informed neural networks(PINNs),a good deep learning neural network framework that can be used to effectively solve the nonlinear evolution partial differential equations(PDEs)and other types of nonlinear physical models,we study the nonlinear Schrodinger equation(NLSE)with the generalized PT-symmetric Scarf-Ⅱpotential,which is an important physical model in many fields of nonlinear physics.Firstly,we choose three different initial values and the same Dinchlet boundaiy conditions to solve the NLSE with the generalized PT-symmetric Scarf-Ⅱpotential via the PINN deep learning method,and the obtained results are compared with ttose denved by the toditional numencal methods.Then,we mvestigate effect of two factors(optimization steps and activation functions)on the performance of the PINN deep learning method in the NLSE with the generalized PT-symmetric Scarf-Ⅱpotential.Ultimately,the data-driven coefficient discovery of the generalized PT-symmetric Scarf-Ⅱpotential or the dispersion and nonlinear items of the NLSE with the generalized PT-symmetric Scarf-Ⅱpotential can be approximately ascertained by using the PINN deep learning method.Our results may be meaningful for further investigation of the nonlinear Schrodmger equation with the generalized PT-symmetric Scarf-Ⅱpotential in the deep learning.