The Lhasa River Basin forms an essential human settlement area in the southern part of the Qinghai-Tibet Plateau.This study employed ecosystem service value(ESV)evaluation model,terrain gradient grading,and Geodetecto...The Lhasa River Basin forms an essential human settlement area in the southern part of the Qinghai-Tibet Plateau.This study employed ecosystem service value(ESV)evaluation model,terrain gradient grading,and Geodetector to analyze land use and ESV in the Lhasa River Basin from 1985 to 2020.The findings reveal that:(1)From 1985 to 2020,grassland was the dominant land use.There was a trend of grassland reduction and the expansion of other land types.(2)ESV has increased over the research period(with a total increase of 0.84%),with higher values in the southeast and lower values in the northwest.Grassland contributed the most to ESV,and climate regulation and hydrological regulation were the ecosystem services that contribute the most to ESV.(3)Natural factors like NDVI and altitude,as well as economic factors like population density and distance from roads,influenced the spatial differentiation of ESV,the explanatory power of NDVI reached up to 0.47.The interaction between factors had a greater impact than individual factors.These research results can provide theoretical support for national spatial planning and ecological environment protection in the Lhasa River Basin and other similar areas.展开更多
New existence results are presented for the singular second-order nonlinear boundary value problems u ' + g(t)f(u) = 0, 0 < t < 1, au(0) - betau ' (0) = 0, gammau(1) + deltau ' (1) = 0 under the cond...New existence results are presented for the singular second-order nonlinear boundary value problems u ' + g(t)f(u) = 0, 0 < t < 1, au(0) - betau ' (0) = 0, gammau(1) + deltau ' (1) = 0 under the conditions 0 less than or equal to f(0)(+) < M-1, m(1) < f(infinity)(-)less than or equal to infinity or 0 less than or equal to f(infinity)(+)< M-1, m(1) < f (-)(0)less than or equal to infinity where f(0)(+) = lim(u -->0)f(u)/u, f(infinity)(-)= lim(u --> infinity)f(u)/u, f(0)(-)= lim(u -->0)f(u)/u, f(infinity)(+) = lim(u --> infinity)f(u)/u, g may be singular at t = 0 and/or t = 1. The proof uses a fixed point theorem in cone theory.展开更多
Using a fixed point theorem in cones, the paper consider the existence of positive solutions for a class of second-order m-point boundary value problem. Sufficient conditions to ensure the existence of double positive...Using a fixed point theorem in cones, the paper consider the existence of positive solutions for a class of second-order m-point boundary value problem. Sufficient conditions to ensure the existence of double positive solutions are obtained. The associated Green function of this problem is also given.展开更多
From 700 m to 1900 m on the northern slope of Changbai Mountain, 13 plots with an interval of 100 m in elevation were investigated to study the variations of population structure and important value of the main edific...From 700 m to 1900 m on the northern slope of Changbai Mountain, 13 plots with an interval of 100 m in elevation were investigated to study the variations of population structure and important value of the main edificators along the elevation gradient. In their core distribution areas, most of the edificators had healthy population structure and could regenerate smoothly except Larix olgensis, but important value of Larix olgensis had no obvious variations with elevation changes, which showed that Larix olgensis had its own particularity and strong adaptability. At high elevation above 1800 m, Betula ermanii was the only species that could form a mono-dominant community. Important values of Pinus koraiensis and Acer mono had similar changing trends, and they had the similar ecological adaptabilities.展开更多
This work was to generate landslide susceptibility maps for the Three Gorges Reservoir(TGR) area, China by using different machine learning models. Three advanced machine learning methods, namely, gradient boosting de...This work was to generate landslide susceptibility maps for the Three Gorges Reservoir(TGR) area, China by using different machine learning models. Three advanced machine learning methods, namely, gradient boosting decision tree(GBDT), random forest(RF) and information value(InV) models, were used, and the performances were assessed and compared. In total, 202 landslides were mapped by using a series of field surveys, aerial photographs, and reviews of historical and bibliographical data. Nine causative factors were then considered in landslide susceptibility map generation by using the GBDT, RF and InV models. All of the maps of the causative factors were resampled to a resolution of 28.5 m. Of the 486289 pixels in the area,28526 pixels were landslide pixels, and 457763 pixels were non-landslide pixels. Finally, landslide susceptibility maps were generated by using the three machine learning models, and their performances were assessed through receiver operating characteristic(ROC) curves, the sensitivity, specificity,overall accuracy(OA), and kappa coefficient(KAPPA). The results showed that the GBDT, RF and In V models in overall produced reasonable accurate landslide susceptibility maps. Among these three methods, the GBDT method outperforms the other two machine learning methods, which can provide strong technical support for producing landslide susceptibility maps in TGR.展开更多
The Da-Xiao Liangshan mountains are critical ecological function areas and essential ecological barriers in the upper reaches of the Yangtze River in China.This study selected a total of six periods of land use land c...The Da-Xiao Liangshan mountains are critical ecological function areas and essential ecological barriers in the upper reaches of the Yangtze River in China.This study selected a total of six periods of land use land cover(LULC)data from 1995 to 2020,to estimate the ecosystem service value(ESV)and analyzed its spatiotemporal evolution and topographic gradient divergence.The results showed that:(1)The ESV increased by 1.1 billion yuan,with an increase rate of 1.47%from 1995 to 2020.Two time periods,2005–2010 and 2015–2020,showed more significant increases than other periods.(2)The elevation and slope of mountainous areas determine the type of land use and further influence the spatial pattern of ESV.(3)Although woodland and grassland are the main land use types of the study area(more than 90%),the hydrological regulation function of the water area partially compensated for the impact of the encroachment of the built-up area on the ESV of grassland.(4)The spatial distribution of ESVs showed an inverted V-shaped characteristic as the topographic gradient increased,with the dominant position being the 5th topographic gradient zone.Finally,this study provided relevant recommendations for ecosystem protection and optimization.The findings of this study clarified the influence of topographical factors on the spatial differentiation of ESV and provided novel insights into ecosystem protection.展开更多
In this article we consider the asymptotic behavior of extreme distribution with the extreme value index γ>0 . The rates of uniform convergence for Fréchet distribution are constructed under the second-order ...In this article we consider the asymptotic behavior of extreme distribution with the extreme value index γ>0 . The rates of uniform convergence for Fréchet distribution are constructed under the second-order regular variation condition.展开更多
The metal and nano-ceramic composite coatings were formed on gray cast iron surface by electrodepositon method.The Ni-Co was used as metal matrix,and the nano-Al2O3 was chosen as second-phase particulates.The gradient...The metal and nano-ceramic composite coatings were formed on gray cast iron surface by electrodepositon method.The Ni-Co was used as metal matrix,and the nano-Al2O3 was chosen as second-phase particulates.The gradient structure of biology material was the model to avoid bad interface bonding and stress distribution,therefore the gradient composite coating was prepared.The morphology of composite coatings was flatter and the microstructure was more com-pact than the pure Ni-Co coatings.Effect of pH value on surface morphology was analyzed,which some microcracks appeared when pH value was greater than 5.The content of codeposition na-no-Al2O3 reached a maximum value at pH value of 3-4,at the same time the properties including mi-crohardness and wear-resistance were analyzed.The result indicated that the mirohardness reached a maximum value and the wear loss volume was less at pH value 3-4.展开更多
The wavelet multiresolution interpolation for continuous functions defined on a finite interval is developed in this study by using a simple alternative of transformation matrix.The wavelet multiresolution interpolati...The wavelet multiresolution interpolation for continuous functions defined on a finite interval is developed in this study by using a simple alternative of transformation matrix.The wavelet multiresolution interpolation Galerkin method that applies this interpolation to represent the unknown function and nonlinear terms independently is proposed to solve the boundary value problems with the mixed Dirichlet-Robin boundary conditions and various nonlinearities,including transcendental ones,in which the discretization process is as simple as that in solving linear problems,and only common two-term connection coefficients are needed.All matrices are independent of unknown node values and lead to high efficiency in the calculation of the residual and Jacobian matrices needed in Newton’s method,which does not require numerical integration in the resulting nonlinear discrete system.The validity of the proposed method is examined through several nonlinear problems with interior or boundary layers.The results demonstrate that the proposed wavelet method shows excellent accuracy and stability against nonuniform grids,and high resolution of localized steep gradients can be achieved by using local refined multiresolution grids.In addition,Newton’s method converges rapidly in solving the nonlinear discrete system created by the proposed wavelet method,including the initial guess far from real solutions.展开更多
This paper considers eigenstructure assignment in second-order linear systems via proportional plus derivative feedback. It is shown that the problem is closely related to a type of so-called second-order Sylvester ma...This paper considers eigenstructure assignment in second-order linear systems via proportional plus derivative feedback. It is shown that the problem is closely related to a type of so-called second-order Sylvester matrix equations. Through establishing two general parametric solutions to this type of matrix equations, two complete parametric methods for the proposed eigenstructure assignment problem are presented. Both methods give simple complete parametric expressions for the feedback gains and the closed-loop eigenvector matrices. The first one mainly depends on a series of singular value decompositions, and is thus numerically simple and reliable; the second one utilizes the right factorization of the system, and allows the closed-loop eigenvalues to be set undetermined and sought via certain optimization procedures. An example shows the effectiveness of the proposed approaches. Keywords Second-order linear systems - Eigenstructure assignment - Proportional plus derivative feedback - Parametric solution - Singular value decompoition - Right factorization This work was supported in part by the Chinese Outstanding Youth Foundation (No.69504002).展开更多
A second-order dual problem is formulated for a class of continuous programming problem in which both objective and constrained functions contain support functions, hence it is nondifferentiable. Under second-order in...A second-order dual problem is formulated for a class of continuous programming problem in which both objective and constrained functions contain support functions, hence it is nondifferentiable. Under second-order invexity and second-order pseudoinvexity, weak, strong and converse duality theorems are established for this pair of dual problems. Special cases are deduced and a pair of dual continuous problems with natural boundary values is constructed. A close relationship between duality results of our problems and those of the corresponding (static) nonlinear programming problem with support functions is briefly outlined.展开更多
A type of high-order integral observers for matrix second-order linear systems is proposed on the basis of generalized eigenstructure assignment via unified parametric approaches. Through establishing two general para...A type of high-order integral observers for matrix second-order linear systems is proposed on the basis of generalized eigenstructure assignment via unified parametric approaches. Through establishing two general parametric solutions to this type of generalized matrix second-order Sylvester matrix equations, two unified complete parametric methods for the proposed observer design problem are presented. Both methods give simple complete parametric expressions for the observer gain matrices. The first one mainly depends on a series of singular value decompositions, and is thus numerically simple and reliable; the second one utilizes the fight factorization of the system, and allows eigenvalues of the error system to be set undetermined and sought via certain optimization procedures. A spring-mass-dashpot system is utilized to illustrate the design procedure and show the effect of the proposed approach.展开更多
This paper studies the Browder-Tikhonov regularization of a second-order evolution hemivariational inequality （SOEHVI） with non-coercive operators. With duality mapping, the regularized formulations and a derived fi...This paper studies the Browder-Tikhonov regularization of a second-order evolution hemivariational inequality （SOEHVI） with non-coercive operators. With duality mapping, the regularized formulations and a derived first-order evolution hemivariational inequality （FOEHVI） for the problem considered are presented. By applying the Browder-Tikhonov regularization method to the derived FOEHVI, a sequence of regularized solutions to the regularized SOEHVI is constructed, and the strong convergence of the whole sequence of regularized solutions to a solution to the problem is proved.展开更多
In this paper, we are concerned with the numerical solution of second-order partial differential equations. We analyse the use of the Sine Transform precondilioners for the solution of linear systems arising from the ...In this paper, we are concerned with the numerical solution of second-order partial differential equations. We analyse the use of the Sine Transform precondilioners for the solution of linear systems arising from the discretization of p.d.e. via the preconditioned conjugate gradient method. For the second-order partial differential equations with Dirichlel boundary conditions, we prove that the condition number of the preconditioned system is O(1) while the condition number of the original system is O(m 2) Here m is the number of interior gridpoints in each direction. Such condition number produces a linear convergence rale.展开更多
We establish the existence of positive solutions for singular boundary value problems of coupled systems? The proof relies on Schauder’s fixed point theorem. Some recent results in the literature are generalized and ...We establish the existence of positive solutions for singular boundary value problems of coupled systems? The proof relies on Schauder’s fixed point theorem. Some recent results in the literature are generalized and improved.展开更多
We propose and analyze a single-interval Legendre-Gauss-Radau(LGR)spectral collocation method for nonlinear second-order initial value problems of ordinary differential equations.We design an efficient iterative algor...We propose and analyze a single-interval Legendre-Gauss-Radau(LGR)spectral collocation method for nonlinear second-order initial value problems of ordinary differential equations.We design an efficient iterative algorithm and prove spectral convergence for the single-interval LGR collocation method.For more effective implementation,we propose a multi-interval LGR spectral collocation scheme,which provides us great flexibility with respect to the local time steps and local approximation degrees.Moreover,we combine the multi-interval LGR collocation method in time with the Legendre-Gauss-Lobatto collocation method in space to obtain a space-time spectral collocation approximation for nonlinear second-order evolution equations.Numerical results show that the proposed methods have high accuracy and excellent long-time stability.Numerical comparison between our methods and several commonly used methods are also provided.展开更多
基金supported by the National Natural Science Foundation of China(Grant No.U20A20112)Construction of Talent Innovation Team and Laboratory Platform of Tibet University-Construction of Plateau Geothermal New Energy Innovation Team and Laboratory Platform(Grant No.2022ZDTD10)Central Support for Local Ministry and Regional Joint Construction/First-class Everest Construction Project-Construction of Geological Resources and Geological Engineering Characteristics(Grant No.Tibetan Finance Pre-indication[2022]No.1).
文摘The Lhasa River Basin forms an essential human settlement area in the southern part of the Qinghai-Tibet Plateau.This study employed ecosystem service value(ESV)evaluation model,terrain gradient grading,and Geodetector to analyze land use and ESV in the Lhasa River Basin from 1985 to 2020.The findings reveal that:(1)From 1985 to 2020,grassland was the dominant land use.There was a trend of grassland reduction and the expansion of other land types.(2)ESV has increased over the research period(with a total increase of 0.84%),with higher values in the southeast and lower values in the northwest.Grassland contributed the most to ESV,and climate regulation and hydrological regulation were the ecosystem services that contribute the most to ESV.(3)Natural factors like NDVI and altitude,as well as economic factors like population density and distance from roads,influenced the spatial differentiation of ESV,the explanatory power of NDVI reached up to 0.47.The interaction between factors had a greater impact than individual factors.These research results can provide theoretical support for national spatial planning and ecological environment protection in the Lhasa River Basin and other similar areas.
文摘New existence results are presented for the singular second-order nonlinear boundary value problems u ' + g(t)f(u) = 0, 0 < t < 1, au(0) - betau ' (0) = 0, gammau(1) + deltau ' (1) = 0 under the conditions 0 less than or equal to f(0)(+) < M-1, m(1) < f(infinity)(-)less than or equal to infinity or 0 less than or equal to f(infinity)(+)< M-1, m(1) < f (-)(0)less than or equal to infinity where f(0)(+) = lim(u -->0)f(u)/u, f(infinity)(-)= lim(u --> infinity)f(u)/u, f(0)(-)= lim(u -->0)f(u)/u, f(infinity)(+) = lim(u --> infinity)f(u)/u, g may be singular at t = 0 and/or t = 1. The proof uses a fixed point theorem in cone theory.
文摘Using a fixed point theorem in cones, the paper consider the existence of positive solutions for a class of second-order m-point boundary value problem. Sufficient conditions to ensure the existence of double positive solutions are obtained. The associated Green function of this problem is also given.
基金This research was supported by Chinese Academy of Sciences '100 people'project National Natural Science Foundation of China (39970123) and Changbai Mountain Open Research Station.
文摘From 700 m to 1900 m on the northern slope of Changbai Mountain, 13 plots with an interval of 100 m in elevation were investigated to study the variations of population structure and important value of the main edificators along the elevation gradient. In their core distribution areas, most of the edificators had healthy population structure and could regenerate smoothly except Larix olgensis, but important value of Larix olgensis had no obvious variations with elevation changes, which showed that Larix olgensis had its own particularity and strong adaptability. At high elevation above 1800 m, Betula ermanii was the only species that could form a mono-dominant community. Important values of Pinus koraiensis and Acer mono had similar changing trends, and they had the similar ecological adaptabilities.
基金This work was supported in part by the National Natural Science Foundation of China(61601418,41602362,61871259)in part by the Opening Foundation of Hunan Engineering and Research Center of Natural Resource Investigation and Monitoring(2020-5)+1 种基金in part by the Qilian Mountain National Park Research Center(Qinghai)(grant number:GKQ2019-01)in part by the Geomatics Technology and Application Key Laboratory of Qinghai Province,Grant No.QHDX-2019-01.
文摘This work was to generate landslide susceptibility maps for the Three Gorges Reservoir(TGR) area, China by using different machine learning models. Three advanced machine learning methods, namely, gradient boosting decision tree(GBDT), random forest(RF) and information value(InV) models, were used, and the performances were assessed and compared. In total, 202 landslides were mapped by using a series of field surveys, aerial photographs, and reviews of historical and bibliographical data. Nine causative factors were then considered in landslide susceptibility map generation by using the GBDT, RF and InV models. All of the maps of the causative factors were resampled to a resolution of 28.5 m. Of the 486289 pixels in the area,28526 pixels were landslide pixels, and 457763 pixels were non-landslide pixels. Finally, landslide susceptibility maps were generated by using the three machine learning models, and their performances were assessed through receiver operating characteristic(ROC) curves, the sensitivity, specificity,overall accuracy(OA), and kappa coefficient(KAPPA). The results showed that the GBDT, RF and In V models in overall produced reasonable accurate landslide susceptibility maps. Among these three methods, the GBDT method outperforms the other two machine learning methods, which can provide strong technical support for producing landslide susceptibility maps in TGR.
基金supported by the National Natural Science Foundation of China(Grant No.52078423)the Sichuan Province Science and Technology Support Program(Grant No.2020YFS0309 and 2020YFS0054)+1 种基金the China Engineering Science and Technology Strategic Consulting Project(Grant No.2022JDR0356)the Key Research Institution of Philosophy and Social Sciences in Sichuan Province:Research Center of National Parks(Grant No.GJGY2023-YB001)。
文摘The Da-Xiao Liangshan mountains are critical ecological function areas and essential ecological barriers in the upper reaches of the Yangtze River in China.This study selected a total of six periods of land use land cover(LULC)data from 1995 to 2020,to estimate the ecosystem service value(ESV)and analyzed its spatiotemporal evolution and topographic gradient divergence.The results showed that:(1)The ESV increased by 1.1 billion yuan,with an increase rate of 1.47%from 1995 to 2020.Two time periods,2005–2010 and 2015–2020,showed more significant increases than other periods.(2)The elevation and slope of mountainous areas determine the type of land use and further influence the spatial pattern of ESV.(3)Although woodland and grassland are the main land use types of the study area(more than 90%),the hydrological regulation function of the water area partially compensated for the impact of the encroachment of the built-up area on the ESV of grassland.(4)The spatial distribution of ESVs showed an inverted V-shaped characteristic as the topographic gradient increased,with the dominant position being the 5th topographic gradient zone.Finally,this study provided relevant recommendations for ecosystem protection and optimization.The findings of this study clarified the influence of topographical factors on the spatial differentiation of ESV and provided novel insights into ecosystem protection.
文摘In this article we consider the asymptotic behavior of extreme distribution with the extreme value index γ>0 . The rates of uniform convergence for Fréchet distribution are constructed under the second-order regular variation condition.
基金Funded by the National Natural Science Foundation of China(No.50635030)the National Basic Research of China (No.2007CB616913)
文摘The metal and nano-ceramic composite coatings were formed on gray cast iron surface by electrodepositon method.The Ni-Co was used as metal matrix,and the nano-Al2O3 was chosen as second-phase particulates.The gradient structure of biology material was the model to avoid bad interface bonding and stress distribution,therefore the gradient composite coating was prepared.The morphology of composite coatings was flatter and the microstructure was more com-pact than the pure Ni-Co coatings.Effect of pH value on surface morphology was analyzed,which some microcracks appeared when pH value was greater than 5.The content of codeposition na-no-Al2O3 reached a maximum value at pH value of 3-4,at the same time the properties including mi-crohardness and wear-resistance were analyzed.The result indicated that the mirohardness reached a maximum value and the wear loss volume was less at pH value 3-4.
基金supported by the National Natural Science Foundation of China(Nos.12172154 and 11925204)the 111 Project of China(No.B14044)the National Key Project of China(No.GJXM92579)。
文摘The wavelet multiresolution interpolation for continuous functions defined on a finite interval is developed in this study by using a simple alternative of transformation matrix.The wavelet multiresolution interpolation Galerkin method that applies this interpolation to represent the unknown function and nonlinear terms independently is proposed to solve the boundary value problems with the mixed Dirichlet-Robin boundary conditions and various nonlinearities,including transcendental ones,in which the discretization process is as simple as that in solving linear problems,and only common two-term connection coefficients are needed.All matrices are independent of unknown node values and lead to high efficiency in the calculation of the residual and Jacobian matrices needed in Newton’s method,which does not require numerical integration in the resulting nonlinear discrete system.The validity of the proposed method is examined through several nonlinear problems with interior or boundary layers.The results demonstrate that the proposed wavelet method shows excellent accuracy and stability against nonuniform grids,and high resolution of localized steep gradients can be achieved by using local refined multiresolution grids.In addition,Newton’s method converges rapidly in solving the nonlinear discrete system created by the proposed wavelet method,including the initial guess far from real solutions.
文摘This paper considers eigenstructure assignment in second-order linear systems via proportional plus derivative feedback. It is shown that the problem is closely related to a type of so-called second-order Sylvester matrix equations. Through establishing two general parametric solutions to this type of matrix equations, two complete parametric methods for the proposed eigenstructure assignment problem are presented. Both methods give simple complete parametric expressions for the feedback gains and the closed-loop eigenvector matrices. The first one mainly depends on a series of singular value decompositions, and is thus numerically simple and reliable; the second one utilizes the right factorization of the system, and allows the closed-loop eigenvalues to be set undetermined and sought via certain optimization procedures. An example shows the effectiveness of the proposed approaches. Keywords Second-order linear systems - Eigenstructure assignment - Proportional plus derivative feedback - Parametric solution - Singular value decompoition - Right factorization This work was supported in part by the Chinese Outstanding Youth Foundation (No.69504002).
文摘A second-order dual problem is formulated for a class of continuous programming problem in which both objective and constrained functions contain support functions, hence it is nondifferentiable. Under second-order invexity and second-order pseudoinvexity, weak, strong and converse duality theorems are established for this pair of dual problems. Special cases are deduced and a pair of dual continuous problems with natural boundary values is constructed. A close relationship between duality results of our problems and those of the corresponding (static) nonlinear programming problem with support functions is briefly outlined.
基金This work was supported by the Chinese National Natural Science Foundation ( No. 69925308).
文摘A type of high-order integral observers for matrix second-order linear systems is proposed on the basis of generalized eigenstructure assignment via unified parametric approaches. Through establishing two general parametric solutions to this type of generalized matrix second-order Sylvester matrix equations, two unified complete parametric methods for the proposed observer design problem are presented. Both methods give simple complete parametric expressions for the observer gain matrices. The first one mainly depends on a series of singular value decompositions, and is thus numerically simple and reliable; the second one utilizes the fight factorization of the system, and allows eigenvalues of the error system to be set undetermined and sought via certain optimization procedures. A spring-mass-dashpot system is utilized to illustrate the design procedure and show the effect of the proposed approach.
基金supported by the National Natural Science Foundation of China(Nos.11101069,11171237,11471059,and 81171411)the China Postdoctoral Science Foundation(Nos.2014M552328 and2015T80967)the Scientific Research Foundation for the Returned Overseas Chinese Scholars,State Education Ministry
文摘This paper studies the Browder-Tikhonov regularization of a second-order evolution hemivariational inequality （SOEHVI） with non-coercive operators. With duality mapping, the regularized formulations and a derived first-order evolution hemivariational inequality （FOEHVI） for the problem considered are presented. By applying the Browder-Tikhonov regularization method to the derived FOEHVI, a sequence of regularized solutions to the regularized SOEHVI is constructed, and the strong convergence of the whole sequence of regularized solutions to a solution to the problem is proved.
文摘In this paper, we are concerned with the numerical solution of second-order partial differential equations. We analyse the use of the Sine Transform precondilioners for the solution of linear systems arising from the discretization of p.d.e. via the preconditioned conjugate gradient method. For the second-order partial differential equations with Dirichlel boundary conditions, we prove that the condition number of the preconditioned system is O(1) while the condition number of the original system is O(m 2) Here m is the number of interior gridpoints in each direction. Such condition number produces a linear convergence rale.
文摘We establish the existence of positive solutions for singular boundary value problems of coupled systems? The proof relies on Schauder’s fixed point theorem. Some recent results in the literature are generalized and improved.
基金supported in part by the National Natural Science Foundation of China(Grant Nos.12171322,11771298 and 11871043)the Natural Science Foundation of Shanghai(Grant Nos.21ZR1447200,20ZR1441200 and 22ZR1445500)the Science and Technology Innovation Plan of Shanghai(Grant No.20JC1414200).
文摘We propose and analyze a single-interval Legendre-Gauss-Radau(LGR)spectral collocation method for nonlinear second-order initial value problems of ordinary differential equations.We design an efficient iterative algorithm and prove spectral convergence for the single-interval LGR collocation method.For more effective implementation,we propose a multi-interval LGR spectral collocation scheme,which provides us great flexibility with respect to the local time steps and local approximation degrees.Moreover,we combine the multi-interval LGR collocation method in time with the Legendre-Gauss-Lobatto collocation method in space to obtain a space-time spectral collocation approximation for nonlinear second-order evolution equations.Numerical results show that the proposed methods have high accuracy and excellent long-time stability.Numerical comparison between our methods and several commonly used methods are also provided.
