10 Heat Equation: Interpretation of the Solution
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Laplace's Equation, the Wave Equation and More
Lecture Notes on PDEs, part II: Laplace's equation, the wave equation and more Fall 2018 Contents 1 The wave equation (introduction)2 1.1 Solution (separation of variables)........................3 1.2 Standing waves..................................4 1.3 Solution (eigenfunction expansion).......................6 2 Laplace's equation8 2.1 Solution in a rectangle..............................9 2.2 Rectangle, with more boundary conditions................... 11 2.2.1 Remarks on separation of variables for Laplace's equation....... 13 3 More solution techniques 14 3.1 Steady states................................... 14 3.2 Moving inhomogeneous BCs into a source................... 17 4 Inhomogeneous boundary conditions 20 4.1 A useful lemma.................................. 20 4.2 PDE with Inhomogeneous BCs: example.................... 21 4.2.1 The new part: equations for the coefficients.............. 21 4.2.2 The rest of the solution.......................... 22 4.3 Theoretical note: What does it mean to be a solution?............ 23 5 Robin boundary conditions 24 5.1 Eigenvalues in nasty cases, graphically..................... 24 5.2 Solving the PDE................................. 26 6 Equations in other geometries 29 6.1 Laplace's equation in a disk........................... 29 7 Appendix 31 7.1 PDEs with source terms; example........................ 31 7.2 Good basis functions for Laplace's equation.................. 33 7.3 Compatibility conditions............................. 34 1 1 The wave equation (introduction) The wave equation is the third of the essential linear PDEs in applied mathematics. In one dimension, it has the form 2 utt = c uxx for u(x; t): As the name suggests, the wave equation describes the propagation of waves, so it is of fundamental importance to many fields. -
THE ONE-DIMENSIONAL HEAT EQUATION. 1. Derivation. Imagine a Dilute Material Species Free to Diffuse Along One Dimension
THE ONE-DIMENSIONAL HEAT EQUATION. 1. Derivation. Imagine a dilute material species free to diffuse along one dimension; a gas in a cylindrical cavity, for example. To model this mathematically, we consider the concentration of the given species as a function of the linear dimension and time: u(x; t), defined so that the total amount Q of diffusing material contained in a given interval [a; b] is equal to (or at least proportional to) the integral of u: Z b Q[a; b](t) = u(x; t)dx: a Then the conservation law of the species in question says the rate of change of Q[a; b] in time equals the net amount of the species flowing into [a; b] through its boundary points; that is, there exists a function d(x; t), the ‘diffusion rate from right to left', so that: dQ[a; b] = d(b; t) − d(a; t): dt It is an experimental fact about diffusion phenomena (which can be under- stood in terms of collisions of large numbers of particles) that the diffusion rate at any point is proportional to the gradient of concentration, with the right-to-left flow rate being positive if the concentration is increasing from left to right at (x; t), that is: d(x; t) > 0 when ux(x; t) > 0. So as a first approximation, it is natural to set: d(x; t) = kux(x; t); k > 0 (`Fick's law of diffusion’ ): Combining these two assumptions we have, for any bounded interval [a; b]: d Z b u(x; t)dx = k(ux(b; t) − ux(a; t)): dt a Since b is arbitrary, differentiating both sides as a function of b we find: ut(x; t) = kuxx(x; t); the diffusion (or heat) equation in one dimension. -
Discontinuous Forcing Functions
Math 135A, Winter 2012 Discontinuous forcing functions 1 Preliminaries If f(t) is defined on the interval [0; 1), then its Laplace transform is defined to be Z 1 F (s) = L (f(t)) = e−stf(t) dt; 0 as long as this integral is defined and converges. In particular, if f is of exponential order and is piecewise continuous, the Laplace transform of f(t) will be defined. • f is of exponential order if there are constants M and c so that jf(t)j ≤ Mect: Z 1 Since the integral e−stMect dt converges if s > c, then by a comparison test (like (11.7.2) 0 in Salas-Hille-Etgen), the integral defining the Laplace transform of f(t) will converge. • f is piecewise continuous if over each interval [0; b], f(t) has only finitely many discontinuities, and at each point a in [0; b], both of the limits lim f(t) and lim f(t) t!a− t!a+ exist { they need not be equal, but they must exist. (At the endpoints 0 and b, the appropriate one-sided limits must exist.) 2 Step functions Define u(t) to be the function ( 0 if t < 0; u(t) = 1 if t ≥ 0: Then u(t) is called the step function, or sometimes the Heaviside step function: it jumps from 0 to 1 at t = 0. Note that for any number a > 0, the graph of the function u(t − a) is the same as the graph of u(t), but translated right by a: u(t − a) jumps from 0 to 1 at t = a. -
On Stochastic Distributions and Currents
NISSUNA UMANA INVESTIGAZIONE SI PUO DIMANDARE VERA SCIENZIA S’ESSA NON PASSA PER LE MATEMATICHE DIMOSTRAZIONI LEONARDO DA VINCI vol. 4 no. 3-4 2016 Mathematics and Mechanics of Complex Systems VINCENZO CAPASSO AND FRANCO FLANDOLI ON STOCHASTIC DISTRIBUTIONS AND CURRENTS msp MATHEMATICS AND MECHANICS OF COMPLEX SYSTEMS Vol. 4, No. 3-4, 2016 dx.doi.org/10.2140/memocs.2016.4.373 ∩ MM ON STOCHASTIC DISTRIBUTIONS AND CURRENTS VINCENZO CAPASSO AND FRANCO FLANDOLI Dedicated to Lucio Russo, on the occasion of his 70th birthday In many applications, it is of great importance to handle random closed sets of different (even though integer) Hausdorff dimensions, including local infor- mation about initial conditions and growth parameters. Following a standard approach in geometric measure theory, such sets may be described in terms of suitable measures. For a random closed set of lower dimension with respect to the environment space, the relevant measures induced by its realizations are sin- gular with respect to the Lebesgue measure, and so their usual Radon–Nikodym derivatives are zero almost everywhere. In this paper, how to cope with these difficulties has been suggested by introducing random generalized densities (dis- tributions) á la Dirac–Schwarz, for both the deterministic case and the stochastic case. For the last one, mean generalized densities are analyzed, and they have been related to densities of the expected values of the relevant measures. Ac- tually, distributions are a subclass of the larger class of currents; in the usual Euclidean space of dimension d, currents of any order k 2 f0; 1;:::; dg or k- currents may be introduced. -
SOME ALGEBRAIC DEFINITIONS and CONSTRUCTIONS Definition
SOME ALGEBRAIC DEFINITIONS AND CONSTRUCTIONS Definition 1. A monoid is a set M with an element e and an associative multipli- cation M M M for which e is a two-sided identity element: em = m = me for all m M×. A−→group is a monoid in which each element m has an inverse element m−1, so∈ that mm−1 = e = m−1m. A homomorphism f : M N of monoids is a function f such that f(mn) = −→ f(m)f(n) and f(eM )= eN . A “homomorphism” of any kind of algebraic structure is a function that preserves all of the structure that goes into the definition. When M is commutative, mn = nm for all m,n M, we often write the product as +, the identity element as 0, and the inverse of∈m as m. As a convention, it is convenient to say that a commutative monoid is “Abelian”− when we choose to think of its product as “addition”, but to use the word “commutative” when we choose to think of its product as “multiplication”; in the latter case, we write the identity element as 1. Definition 2. The Grothendieck construction on an Abelian monoid is an Abelian group G(M) together with a homomorphism of Abelian monoids i : M G(M) such that, for any Abelian group A and homomorphism of Abelian monoids−→ f : M A, there exists a unique homomorphism of Abelian groups f˜ : G(M) A −→ −→ such that f˜ i = f. ◦ We construct G(M) explicitly by taking equivalence classes of ordered pairs (m,n) of elements of M, thought of as “m n”, under the equivalence relation generated by (m,n) (m′,n′) if m + n′ = −n + m′. -
LECTURE 7: HEAT EQUATION and ENERGY METHODS Readings
LECTURE 7: HEAT EQUATION AND ENERGY METHODS Readings: • Section 2.3.4: Energy Methods • Convexity (see notes) • Section 2.3.3a: Strong Maximum Principle (pages 57-59) This week we'll discuss more properties of the heat equation, in partic- ular how to apply energy methods to the heat equation. In fact, let's start with energy methods, since they are more fun! Recall the definition of parabolic boundary and parabolic cylinder from last time: Notation: UT = U × (0;T ] ΓT = UT nUT Note: Think of ΓT like a cup: It contains the sides and bottoms, but not the top. And think of UT as water: It contains the inside and the top. Date: Monday, May 11, 2020. 1 2 LECTURE 7: HEAT EQUATION AND ENERGY METHODS Weak Maximum Principle: max u = max u UT ΓT 1. Uniqueness Reading: Section 2.3.4: Energy Methods Fact: There is at most one solution of: ( ut − ∆u = f in UT u = g on ΓT LECTURE 7: HEAT EQUATION AND ENERGY METHODS 3 We will present two proofs of this fact: One using energy methods, and the other one using the maximum principle above Proof: Suppose u and v are solutions, and let w = u − v, then w solves ( wt − ∆w = 0 in UT u = 0 on ΓT Proof using Energy Methods: Now for fixed t, consider the following energy: Z E(t) = (w(x; t))2 dx U Then: Z Z Z 0 IBP 2 E (t) = 2w (wt) = 2w∆w = −2 jDwj ≤ 0 U U U 4 LECTURE 7: HEAT EQUATION AND ENERGY METHODS Therefore E0(t) ≤ 0, so the energy is decreasing, and hence: Z Z (0 ≤)E(t) ≤ E(0) = (w(x; 0))2 dx = 0 = 0 U And hence E(t) = R w2 ≡ 0, which implies w ≡ 0, so u − v ≡ 0, and hence u = v . -
Lecture 24: Laplace's Equation
Introductory lecture notes on Partial Differential Equations - ⃝c Anthony Peirce. 1 Not to be copied, used, or revised without explicit written permission from the copyright owner. Lecture 24: Laplace's Equation (Compiled 26 April 2019) In this lecture we start our study of Laplace's equation, which represents the steady state of a field that depends on two or more independent variables, which are typically spatial. We demonstrate the decomposition of the inhomogeneous Dirichlet Boundary value problem for the Laplacian on a rectangular domain into a sequence of four boundary value problems each having only one boundary segment that has inhomogeneous boundary conditions and the remainder of the boundary is subject to homogeneous boundary conditions. These latter problems can then be solved by separation of variables. Key Concepts: Laplace's equation; Steady State boundary value problems in two or more dimensions; Linearity; Decomposition of a complex boundary value problem into subproblems Reference Section: Boyce and Di Prima Section 10.8 24 Laplace's Equation 24.1 Summary of the equations we have studied thus far In this course we have studied the solution of the second order linear PDE. @u = α2∆u Heat equation: Parabolic T = α2X2 Dispersion Relation σ = −α2k2 @t @2u (24.1) = c2∆u Wave equation: Hyperbolic T 2 − c2X2 = A Dispersion Relation σ = ick @t2 ∆u = 0 Laplace's equation: Elliptic X2 + Y 2 = A Dispersion Relation σ = k Important: (1) These equations are second order because they have at most 2nd partial derivatives. (2) These equations are all linear so that a linear combination of solutions is again a solution. -
Laplace Transforms: Theory, Problems, and Solutions
Laplace Transforms: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved 1 Contents 43 The Laplace Transform: Basic Definitions and Results 3 44 Further Studies of Laplace Transform 15 45 The Laplace Transform and the Method of Partial Fractions 28 46 Laplace Transforms of Periodic Functions 35 47 Convolution Integrals 45 48 The Dirac Delta Function and Impulse Response 53 49 Solving Systems of Differential Equations Using Laplace Trans- form 61 50 Solutions to Problems 68 2 43 The Laplace Transform: Basic Definitions and Results Laplace transform is yet another operational tool for solving constant coeffi- cients linear differential equations. The process of solution consists of three main steps: • The given \hard" problem is transformed into a \simple" equation. • This simple equation is solved by purely algebraic manipulations. • The solution of the simple equation is transformed back to obtain the so- lution of the given problem. In this way the Laplace transformation reduces the problem of solving a dif- ferential equation to an algebraic problem. The third step is made easier by tables, whose role is similar to that of integral tables in integration. The above procedure can be summarized by Figure 43.1 Figure 43.1 In this section we introduce the concept of Laplace transform and discuss some of its properties. The Laplace transform is defined in the following way. Let f(t) be defined for t ≥ 0: Then the Laplace transform of f; which is denoted by L[f(t)] or by F (s), is defined by the following equation Z T Z 1 L[f(t)] = F (s) = lim f(t)e−stdt = f(t)e−stdt T !1 0 0 The integral which defined a Laplace transform is an improper integral. -
Greens' Function and Dual Integral Equations Method to Solve Heat Equation for Unbounded Plate
Gen. Math. Notes, Vol. 23, No. 2, August 2014, pp. 108-115 ISSN 2219-7184; Copyright © ICSRS Publication, 2014 www.i-csrs.org Available free online at http://www.geman.in Greens' Function and Dual Integral Equations Method to Solve Heat Equation for Unbounded Plate N.A. Hoshan Tafila Technical University, Tafila, Jordan E-mail: [email protected] (Received: 16-3-13 / Accepted: 14-5-14) Abstract In this paper we present Greens function for solving non-homogeneous two dimensional non-stationary heat equation in axially symmetrical cylindrical coordinates for unbounded plate high with discontinuous boundary conditions. The solution of the given mixed boundary value problem is obtained with the aid of the dual integral equations (DIE), Greens function, Laplace transform (L-transform) , separation of variables and given in form of functional series. Keywords: Greens function, dual integral equations, mixed boundary conditions. 1 Introduction Several problems involving homogeneous mixed boundary value problems of a non- stationary heat equation and Helmholtz equation were discussed with details in monographs [1, 4-7]. In this paper we present the use of Greens function and DIE for solving non- homogeneous and non-stationary heat equation in axially symmetrical cylindrical coordinates for unbounded plate of a high h. The goal of this paper is to extend the applications DIE which is widely used for solving mathematical physics equations of the 109 N.A. Hoshan elliptic and parabolic types in different coordinate systems subject to mixed boundary conditions, to solve non-homogeneous heat conduction equation, related to both first and second mixed boundary conditions acted on the surface of unbounded cylindrical plate. -
MTH 620: 2020-04-28 Lecture
MTH 620: 2020-04-28 lecture Alexandru Chirvasitu Introduction In this (last) lecture we'll mix it up a little and do some topology along with the homological algebra. I wanted to bring up the cohomology of profinite groups if only briefly, because we discussed these in MTH619 in the context of Galois theory. Consider this as us connecting back to that material. 1 Topological and profinite groups This is about the cohomology of profinite groups; we discussed these briefly in MTH619 during Fall 2019, so this will be a quick recollection. First things first though: Definition 1.1 A topological group is a group G equipped with a topology, such that both the multiplication G × G ! G and the inverse map g 7! g−1 are continuous. Unless specified otherwise, all of our topological groups will be assumed separated (i.e. Haus- dorff) as topological spaces. Ordinary, plain old groups can be regarded as topological groups equipped with the discrete topology (i.e. so that all subsets are open). When I want to be clear we're considering plain groups I might emphasize that by referring to them as `discrete groups'. The main concepts we will work with are covered by the following two definitions (or so). Definition 1.2 A profinite group is a compact (and as always for us, Hausdorff) topological group satisfying any of the following equivalent conditions: Q G embeds as a closed subgroup into a product i2I Gi of finite groups Gi, equipped with the usual product topology; For every open neighborhood U of the identity 1 2 G there is a normal, open subgroup N E G contained in U. -
Arxiv:2011.03427V2 [Math.AT] 12 Feb 2021 Riae Fmp Ffiiest.W Eosrt Hscategor This Demonstrate We Sets
HYPEROCTAHEDRAL HOMOLOGY FOR INVOLUTIVE ALGEBRAS DANIEL GRAVES Abstract. Hyperoctahedral homology is the homology theory associated to the hyperoctahe- dral crossed simplicial group. It is defined for involutive algebras over a commutative ring using functor homology and the hyperoctahedral bar construction of Fiedorowicz. The main result of the paper proves that hyperoctahedral homology is related to equivariant stable homotopy theory: for a discrete group of odd order, the hyperoctahedral homology of the group algebra is isomorphic to the homology of the fixed points under the involution of an equivariant infinite loop space built from the classifying space of the group. Introduction Hyperoctahedral homology for involutive algebras was introduced by Fiedorowicz [Fie, Sec- tion 2]. It is the homology theory associated to the hyperoctahedral crossed simplicial group [FL91, Section 3]. Fiedorowicz and Loday [FL91, 6.16] had shown that the homology theory constructed from the hyperoctahedral crossed simplicial group via a contravariant bar construc- tion, analogously to cyclic homology, was isomorphic to Hochschild homology and therefore did not detect the action of the hyperoctahedral groups. Fiedorowicz demonstrated that a covariant bar construction did detect this action and sketched results connecting the hyperoctahedral ho- mology of monoid algebras and group algebras to May’s two-sided bar construction and infinite loop spaces, though these were never published. In Section 1 we recall the hyperoctahedral groups. We recall the hyperoctahedral category ∆H associated to the hyperoctahedral crossed simplicial group. This category encodes an involution compatible with an order-preserving multiplication. We introduce the category of involutive non-commutative sets, which encodes the same information by adding data to the preimages of maps of finite sets. -
Chapter H1: 1.1–1.2 – Introduction, Heat Equation
Name Math 3150 Problems Haberman Chapter H1 Due Date: Problems are collected on Wednesday. Submitted work. Please submit one stapled package per problem set. Label each problem with its corresponding problem number, e.g., Problem H3.1-4 . Labels like Problem XC-H1.2-4 are extra credit problems. Attach this printed sheet to simplify your work. Labels Explained. The label Problem Hx.y-z means the problem is for Haberman 4E or 5E, chapter x , section y , problem z . Label Problem H2.0-3 is background problem 3 in Chapter 2 (take y = 0 in label Problem Hx.y-z ). If not a background problem, then the problem number should match a corresponding problem in the textbook. The chapters corresponding to 1,2,3,4,10 in Haberman 4E or 5E appear in the hybrid textbook Edwards-Penney-Haberman as chapters 12,13,14,15,16. Required Background. Only calculus I and II plus engineering differential equations are assumed. If you cannot understand a problem, then read the references and return to the problem afterwards. Your job is to understand the Heat Equation. Chapter H1: 1.1{1.2 { Introduction, Heat Equation Problem H1.0-1. (Partial Derivatives) Compute the partial derivatives. @ @ @ (1 + t + x2) (1 + xt + (tx)2) (sin xt + t2 + tx2) @x @x @x @ @ @ (sinh t sin 2x) (e2t+x sin(t + x)) (e3t+2x(et sin x + ex cos t)) @t @t @t Problem H1.0-2. (Jacobian) f1 Find the Jacobian matrix J and the Jacobian determinant jJj for the transformation F(x; y) = , where f1(x; y) = f2 y cos(2y) sin(3x), f2(x; y) = e sin(y + x).