A relatively high formation pressure gradient can exist in seepage flow in low-permeable porous media with a threshold pressure gradient, and a significant error may then be caused in the model computation by neglecti...A relatively high formation pressure gradient can exist in seepage flow in low-permeable porous media with a threshold pressure gradient, and a significant error may then be caused in the model computation by neglecting the quadratic pressure gradient term in the governing equations. Based on these concerns, in consideration of the quadratic pressure gradient term, a basic moving boundary model is constructed for a one-dimensional seepage flow problem with a threshold pressure gradient. Owing to a strong nonlinearity and the existing moving boundary in the mathematical model, a corresponding numerical solution method is presented. First, a spatial coordinate transformation method is adopted in order to transform the system of partial differential equa- tions with moving boundary conditions into a closed system with fixed boundary conditions; then the solution can be sta- bly numerically obtained by a fully implicit finite-difference method. The validity of the numerical method is verified by a published exact analytical solution. Furthermore, to compare with Darcy＇s flow problem, the exact analytical solution for the case of Darcy＇s flow considering the quadratic pressure gradient term is also derived by an inverse Laplace transform. A comparison of these model solutions leads to the conclu- sion that such moving boundary problems must incorporate the quadratic pressure gradient term in their governing equa- tions; the sensitive effects of the quadratic pressure gradient term tend to diminish, with the dimensionless threshold pres- sure gradient increasing for the one-dimensional problem.展开更多
We consider a class of nonlinear parabolic equations whose prototype is ut-Δu=b(x,t)·■+γ|■u|^(2)-divF(x,t)+f(x,t),(x,t)∈ΩT,u(x,t)=∈ГT,u(x,0)=u0(x),x∈Ω where the functions|b(x,t)|^(2),|F(x,t)|^(2),f(x,t)...We consider a class of nonlinear parabolic equations whose prototype is ut-Δu=b(x,t)·■+γ|■u|^(2)-divF(x,t)+f(x,t),(x,t)∈ΩT,u(x,t)=∈ГT,u(x,0)=u0(x),x∈Ω where the functions|b(x,t)|^(2),|F(x,t)|^(2),f(x,t)lie in the space Lr(0,T;Lq(Ω)),γis a positive constant.The purpose of this paper is to prove,under suitable assumptions on the integrability of the space Lr(0,T;Lq(Ω))for the source terms and the coefficient of the gradient term,a priori L^(∞)estimate and the existence of bounded solutions.The methods consist of constructing a family of perturbation problems by regularization,Stampacchia’s iterative technique fulfilled by an appropriate nonlinear test function and compactness argument for the limit process.展开更多
The authors study the existence of solution to p-Laplacian equation with nonlinear forcing term under optimal assumptions on the initial data, which are assumed to be measures. The existence of local solution is obtai...The authors study the existence of solution to p-Laplacian equation with nonlinear forcing term under optimal assumptions on the initial data, which are assumed to be measures. The existence of local solution is obtained.展开更多
In this article, we consider the controllability of a quasi-linear heat equation involving gradient terms with Dirichlet boundary conditions in a bounded domain of RN.The results are established by using the variation...In this article, we consider the controllability of a quasi-linear heat equation involving gradient terms with Dirichlet boundary conditions in a bounded domain of RN.The results are established by using the variational methods, the related duality theory and Kakutani Fixed-point Theorem.展开更多
This paper is concerned with exact boundary behavior of large solutions to semilinear elliptic equations △u=b(x)f(u)(C0+|▽u|q),x∈Ω,where Ω is a bounded domain with a smooth boundary in RN,C0≥0,q E [0,2),b∈Cloc...This paper is concerned with exact boundary behavior of large solutions to semilinear elliptic equations △u=b(x)f(u)(C0+|▽u|q),x∈Ω,where Ω is a bounded domain with a smooth boundary in RN,C0≥0,q E [0,2),b∈Clocα(Ω) is positive in but may be vanishing or appropriately singular on the boundary,f∈C[0,∞),f(0)=0,and f is strictly increasing on [0,∞)(or f∈C(R),f(s)> 0,■s∈R,f is strictly increasing on R).We show unified boundary behavior of such solutions to the problem under a new structure condition on f.展开更多
In this paper, a new class of three term memory gradient method with non-monotone line search technique for unconstrained optimization is presented. Global convergence properties of the new methods are discussed. Comb...In this paper, a new class of three term memory gradient method with non-monotone line search technique for unconstrained optimization is presented. Global convergence properties of the new methods are discussed. Combining the quasi-Newton method with the new method, the former is modified to have global convergence property. Numerical results show that the new algorithm is efficient.展开更多
A novel image restoration model coupling with a gradient fidelity term based on adaptive total variation is proposed in this paper. In order to choose proper parameters, the selection criteria were analyzed theoretica...A novel image restoration model coupling with a gradient fidelity term based on adaptive total variation is proposed in this paper. In order to choose proper parameters, the selection criteria were analyzed theoretically, and a simple scheme to demonstrate its validity was adopted experimentally. To make fair comparisons of performances of three models, the same numerical algorithm was used to solve partial differential equations. Both the international standard test image on Lena and HR image of CBERS-02B of Dalian city were used to verify the performance of the model. Experimental results illustrate that the new model not only preserved the edge and important details but also alleviated the staircase effect effectively.展开更多
Online gradient algorithm has been widely used as a learning algorithm for feedforward neural network training. In this paper, we prove a weak convergence theorem of an online gradient algorithm with a penalty term, a...Online gradient algorithm has been widely used as a learning algorithm for feedforward neural network training. In this paper, we prove a weak convergence theorem of an online gradient algorithm with a penalty term, assuming that the training examples are input in a stochastic way. The monotonicity of the error function in the iteration and the boundedness of the weight are both guaranteed. We also present a numerical experiment to support our results.展开更多
In this note,by combining the nice numerical performance of PR and HS methods with the global convergence property of FR method,a class of new restarting three terms conjugate gradient methods is presented.Global conv...In this note,by combining the nice numerical performance of PR and HS methods with the global convergence property of FR method,a class of new restarting three terms conjugate gradient methods is presented.Global convergence properties of the new method with two kinds of common line searches are proved.展开更多
Online gradient methods are widely used for training the weight of neural networks and for other engineering computations. In certain cases, the resulting weight may become very large, causing difficulties in the impl...Online gradient methods are widely used for training the weight of neural networks and for other engineering computations. In certain cases, the resulting weight may become very large, causing difficulties in the implementation of the network by electronic circuits. In this paper we introduce a punishing term into the error function of the training procedure to prevent this situation. The corresponding convergence of the iterative training procedure and the boundedness of the weight sequence are proved. A supporting numerical example is also provided.展开更多
The combined application of organic fertilizer and chemical fertilizer is an effective measure to increase nutrient content of soil plough layer, which must have a profound impact on the deep soil nutrients, especiall...The combined application of organic fertilizer and chemical fertilizer is an effective measure to increase nutrient content of soil plough layer, which must have a profound impact on the deep soil nutrients, especially the contents of nitrogen forms. The purpose of this study was to explore the characteristics of soil nitrogen forms in plough layer and along depth gradient in different fertilization treatments, so as to evaluate the soil quality in spatial dimension, further provid</span><span style="font-family:Verdana;">ing</span><span style="font-family:Verdana;"> a theoretical basis for scientific fertilization and improvement of paddy soil fertility. Here, a 34-year field experiment was conducted with three treatments: without any fertilizer (CK), pure chemical fertilizer (NPK) and chemical fertilizer combined with organic fertilizer (NPKM). We analyzed the content of nitrogen forms in 0 - 100 cm soil depth and their ratios to total nitrogen (TN), and discussed the correlation between nitrogen forms contents and pH, CEC. Results showed that, compared with CK, both NPK and NPKM significantly increased the contents of nitrogen forms in topsoil (soil layer of 0 - 20 cm), especially nitrate nitrogen (NO<sub>3</sub><sup style="margin-left:-6px;">-</sup>-N) content increased by 70% (NPK) and 111% (NPKM), respectively. Although the contents of different nitrogen forms decreased gradually along soil depth gradient, NPKS slowed down the decline rate of TN and alkali-hydrolysable nitrogen (AN) in 0 - 60 cm soil layer, compared to CK. Compared to NPK, NPKM significantly increased the NO<sub>3</sub><sup style="margin-left:-6px;">-</sup>-N/TN ratio in 0 - 20 cm soil layer, but also decreased the content of </span><span><span></span><span style="font-family:Verdana;">-N in 20 - 40 cm, which was beneficial to reduce the risk of nitrogen leaching caused by nitrate leaching into deep layer. The increase of soil pH in NPKM treatment obviously alleviated the problem of soil acidification caused by long-term application of chemical fertilizer. Correlation analysis showed that there was a significant positive correlation between soil nitrogen forms and cation exchange capacity (CEC), but no significant correlation with soil pH. In conclusion, NPKM ensured the nutrients of soil plough layer (0 - 20 cm), also reduced the risk of nitrogen infiltration and nitrogen loss, thus ensur</span><span style="font-family:Verdana;">ing</span><span style="font-family:Verdana;"> the fertility of soil profile.展开更多
In this article, we consider existence and nonexistence of solutions to problem {-△pu＋g（x,u）｜↓△｜^p=f in -Ω,u=0 on Ω with 1〈p〈∞ where f is a positive measurable function which is bounded away from 0 in Ω,...In this article, we consider existence and nonexistence of solutions to problem {-△pu＋g（x,u）｜↓△｜^p=f in -Ω,u=0 on Ω with 1〈p〈∞ where f is a positive measurable function which is bounded away from 0 in Ω, and the domain Ω is a smooth bounded open set in R^N（N≥2）. Especially, under the condition that g（x, s） = 1/｜s｜^α （α〉0） is singular at s = 0, we obtain that α 〈 p is necessary and sufficient for the existence of solutions in W0^1,p（Ω） to problem （0.1） when f is sufficiently regular.展开更多
In this paper, a gradient method with momentum for sigma-pi-sigma neural networks (SPSNN) is considered in order to accelerate the convergence of the learning procedure for the network weights. The momentum coefficien...In this paper, a gradient method with momentum for sigma-pi-sigma neural networks (SPSNN) is considered in order to accelerate the convergence of the learning procedure for the network weights. The momentum coefficient is chosen in an adaptive manner, and the corresponding weak convergence and strong convergence results are proved.展开更多
基金funding by the project (Grant 51404232) sponsored by the National Natural Science Foundation of Chinathe National Science and Technology Major Project (Grant 2011ZX05038003)+1 种基金the China Postdoctoral Science Foundation project (Grant 2014M561074)the China Scholarship Council for its generous financial support of the research
文摘A relatively high formation pressure gradient can exist in seepage flow in low-permeable porous media with a threshold pressure gradient, and a significant error may then be caused in the model computation by neglecting the quadratic pressure gradient term in the governing equations. Based on these concerns, in consideration of the quadratic pressure gradient term, a basic moving boundary model is constructed for a one-dimensional seepage flow problem with a threshold pressure gradient. Owing to a strong nonlinearity and the existing moving boundary in the mathematical model, a corresponding numerical solution method is presented. First, a spatial coordinate transformation method is adopted in order to transform the system of partial differential equa- tions with moving boundary conditions into a closed system with fixed boundary conditions; then the solution can be sta- bly numerically obtained by a fully implicit finite-difference method. The validity of the numerical method is verified by a published exact analytical solution. Furthermore, to compare with Darcy＇s flow problem, the exact analytical solution for the case of Darcy＇s flow considering the quadratic pressure gradient term is also derived by an inverse Laplace transform. A comparison of these model solutions leads to the conclu- sion that such moving boundary problems must incorporate the quadratic pressure gradient term in their governing equa- tions; the sensitive effects of the quadratic pressure gradient term tend to diminish, with the dimensionless threshold pres- sure gradient increasing for the one-dimensional problem.
基金Supported by the National Natural Science Foundation of China(Grant No.11901131)the University-Level Research Fund Project in Guizhou University of Finance and Economics(Grant No.2019XYB08)。
文摘We consider a class of nonlinear parabolic equations whose prototype is ut-Δu=b(x,t)·■+γ|■u|^(2)-divF(x,t)+f(x,t),(x,t)∈ΩT,u(x,t)=∈ГT,u(x,0)=u0(x),x∈Ω where the functions|b(x,t)|^(2),|F(x,t)|^(2),f(x,t)lie in the space Lr(0,T;Lq(Ω)),γis a positive constant.The purpose of this paper is to prove,under suitable assumptions on the integrability of the space Lr(0,T;Lq(Ω))for the source terms and the coefficient of the gradient term,a priori L^(∞)estimate and the existence of bounded solutions.The methods consist of constructing a family of perturbation problems by regularization,Stampacchia’s iterative technique fulfilled by an appropriate nonlinear test function and compactness argument for the limit process.
基金supported by the Fujian Provincial Natural Science Foundation of China (No. Z0511048)
文摘The authors study the existence of solution to p-Laplacian equation with nonlinear forcing term under optimal assumptions on the initial data, which are assumed to be measures. The existence of local solution is obtained.
基金supported financially by the National Natural Science Foundation of China(10971019)supported financially by the Scientific Research Fund of Hunan Provincial Educational Department(09C852)
文摘In this article, we consider the controllability of a quasi-linear heat equation involving gradient terms with Dirichlet boundary conditions in a bounded domain of RN.The results are established by using the variational methods, the related duality theory and Kakutani Fixed-point Theorem.
基金supported by National Natural Science Foundation of China (Grant No. 11571295)
文摘This paper is concerned with exact boundary behavior of large solutions to semilinear elliptic equations △u=b(x)f(u)(C0+|▽u|q),x∈Ω,where Ω is a bounded domain with a smooth boundary in RN,C0≥0,q E [0,2),b∈Clocα(Ω) is positive in but may be vanishing or appropriately singular on the boundary,f∈C[0,∞),f(0)=0,and f is strictly increasing on [0,∞)(or f∈C(R),f(s)> 0,■s∈R,f is strictly increasing on R).We show unified boundary behavior of such solutions to the problem under a new structure condition on f.
文摘In this paper, a new class of three term memory gradient method with non-monotone line search technique for unconstrained optimization is presented. Global convergence properties of the new methods are discussed. Combining the quasi-Newton method with the new method, the former is modified to have global convergence property. Numerical results show that the new algorithm is efficient.
基金Supported by the National Basic Research Program of China ("973"Program)(2009CB72400603) the National Natural Science Foundation of China(6102700260972100)
文摘A novel image restoration model coupling with a gradient fidelity term based on adaptive total variation is proposed in this paper. In order to choose proper parameters, the selection criteria were analyzed theoretically, and a simple scheme to demonstrate its validity was adopted experimentally. To make fair comparisons of performances of three models, the same numerical algorithm was used to solve partial differential equations. Both the international standard test image on Lena and HR image of CBERS-02B of Dalian city were used to verify the performance of the model. Experimental results illustrate that the new model not only preserved the edge and important details but also alleviated the staircase effect effectively.
基金Partly supported by the National Natural Science Foundation of China,and the Basic Research Program of the Committee of ScienceTechnology and Industry of National Defense of China.
文摘Online gradient algorithm has been widely used as a learning algorithm for feedforward neural network training. In this paper, we prove a weak convergence theorem of an online gradient algorithm with a penalty term, assuming that the training examples are input in a stochastic way. The monotonicity of the error function in the iteration and the boundedness of the weight are both guaranteed. We also present a numerical experiment to support our results.
基金Supported by the National Natural Science Foundation of China（10571106） Supported by the Fundamental Research Funds for the Central Universities（10CX04044A）
文摘In this note,by combining the nice numerical performance of PR and HS methods with the global convergence property of FR method,a class of new restarting three terms conjugate gradient methods is presented.Global convergence properties of the new method with two kinds of common line searches are proved.
文摘Online gradient methods are widely used for training the weight of neural networks and for other engineering computations. In certain cases, the resulting weight may become very large, causing difficulties in the implementation of the network by electronic circuits. In this paper we introduce a punishing term into the error function of the training procedure to prevent this situation. The corresponding convergence of the iterative training procedure and the boundedness of the weight sequence are proved. A supporting numerical example is also provided.
文摘The combined application of organic fertilizer and chemical fertilizer is an effective measure to increase nutrient content of soil plough layer, which must have a profound impact on the deep soil nutrients, especially the contents of nitrogen forms. The purpose of this study was to explore the characteristics of soil nitrogen forms in plough layer and along depth gradient in different fertilization treatments, so as to evaluate the soil quality in spatial dimension, further provid</span><span style="font-family:Verdana;">ing</span><span style="font-family:Verdana;"> a theoretical basis for scientific fertilization and improvement of paddy soil fertility. Here, a 34-year field experiment was conducted with three treatments: without any fertilizer (CK), pure chemical fertilizer (NPK) and chemical fertilizer combined with organic fertilizer (NPKM). We analyzed the content of nitrogen forms in 0 - 100 cm soil depth and their ratios to total nitrogen (TN), and discussed the correlation between nitrogen forms contents and pH, CEC. Results showed that, compared with CK, both NPK and NPKM significantly increased the contents of nitrogen forms in topsoil (soil layer of 0 - 20 cm), especially nitrate nitrogen (NO<sub>3</sub><sup style="margin-left:-6px;">-</sup>-N) content increased by 70% (NPK) and 111% (NPKM), respectively. Although the contents of different nitrogen forms decreased gradually along soil depth gradient, NPKS slowed down the decline rate of TN and alkali-hydrolysable nitrogen (AN) in 0 - 60 cm soil layer, compared to CK. Compared to NPK, NPKM significantly increased the NO<sub>3</sub><sup style="margin-left:-6px;">-</sup>-N/TN ratio in 0 - 20 cm soil layer, but also decreased the content of </span><span><span></span><span style="font-family:Verdana;">-N in 20 - 40 cm, which was beneficial to reduce the risk of nitrogen leaching caused by nitrate leaching into deep layer. The increase of soil pH in NPKM treatment obviously alleviated the problem of soil acidification caused by long-term application of chemical fertilizer. Correlation analysis showed that there was a significant positive correlation between soil nitrogen forms and cation exchange capacity (CEC), but no significant correlation with soil pH. In conclusion, NPKM ensured the nutrients of soil plough layer (0 - 20 cm), also reduced the risk of nitrogen infiltration and nitrogen loss, thus ensur</span><span style="font-family:Verdana;">ing</span><span style="font-family:Verdana;"> the fertility of soil profile.
基金supported by the Natural Science Foundation of Henan Province(15A110050)
文摘In this article, we consider existence and nonexistence of solutions to problem {-△pu＋g（x,u）｜↓△｜^p=f in -Ω,u=0 on Ω with 1〈p〈∞ where f is a positive measurable function which is bounded away from 0 in Ω, and the domain Ω is a smooth bounded open set in R^N（N≥2）. Especially, under the condition that g（x, s） = 1/｜s｜^α （α〉0） is singular at s = 0, we obtain that α 〈 p is necessary and sufficient for the existence of solutions in W0^1,p（Ω） to problem （0.1） when f is sufficiently regular.
文摘In this paper, a gradient method with momentum for sigma-pi-sigma neural networks (SPSNN) is considered in order to accelerate the convergence of the learning procedure for the network weights. The momentum coefficient is chosen in an adaptive manner, and the corresponding weak convergence and strong convergence results are proved.
