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non-negative solutions FOR A CLASS OF non-POSITONE PROBLEMS

Annals of Differential Equations 1997年第3期13卷 304-312页

作者：杨作东杨会生崔光云河南师范大学 453002 平原大学 453002

In the recent past maily results have been established on non-negative solu tions to boundry value problems of the form u(0) = 0= u(1) where λ > 0, f(0) > 0 (positone problem). In this paper we consider the impact on the non-negative solutions when f(0) solutions and f(u) to be appropriately concave for multiple positive solutions.

关键词： Bonndary value problems non-negative solutions uniquenness

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EXISTENCE OF solutions FOR THE FRACTIONAL(p,q)-LAPLACIAN PROBLEMS INVOLVING A CRITICAL SOBOLEV EXPONENT

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Acta Mathematica Scientia 2020年第6期40卷 1666-1678页

作者： Fanfan CHEN Yang YANG School of Science Jiangnan UniversityWuxi 214122China

In this article,we study the following fractional(p,q)-Laplacian equations involving the critical Sobolev exponent:(Pμ,λ){(−Δ)s 1 p u+(−Δ)s 2 q u=μ|u|q−2 u+λ|u|p−2 u+|u|p∗s 1−2 u,u=0,inΩ,in R N∖Ω,whereΩ⊂R N is a smooth and bounded domain,λ,μ>0,0non-negative nontrivial weak solution to(Pμ,λ)by using the Mountain Pass *** lack of compactness associated with problems involving critical Sobolev exponents is overcome by working with certain asymptotic estimates for minimizers.

关键词： fractional(p,q)-Laplacian non-negative solutions critical Sobolev exponents

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A Deep Learning Modeling Framework to Capture Mixing Patterns in Reactive-Transport Systems

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Communications in Computational Physics 2022年第1期31卷 188-223页

作者： N.V.Jagtap M.K.Mudunuru K.B.Nakshatrala Department of Mechanical Engineering University of HoustonTexas 77204USA Atmospheric Sciences&Global Change Division Pacific Northwest National LaboratoryRichlandWashington 99352USA Department of Civil&Environmental Engineering University of HoustonHoustonTexas 77204USA

Prediction and control of chemical mixing are vital for many scientific areas such as subsurface reactive transport, climate modeling, combustion, epidemiology, and pharmacology. Due to the complex nature of mixing in heterogeneous andanisotropic media, the mathematical models related to this phenomenon are not analytically tractable. Numerical simulations often provide a viable route to predict chemical mixing accurately. However, contemporary modeling approaches for mixing cannot utilize available spatial-temporal data to improve the accuracy of the future prediction and can be compute-intensive, especially when the spatial domain is large andfor long-term temporal predictions. To address this knowledge gap, we will present inthis paper a deep learning (DL) modeling framework applied to predict the progress ofchemical mixing under fast bimolecular reactions. This framework uses convolutionalneural networks (CNN) for capturing spatial patterns and long short-term memory(LSTM) networks for forecasting temporal variations in mixing. By careful design ofthe framework—placement of non-negative constraint on the weights of the CNN andthe selection of activation function, the framework ensures non-negativity of the chemical species at all spatial points and for all times. Our DL-based framework is fast,accurate, and requires minimal data for training. The time needed to obtain a forecastusing the model is a fraction (≈ O(10−6)) of the time needed to obtain the result using a high-fidelity simulation. To achieve an error of 10% (measured using the infinitynorm) for capturing local-scale mixing features such as interfacial mixing, only 24%to 32% of the sequence data for model training is required. To achieve the same levelof accuracy for capturing global-scale mixing features, the sequence data required formodel training is 64% to 70% of the total spatial-temporal data. Hence, the proposedapproach—a fast and accurate way to forecast long-time spatial-temporal mixin

关键词： Deep learning reactive-transport non-negative solutions spatial-temporal forecasting pattern recognition convolutional neural networks(CNN) long short-term memory(LSTM) networks

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A Modeling Framework for Coupling Plasticity with Species Diffusion

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Communications in Computational Physics 2022年第6期32卷 83-125页

作者： M.S.Joshaghani K.B.Nakshatrala Department of Civil and Environmental Engineering University of HoustonTexas

This paper presents a modeling framework—mathematical model and computational framework—to study the response of a plastic material due to the presenceand transport of a chemical species in the host material. Such a modeling frameworkis important to a wide variety of problems ranging from Li-ion batteries, moisturediffusion in cementitious materials, hydrogen diffusion in metals, to consolidation ofsoils under severe loading-unloading regimes. The mathematical model incorporatesexperimental observations reported in the literature on how (elastic and plastic) material properties change because of the presence and transport of a chemical ***, the model accounts for one-way (transport affects the deformation but not viceversa) and two-way couplings between deformation and transport subproblems. Theresulting coupled equations are not amenable to analytical solutions;so, we present arobust computational framework for obtaining numerical solutions. Given that popular numerical formulations do not produce nonnegative solutions, the computationalframework uses an optimized-based nonnegative formulation that respects physicalconstraints (e.g., nonnegative concentrations). For completeness, we also show theeffect and propagation of the negative concentrations, often produced by contemporary transport solvers, into the overall predictions of deformation and concentrationfields. Notably, anisotropy of the diffusion process exacerbates these unphysical violations. Using representative numerical examples, we discuss how the concentrationfield affects plastic deformations of a degrading solid. Based on these numerical examples, we also discuss how plastic zones spread because of material *** illustrate how the proposed computational framework performs, we report variousperformance metrics such as optimization iterations and time-to-solution.

关键词： Species diffusion plasticity non-negative solutions damage mechanics degradation/healing coupled problem

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A Numerical Methodology for Enforcing Maximum Principles and the non-negative Constraint for Transient Diffusion Equations

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Communications in Computational Physics 2016年第1期19卷 53-93页

作者： K.B.Nakshatrala H.Nagarajan M.Shabouei Department of Civil&Environmental Engineering University of HoustonHoustonTexas 77204-4003USA Department of Mechanial Engineering Texas A&M UniversityCollege StationTX 77843USA.

Transient diffusion equations arise in many branches of engineering and applied sciences(e.g.,heat transfer and mass transfer),and are parabolic partial differential *** is well-known that these equations satisfy important mathematical properties like maximum principles and the non-negative constraint,which have implications in mathematical ***,existing numerical formulations for these types of equations do not,in general,satisfy maximum principles and the nonnegative *** this paper,we present a methodology for enforcing maximum principles and the non-negative constraint for transient anisotropic diffusion *** proposed methodology is based on the method of horizontal lines in which the time is discretized *** results in solving steady anisotropic diffusion equation with decay equation at every discrete *** also present other plausible temporal discretizations,and illustrate their shortcomings in meeting maximum principles and the non-negative *** proposedmethodology can handle general computational grids with no additional restrictions on the *** illustrate the performance and accuracy of the proposed methodology using representative numerical *** also perform a numerical convergence analysis of the proposed *** comparison,we also present the results from the standard singlefield semi-discrete formulation and the results froma popular software package,which all will violate maximum principles and the non-negative constraint.

关键词： Numerical heat and mass transfer maximum principles non-negative solutions anisotropic diffusion method of horizontal lines convex quadratic programming parabolic PDEs

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