Existence and Uniqueness Theorem For
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Modelling for Aquifer Parameters Determination
Journal of Engineering Science, Vol. 14, 71–99, 2018 Mathematical Implications of Hydrogeological Conditions: Modelling for Aquifer Parameters Determination Sheriffat Taiwo Okusaga1, Mustapha Adewale Usman1* and Niyi Olaonipekun Adebisi2 1Department of Mathematical Sciences, Olabisi Onabanjo University, Ago-Iwoye, Nigeria 2Department of Earth Sciences, Olabisi Onabanjo University, Ago-Iwoye, Nigeria *Corresponding author: [email protected] Published online: 15 April 2018 To cite this article: Sheriffat Taiwo Okusaga, Mustapha Adewale Usman and Niyi Olaonipekun Adebisi. (2018). Mathematical implications of hydrogeological conditions: Modelling for aquifer parameters determination. Journal of Engineering Science, 14: 71–99, https://doi.org/10.21315/jes2018.14.6. To link to this article: https://doi.org/10.21315/jes2018.14.6 Abstract: Formulations of natural hydrogeological phenomena have been derived, from experimentation and observation of Darcy fundamental law. Mathematical methods are also been applied to expand on these formulations, but the management of complexities which relate to subsurface heterogeneity is yet to be developed into better models. This study employs a thorough approach to modelling flow in an aquifer (a porous medium) with phenomenological parameters such as Transmissibility and Storage Coefficient on the basis of mathematical arguments. An overview of the essential components of mathematical background and a basic working knowledge of groundwater flow is presented. Equations obtained through the modelling process -
The Cauchy-Kovalevskaya Theorem
CHAPTER 2 The Cauchy-Kovalevskaya Theorem This chapter deals with the only “general theorem” which can be extended from the theory of ODEs, the Cauchy-Kovalevskaya Theorem. This also will allow us to introduce the notion of non-characteristic data, principal symbol and the basic clas- sification of PDEs. It will also allow us to explore why analyticity is not the proper regularity for studying PDEs most of the time. Acknowledgements. Some parts of this chapter – in particular the four proofs of the Cauchy-Kovalevskaya theorem for ODEs – are strongly inspired from the nice lecture notes of Bruce Diver available on his webpage, and some other parts – in particular the organisation of the proof in the PDE case and the discussion of the counter-examples – are strongly inspired from the nice lectures notes of Tsogtgerel Gantumur available on his webpage. 1. Recalls on analyticity Let us first do some recalls on the notion of real analyticity. Definition 1.1. A function f defined on some open subset U of the real line is said to be real analytic at a point x if it is an infinitely di↵erentiable function 0 2U such that the Taylor series at the point x0 1 f (n)(x ) (1.1) T (x)= 0 (x x )n n! − 0 n=0 X converges to f(x) for x in a neighborhood of x0. A function f is real analytic on an open set of the real line if it is real analytic U at any point x0 . The set of all real analytic functions on a given open set is often denoted by2C!U( ). -
Lipschitz Continuity of Convex Functions
Lipschitz Continuity of Convex Functions Bao Tran Nguyen∗ Pham Duy Khanh† November 13, 2019 Abstract We provide some necessary and sufficient conditions for a proper lower semi- continuous convex function, defined on a real Banach space, to be locally or globally Lipschitz continuous. Our criteria rely on the existence of a bounded selection of the subdifferential mapping and the intersections of the subdifferential mapping and the normal cone operator to the domain of the given function. Moreover, we also point out that the Lipschitz continuity of the given function on an open and bounded (not necessarily convex) set can be characterized via the existence of a bounded selection of the subdifferential mapping on the boundary of the given set and as a consequence it is equivalent to the local Lipschitz continuity at every point on the boundary of that set. Our results are applied to extend a Lipschitz and convex function to the whole space and to study the Lipschitz continuity of its Moreau envelope functions. Keywords Convex function, Lipschitz continuity, Calmness, Subdifferential, Normal cone, Moreau envelope function. Mathematics Subject Classification (2010) 26A16, 46N10, 52A41 1 Introduction arXiv:1911.04886v1 [math.FA] 12 Nov 2019 Lipschitz continuous and convex functions play a significant role in convex and nonsmooth analysis. It is well-known that if the domain of a proper lower semicontinuous convex function defined on a real Banach space has a nonempty interior then the function is continuous over the interior of its domain [3, Proposition 2.111] and as a consequence, it is subdifferentiable (its subdifferential is a nonempty set) and locally Lipschitz continuous at every point in the interior of its domain [3, Proposition 2.107]. -
The Beginnings 2 2. the Topology of Metric Spaces 5 3. Sequences and Completeness 9 4
MATH 3963 NONLINEAR ODES WITH APPLICATIONS R MARANGELL Contents 1. Metric Spaces - The Beginnings 2 2. The Topology of Metric Spaces 5 3. Sequences and Completeness 9 4. The Contraction Mapping Theorem 13 5. Lipschitz Continuity 19 6. Existence and Uniqueness of Solutions to ODEs 21 7. Dependence on Initial Conditions and Parameters 26 8. Maximal Intervals of Existence 29 Date: September 6, 2017. 1 R Marangell Part II - Existence and Uniqueness of ODES 1. Metric Spaces - The Beginnings So.... I am from California, and so I fly through L.A. a lot on my way to my parents house. How far is that from Sydney? Well... Google tells me it's 12000 km (give or take), and in fact this agrees with my airline - they very nicely give me 12000 km on my frequent flyer points. But.... well.... are they really 12000 km apart? What if I drilled a hole through the surface of the Earth? Using a little trigonometry, and the fact that the radius of the earth is R = 6371 km, we have that the distance `as the mole digs' so-to-speak from Sydney to Los Angeles CA is given by (see Figure 1): 12000 x = 2R sin ≈ 10303 km (give or take) 2R Figure 1. A schematic of the earth showing an `equatorial' (or great) circle from Sydney to L.A. So I have two different answers, to the same question, and both are correct. Indeed, Sydney is both 12000 km and around 10303 km away from L.A. What's going on here is pretty obvious, but we're going to make it precise. -
Modifications of the Method of Variation of Parameters
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector International Joumal Available online at www.sciencedirect.com computers & oc,-.c~ ~°..-cT- mathematics with applications Computers and Mathematics with Applications 51 (2006) 451-466 www.elsevier.com/locate/camwa Modifications of the Method of Variation of Parameters ROBERTO BARRIO* GME, Depto. Matem~tica Aplicada Facultad de Ciencias Universidad de Zaragoza E-50009 Zaragoza, Spain rbarrio@unizar, es SERGIO SERRANO GME, Depto. MatemAtica Aplicada Centro Politdcnico Superior Universidad de Zaragoza Maria de Luna 3, E-50015 Zaragoza, Spain sserrano©unizar, es (Received February PO05; revised and accepted October 2005) Abstract--ln spite of being a classical method for solving differential equations, the method of variation of parameters continues having a great interest in theoretical and practical applications, as in astrodynamics. In this paper we analyse this method providing some modifications and generalised theoretical results. Finally, we present an application to the determination of the ephemeris of an artificial satellite, showing the benefits of the method of variation of parameters for this kind of problems. ~) 2006 Elsevier Ltd. All rights reserved. Keywords--Variation of parameters, Lagrange equations, Gauss equations, Satellite orbits, Dif- ferential equations. 1. INTRODUCTION The method of variation of parameters was developed by Leonhard Euler in the middle of XVIII century to describe the mutual perturbations of Jupiter and Saturn. However, his results were not absolutely correct because he did not consider the orbital elements varying simultaneously. In 1766, Lagrange improved the method developed by Euler, although he kept considering some of the orbital elements as constants, which caused some of his equations to be incorrect. -
Review of the Methods of Transition from Partial to Ordinary Differential
Archives of Computational Methods in Engineering https://doi.org/10.1007/s11831-021-09550-5 REVIEW ARTICLE Review of the Methods of Transition from Partial to Ordinary Diferential Equations: From Macro‑ to Nano‑structural Dynamics J. Awrejcewicz1 · V. A. Krysko‑Jr.1 · L. A. Kalutsky2 · M. V. Zhigalov2 · V. A. Krysko2 Received: 23 October 2020 / Accepted: 10 January 2021 © The Author(s) 2021 Abstract This review/research paper deals with the reduction of nonlinear partial diferential equations governing the dynamic behavior of structural mechanical members with emphasis put on theoretical aspects of the applied methods and signal processing. Owing to the rapid development of technology, materials science and in particular micro/nano mechanical systems, there is a need not only to revise approaches to mathematical modeling of structural nonlinear vibrations, but also to choose/pro- pose novel (extended) theoretically based methods and hence, motivating development of numerical algorithms, to get the authentic, reliable, validated and accurate solutions to complex mathematical models derived (nonlinear PDEs). The review introduces the reader to traditional approaches with a broad spectrum of the Fourier-type methods, Galerkin-type methods, Kantorovich–Vlasov methods, variational methods, variational iteration methods, as well as the methods of Vaindiner and Agranovskii–Baglai–Smirnov. While some of them are well known and applied by computational and engineering-oriented community, attention is paid to important (from our point of view) but not widely known and used classical approaches. In addition, the considerations are supported by the most popular and frequently employed algorithms and direct numerical schemes based on the fnite element method (FEM) and fnite diference method (FDM) to validate results obtained. -
Method of Separation of Variables
2.2. Linearity 33 2.2 Linearity As in the study of ordinary differential equations, the concept of linearity will be very important for liS. A linear operator L lJy definition satisfies Chapter 2 (2.2.1) for any two functions '/1,1 and 112, where Cl and ('2 are arhitrary constants. a/at and a 2 /ax' are examples of linear operators since they satisfy (2.2.1): Method of Separation iJ at (el'/11 + C2'/1,) of Variables a' 2 (C1'/11 +C2U 2) ax It can be shown (see Exercise 2.2.1) that any linear combination of linear operators is a linear operator. Thus, the heat operator 2.1 Introduction a iJl In Chapter I we developed from physical principles an lmderstanding of the heat at - k a2·2 eqnation and its corresponding initial and boundary conditions. We are ready is also a linear operator. to pursue the mathematical solution of some typical problems involving partial A linear equation fora it is of the form differential equations. \Ve \-vilt use a technique called the method of separation of variables. You will have to become an expert in this method, and so we will discuss L(u) = J, (2.2.2) quite a fev.; examples. v~,fe will emphasize problem solving techniques, but \ve must also understand how not to misuse the technique. where L is a linear operator and f is known. Examples of linear partial dijjerentinl A relatively simple but typical, problem for the equation of heat conduction equations are occurs for a one - dimensional rod (0 ::; x .::; L) when all the thermal coefficients are constant. -
Convergence Rates for Deterministic and Stochastic Subgradient
Convergence Rates for Deterministic and Stochastic Subgradient Methods Without Lipschitz Continuity Benjamin Grimmer∗ Abstract We extend the classic convergence rate theory for subgradient methods to apply to non-Lipschitz functions. For the deterministic projected subgradient method, we present a global O(1/√T ) convergence rate for any convex function which is locally Lipschitz around its minimizers. This approach is based on Shor’s classic subgradient analysis and implies generalizations of the standard convergence rates for gradient descent on functions with Lipschitz or H¨older continuous gradients. Further, we show a O(1/√T ) convergence rate for the stochastic projected subgradient method on convex functions with at most quadratic growth, which improves to O(1/T ) under either strong convexity or a weaker quadratic lower bound condition. 1 Introduction We consider the nonsmooth, convex optimization problem given by min f(x) x∈Q for some lower semicontinuous convex function f : Rd R and closed convex feasible → ∪{∞} region Q. We assume Q lies in the domain of f and that this problem has a nonempty set of minimizers X∗ (with minimum value denoted by f ∗). Further, we assume orthogonal projection onto Q is computationally tractable (which we denote by PQ( )). arXiv:1712.04104v3 [math.OC] 26 Feb 2018 Since f may be nondifferentiable, we weaken the notion of gradients to· subgradients. The set of all subgradients at some x Q (referred to as the subdifferential) is denoted by ∈ ∂f(x)= g Rd y Rd f(y) f(x)+ gT (y x) . { ∈ | ∀ ∈ ≥ − } We consider solving this problem via a (potentially stochastic) projected subgradient method. -
The Lipschitz Constant of Self-Attention
The Lipschitz Constant of Self-Attention Hyunjik Kim 1 George Papamakarios 1 Andriy Mnih 1 Abstract constraint for neural networks, to control how much a net- Lipschitz constants of neural networks have been work’s output can change relative to its input. Such Lips- explored in various contexts in deep learning, chitz constraints are useful in several contexts. For example, such as provable adversarial robustness, estimat- Lipschitz constraints can endow models with provable ro- ing Wasserstein distance, stabilising training of bustness against adversarial pertubations (Cisse et al., 2017; GANs, and formulating invertible neural net- Tsuzuku et al., 2018; Anil et al., 2019), and guaranteed gen- works. Such works have focused on bounding eralisation bounds (Sokolic´ et al., 2017). Moreover, the dual the Lipschitz constant of fully connected or con- form of the Wasserstein distance is defined as a supremum volutional networks, composed of linear maps over Lipschitz functions with a given Lipschitz constant, and pointwise non-linearities. In this paper, we hence Lipschitz-constrained networks are used for estimat- investigate the Lipschitz constant of self-attention, ing Wasserstein distances (Peyré & Cuturi, 2019). Further, a non-linear neural network module widely used Lipschitz-constrained networks can stabilise training for in sequence modelling. We prove that the stan- GANs, an example being spectral normalisation (Miyato dard dot-product self-attention is not Lipschitz et al., 2018). Finally, Lipschitz-constrained networks are for unbounded input domain, and propose an al- also used to construct invertible models and normalising ternative L2 self-attention that is Lipschitz. We flows. For example, Lipschitz-constrained networks can be derive an upper bound on the Lipschitz constant of used as a building block for invertible residual networks and L2 self-attention and provide empirical evidence hence flow-based generative models (Behrmann et al., 2019; for its asymptotic tightness. -
Hydrogeology and Groundwater Flow
Hydrogeology and Groundwater Flow Hydrogeology (hydro- meaning water, and -geology meaning the study of rocks) is the part of hydrology that deals with the distribution and movement of groundwater in the soil and rocks of the Earth's crust, (commonly in aquifers). The term geohydrology is often used interchangeably. Some make the minor distinction between a hydrologist or engineer applying themselves to geology (geohydrology), and a geologist applying themselves to hydrology (hydrogeology). Hydrogeology (like most earth sciences) is an interdisciplinary subject; it can be difficult to account fully for the chemical, physical, biological and even legal interactions between soil, water, nature and society. Although the basic principles of hydrogeology are very intuitive (e.g., water flows "downhill"), the study of their interaction can be quite complex. Taking into account the interplay of the different facets of a multi-component system often requires knowledge in several diverse fields at both the experimental and theoretical levels. This being said, the following is a more traditional (reductionist viewpoint) introduction to the methods and nomenclature of saturated subsurface hydrology, or simply hydrogeology. © 2014 All Star Training, Inc. 1 Hydrogeology in Relation to Other Fields Hydrogeology, as stated above, is a branch of the earth sciences dealing with the flow of water through aquifers and other shallow porous media (typically less than 450 m or 1,500 ft below the land surface.) The very shallow flow of water in the subsurface (the upper 3 m or 10 ft) is pertinent to the fields of soil science, agriculture and civil engineering, as well as to hydrogeology. -
MTHE / MATH 237 Differential Equations for Engineering Science
i Queen’s University Mathematics and Engineering and Mathematics and Statistics MTHE / MATH 237 Differential Equations for Engineering Science Supplemental Course Notes Serdar Y¨uksel November 26, 2015 ii This document is a collection of supplemental lecture notes used for MTHE / MATH 237: Differential Equations for Engineering Science. Serdar Y¨uksel Contents 1 Introduction to Differential Equations ................................................... .......... 1 1.1 Introduction.................................... ............................................ 1 1.2 Classification of DifferentialEquations ............ ............................................. 1 1.2.1 OrdinaryDifferentialEquations ................. ........................................ 2 1.2.2 PartialDifferentialEquations .................. ......................................... 2 1.2.3 HomogeneousDifferentialEquations .............. ...................................... 2 1.2.4 N-thorderDifferentialEquations................ ........................................ 2 1.2.5 LinearDifferentialEquations ................... ........................................ 2 1.3 SolutionsofDifferentialequations ................ ............................................. 3 1.4 DirectionFields................................. ............................................ 3 1.5 Fundamental Questions on First-Order Differential Equations ...................................... 4 2 First-Order Ordinary Differential Equations .................................................. -
Separation of Variables 1.1 Separation for the Laplace Problem
Physics 342 Lecture 1 Separation of Variables Lecture 1 Physics 342 Quantum Mechanics I Monday, January 28th, 2008 There are three basic mathematical tools we need, and then we can begin working on the physical implications of Schr¨odinger'sequation, which will take up the rest of the semester. So we start with a review of: Linear Algebra, Separation of Variables (SOV), and Probability. There will be no particular completeness to our discussions { at this stage, I want to emphasize those aspects of each of these that will prove useful. Pitfalls and caveats will be addressed as we encounter them in actual problems (where they won't show up, so won't be addressed). Try as one might, it is difficult to relate these three areas, so we will take each one in term, starting from the most familiar, and working to the least (that's a matter of taste, so apologies in advance for the wrong ordering { in anticipation of our section on probability: How many different orderings of these three subjects are there? On average, then, how many people disagree with the current one?). We will begin with separation of variables, a simple technique that you have used recently { we will review the basic electrostatic interest in these solu- tions, and do a few examples of SOV applied to PDE's other than Laplace's equation (r2 V = 0). 1.1 Separation for the Laplace Problem Separation of variables refers to a class of techniques for probing solutions to partial differential equations (PDEs) by turning them into ordinary differen- tial equations (ODEs).