Dual Variational Approach To Nonlinear Diffusion Equations

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Dual Variational Approach To Nonlinear Diffusion Equations

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Author : Gabriela Marinoschi
language : en
Publisher: Springer Nature
Release Date : 2023-03-28

Dual Variational Approach To Nonlinear Diffusion Equations written by Gabriela Marinoschi and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-03-28 with Mathematics categories.

This monograph explores a dual variational formulation of solutions to nonlinear diffusion equations with general nonlinearities as null minimizers of appropriate energy functionals. The author demonstrates how this method can be utilized as a convenient tool for proving the existence of these solutions when others may fail, such as in cases of evolution equations with nonautonomous operators, with low regular data, or with singular diffusion coefficients. By reducing it to a minimization problem, the original problem is transformed into an optimal control problem with a linear state equation. This procedure simplifies the proof of the existence of minimizers and, in particular, the determination of the first-order conditions of optimality. The dual variational formulation is illustrated in the text with specific diffusion equations that have general nonlinearities provided by potentials having various stronger or weaker properties. These equations can represent mathematical models to various real-world physical processes. Inverse problems and optimal control problems are also considered, as this technique is useful in their treatment as well.

Dual Variational Approach To Nonlinear Diffusion Equations

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Author : Gabriela Marinoschi
language : en
Publisher:
Release Date : 2023

Dual Variational Approach To Nonlinear Diffusion Equations written by Gabriela Marinoschi and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023 with categories.

This monograph explores a dual variational formulation of solutions to nonlinear diffusion equations with general nonlinearities as null minimizers of appropriate energy functionals. The author demonstrates how this method can be utilized as a convenient tool for proving the existence of these solutions when others may fail, such as in cases of evolution equations with nonautonomous operators, with low regular data, or with singular diffusion coefficients. By reducing it to a minimization problem, the original problem is transformed into an optimal control problem with a linear state equation. This procedure simplifies the proof of the existence of minimizers and, in particular, the determination of the first-order conditions of optimality. The dual variational formulation is illustrated in the text with specific diffusion equations that have general nonlinearities provided by potentials having various stronger or weaker properties. These equations can represent mathematical models to various real-world physical processes. Inverse problems and optimal control problems are also considered, as this technique is useful in their treatment as well.

Nonlinear Diffusion Equations And Their Equilibrium States Ii

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Author : W.-M. Ni
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

Nonlinear Diffusion Equations And Their Equilibrium States Ii written by W.-M. Ni and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.

In recent years considerable interest has been focused on nonlinear diffu sion problems, the archetypical equation for these being Ut = ~U + f(u). Here ~ denotes the n-dimensional Laplacian, the solution u = u(x, t) is defined over some space-time domain of the form n x [O,T], and f(u) is a given real function whose form is determined by various physical and mathematical applications. These applications have become more varied and widespread as problem after problem has been shown to lead to an equation of this type or to its time-independent counterpart, the elliptic equation of equilibrium ~u+f(u)=O. Particular cases arise, for example, in population genetics, the physics of nu clear stability, phase transitions between liquids and gases, flows in porous media, the Lend-Emden equation of astrophysics, various simplified com bustion models, and in determining metrics which realize given scalar or Gaussian curvatures. In the latter direction, for example, the problem of finding conformal metrics with prescribed curvature leads to a ground state problem involving critical exponents. Thus not only analysts, but geome ters as well, can find common ground in the present work. The corresponding mathematical problem is to determine how the struc ture of the nonlinear function f(u) influences the behavior of the solution.

Solutions Of Nonlinear Differential Equations Existence Results Via The Variational Approach

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Author : Lin Li
language : en
Publisher: World Scientific
Release Date : 2016-04-15

Solutions Of Nonlinear Differential Equations Existence Results Via The Variational Approach written by Lin Li and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-04-15 with Mathematics categories.

Variational methods are very powerful techniques in nonlinear analysis and are extensively used in many disciplines of pure and applied mathematics (including ordinary and partial differential equations, mathematical physics, gauge theory, and geometrical analysis).In our first chapter, we gather the basic notions and fundamental theorems that will be applied throughout the chapters. While many of these items are easily available in the literature, we gather them here both for the convenience of the reader and for the purpose of making this volume somewhat self-contained. Subsequent chapters deal with how variational methods can be used in fourth-order problems, Kirchhoff problems, nonlinear field problems, gradient systems, and variable exponent problems. A very extensive bibliography is also included.

Nonlinear Diffusion Equations And Their Equilibrium States 3

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Author : N.G Lloyd
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

Nonlinear Diffusion Equations And Their Equilibrium States 3 written by N.G Lloyd and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.

Nonlinear diffusion equations have held a prominent place in the theory of partial differential equations, both for the challenging and deep math ematical questions posed by such equations and the important role they play in many areas of science and technology. Examples of current inter est are biological and chemical pattern formation, semiconductor design, environmental problems such as solute transport in groundwater flow, phase transitions and combustion theory. Central to the theory is the equation Ut = ~cp(U) + f(u). Here ~ denotes the n-dimensional Laplacian, cp and f are given functions and the solution is defined on some domain n x [0, T] in space-time. FUn damental questions concern the existence, uniqueness and regularity of so lutions, the existence of interfaces or free boundaries, the question as to whether or not the solution can be continued for all time, the asymptotic behavior, both in time and space, and the development of singularities, for instance when the solution ceases to exist after finite time, either through extinction or through blow up.

Handbook Of Differential Equations Evolutionary Equations

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Author : C.M. Dafermos
language : en
Publisher: Elsevier
Release Date : 2008-10-06

Handbook Of Differential Equations Evolutionary Equations written by C.M. Dafermos and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-10-06 with Mathematics categories.

The material collected in this volume discusses the present as well as expected future directions of development of the field with particular emphasis on applications. The seven survey articles present different topics in Evolutionary PDE's, written by leading experts.- Review of new results in the area- Continuation of previous volumes in the handbook series covering Evolutionary PDEs- Written by leading experts

Nuclear Science Abstracts

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Author :
language : en
Publisher:
Release Date : 1973

Nuclear Science Abstracts written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1973 with Nuclear energy categories.

Estimating The Error Of Numerical Solutions Of Systems Of Reaction Diffusion Equations

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Author : Donald J. Estep
language : en
Publisher: American Mathematical Soc.
Release Date : 2000

Estimating The Error Of Numerical Solutions Of Systems Of Reaction Diffusion Equations written by Donald J. Estep and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000 with Mathematics categories.

This paper is concerned with the computational estimation of the error of numerical solutions of potentially degenerate reaction-diffusion equations. The underlying motivation is a desire to compute accurate estimates as opposed to deriving inaccurate analytic upper bounds. In this paper, we outline, analyze, and test an approach to obtain computational error estimates based on the introduction of the residual error of the numerical solution and in which the effects of the accumulation of errors are estimated computationally. We begin by deriving an a posteriori relationship between the error of a numerical solution and its residual error using a variational argument. This leads to the introduction of stability factors, which measure the sensitivity of solutions to various kinds of perturbations. Next, we perform some general analysis on the residual errors and stability factors to determine when they are defined and to bound their size. Then we describe the practical use of the theory to estimate the errors of numerical solutions computationally. Several key issues arise in the implementation that remain unresolved and we present partial results and numerical experiments about these points. We use this approach to estimate the error of numerical solutions of nine standard reaction-diffusion models and make a systematic comparison of the time scale over which accurate numerical solutions can be computed for these problems. We also perform a numerical test of the accuracy and reliability of the computational error estimate using the bistable equation. Finally, we apply the general theory to the class of problems that admit invariant regions for the solutions, which includes seven of the main examples. Under this additional stability assumption, we obtain a convergence result in the form of an upper bound on the error from the a posteriori error estimate. We conclude by discussing the preservation of invariant regions under discretization.

Stochastic Evolution Equations

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Author : Wilfried Grecksch
language : en
Publisher: De Gruyter Akademie Forschung
Release Date : 1995

Stochastic Evolution Equations written by Wilfried Grecksch and has been published by De Gruyter Akademie Forschung this book supported file pdf, txt, epub, kindle and other format this book has been release on 1995 with Mathematics categories.

The authors give a self-contained exposition of the theory of stochastic evolution equations. Elements of infinite dimensional analysis, martingale theory in Hilbert spaces, stochastic integrals, stochastic convolutions are applied. Existence and uniqueness theorems for stochastic evolution equations in Hilbert spaces in the sense of the semigroup theory, the theory of evolution operators, and monotonous operators in rigged Hilbert spaces are discussed. Relationships between the different concepts are demonstrated. The results are used to concrete stochastic partial differential equations like parabolic and hyperbolic Ito equations and random constitutive equations of elastic viscoplastic materials. Furthermore, stochastic evolution equations in rigged Hilbert spaces are approximated by time discretization methods.

Optimal Control Of Partial Differential Equations

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Author : Andrea Manzoni
language : en
Publisher: Springer Nature
Release Date : 2022-01-01

Optimal Control Of Partial Differential Equations written by Andrea Manzoni and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-01-01 with Mathematics categories.

This is a book on optimal control problems (OCPs) for partial differential equations (PDEs) that evolved from a series of courses taught by the authors in the last few years at Politecnico di Milano, both at the undergraduate and graduate levels. The book covers the whole range spanning from the setup and the rigorous theoretical analysis of OCPs, the derivation of the system of optimality conditions, the proposition of suitable numerical methods, their formulation, their analysis, including their application to a broad set of problems of practical relevance. The first introductory chapter addresses a handful of representative OCPs and presents an overview of the associated mathematical issues. The rest of the book is organized into three parts: part I provides preliminary concepts of OCPs for algebraic and dynamical systems; part II addresses OCPs involving linear PDEs (mostly elliptic and parabolic type) and quadratic cost functions; part III deals with more general classes of OCPs that stand behind the advanced applications mentioned above. Starting from simple problems that allow a “hands-on” treatment, the reader is progressively led to a general framework suitable to face a broader class of problems. Moreover, the inclusion of many pseudocodes allows the reader to easily implement the algorithms illustrated throughout the text. The three parts of the book are suitable to readers with variable mathematical backgrounds, from advanced undergraduate to Ph.D. levels and beyond. We believe that applied mathematicians, computational scientists, and engineers may find this book useful for a constructive approach toward the solution of OCPs in the context of complex applications.