Lecture Notes on Pdes, Part I: the Heat Equation and the Eigenfunction Method
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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. -
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. -
EOF/PC Analysis
ATM 552 Notes: Matrix Methods: EOF, SVD, ETC. D.L.Hartmann Page 68 4. Matrix Methods for Analysis of Structure in Data Sets: Empirical Orthogonal Functions, Principal Component Analysis, Singular Value Decomposition, Maximum Covariance Analysis, Canonical Correlation Analysis, Etc. In this chapter we discuss the use of matrix methods from linear algebra, primarily as a means of searching for structure in data sets. Empirical Orthogonal Function (EOF) analysis seeks structures that explain the maximum amount of variance in a two-dimensional data set. One dimension in the data set represents the dimension in which we are seeking to find structure, and the other dimension represents the dimension in which realizations of this structure are sampled. In seeking characteristic spatial structures that vary with time, for example, we would use space as the structure dimension and time as the sampling dimension. The analysis produces a set of structures in the first dimension, which we call the EOF’s, and which we can think of as being the structures in the spatial dimension. The complementary set of structures in the sampling dimension (e.g. time) we can call the Principal Components (PC’s), and they are related one-to-one to the EOF’s. Both sets of structures are orthogonal in their own dimension. Sometimes it is helpful to sacrifice one or both of these orthogonalities to produce more compact or physically appealing structures, a process called rotation of EOF’s. Singular Value Decomposition (SVD) is a general decomposition of a matrix. It can be used on data matrices to find both the EOF’s and PC’s simultaneously. -
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. -
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). -
Complete Set of Orthogonal Basis Mathematical Physics
R. I. Badran Complete set of orthogonal basis Mathematical Physics Complete set of orthogonal functions Discrete set of vectors: ˆ ˆ ˆ ˆ ˆ ˆ The two vectors A AX i A y j Az k and B BX i B y j Bz k are 3 orthogonal if A B 0 or Ai Bi 0 . i1 Continuous set of functions on an interval (a, b): a) The two continuous functions A(x) and B (x) are orthogonal b on the interval (a, b) if A(x)B(x)dx 0. a b) The two complex functions A (x) and B (x) are orthogonal on b the interval (a, b) if A (x)B(x)dx 0 , where A*(x) is the a complex conjugate of A (x). c) For a whole set of functions An (x) (where n= 1, 2, 3,) and on the interval (a, b) b 0 if m n An (x)Am (x)dx a , const.t 0 if m n An (x) is called a set of orthogonal functions. Examples: 0 m n sin nx sin mxdx i) , m n 0 where sin nx is a set of orthogonal functions on the interval (-, ). Similarly 0 m n cos nx cos mxdx if if m n 0 R. I. Badran Complete set of orthogonal basis Mathematical Physics ii) sin nx cos mxdx 0 for any n and m 0 (einx ) eimxdx iii) 2 1 vi) P (x)Pm (x)dx 0 unless. m 1 [Try to prove this; also solve problems (2, 5) of section 6]. -
10 Heat Equation: Interpretation of the Solution
Math 124A { October 26, 2011 «Viktor Grigoryan 10 Heat equation: interpretation of the solution Last time we considered the IVP for the heat equation on the whole line u − ku = 0 (−∞ < x < 1; 0 < t < 1); t xx (1) u(x; 0) = φ(x); and derived the solution formula Z 1 u(x; t) = S(x − y; t)φ(y) dy; for t > 0; (2) −∞ where S(x; t) is the heat kernel, 1 2 S(x; t) = p e−x =4kt: (3) 4πkt Substituting this expression into (2), we can rewrite the solution as 1 1 Z 2 u(x; t) = p e−(x−y) =4ktφ(y) dy; for t > 0: (4) 4πkt −∞ Recall that to derive the solution formula we first considered the heat IVP with the following particular initial data 1; x > 0; Q(x; 0) = H(x) = (5) 0; x < 0: Then using dilation invariance of the Heaviside step function H(x), and the uniquenessp of solutions to the heat IVP on the whole line, we deduced that Q depends only on the ratio x= t, which lead to a reduction of the heat equation to an ODE. Solving the ODE and checking the initial condition (5), we arrived at the following explicit solution p x= 4kt 1 1 Z 2 Q(x; t) = + p e−p dp; for t > 0: (6) 2 π 0 The heat kernel S(x; t) was then defined as the spatial derivative of this particular solution Q(x; t), i.e. @Q S(x; t) = (x; t); (7) @x and hence it also solves the heat equation by the differentiation property. -
Chapter 2. Steady States and Boundary Value Problems
“rjlfdm” 2007/4/10 i i page 13 i i Copyright ©2007 by the Society for Industrial and Applied Mathematics This electronic version is for personal use and may not be duplicated or distributed. Chapter 2 Steady States and Boundary Value Problems We will first consider ordinary differential equations (ODEs) that are posed on some in- terval a < x < b, together with some boundary conditions at each end of the interval. In the next chapter we will extend this to more than one space dimension and will study elliptic partial differential equations (ODEs) that are posed in some region of the plane or Q5 three-dimensional space and are solved subject to some boundary conditions specifying the solution and/or its derivatives around the boundary of the region. The problems considered in these two chapters are generally steady-state problems in which the solution varies only with the spatial coordinates but not with time. (But see Section 2.16 for a case where Œa; b is a time interval rather than an interval in space.) Steady-state problems are often associated with some time-dependent problem that describes the dynamic behavior, and the 2-point boundary value problem (BVP) or elliptic equation results from considering the special case where the solution is steady in time, and hence the time-derivative terms are equal to zero, simplifying the equations. 2.1 The heat equation As a specific example, consider the flow of heat in a rod made out of some heat-conducting material, subject to some external heat source along its length and some boundary condi- tions at each end. -
Solving the Boundary Value Problems for Differential Equations with Fractional Derivatives by the Method of Separation of Variables
mathematics Article Solving the Boundary Value Problems for Differential Equations with Fractional Derivatives by the Method of Separation of Variables Temirkhan Aleroev National Research Moscow State University of Civil Engineering (NRU MGSU), Yaroslavskoe Shosse, 26, 129337 Moscow, Russia; [email protected] Received: 16 September 2020; Accepted: 22 October 2020; Published: 29 October 2020 Abstract: This paper is devoted to solving boundary value problems for differential equations with fractional derivatives by the Fourier method. The necessary information is given (in particular, theorems on the completeness of the eigenfunctions and associated functions, multiplicity of eigenvalues, and questions of the localization of root functions and eigenvalues are discussed) from the spectral theory of non-self-adjoint operators generated by differential equations with fractional derivatives and boundary conditions of the Sturm–Liouville type, obtained by the author during implementation of the method of separation of variables (Fourier). Solutions of boundary value problems for a fractional diffusion equation and wave equation with a fractional derivative are presented with respect to a spatial variable. Keywords: eigenvalue; eigenfunction; function of Mittag–Leffler; fractional derivative; Fourier method; method of separation of variables In memoriam of my Father Sultan and my son Bibulat 1. Introduction Let j(x) L (0, 1). Then the function 2 1 x a d− 1 a 1 a j(x) (x t) − j(t) dt L1(0, 1) dx− ≡ G(a) − 2 Z0 is known as a fractional integral of order a > 0 beginning at x = 0 [1]. Here G(a) is the Euler gamma-function. As is known (see [1]), the function y(x) L (0, 1) is called the fractional derivative 2 1 of the function j(x) L (0, 1) of order a > 0 beginning at x = 0, if 2 1 a d− j(x) = a y(x), dx− which is written da y(x) = j(x). -
A Complete System of Orthogonal Step Functions
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 132, Number 12, Pages 3491{3502 S 0002-9939(04)07511-2 Article electronically published on July 22, 2004 A COMPLETE SYSTEM OF ORTHOGONAL STEP FUNCTIONS HUAIEN LI AND DAVID C. TORNEY (Communicated by David Sharp) Abstract. We educe an orthonormal system of step functions for the interval [0; 1]. This system contains the Rademacher functions, and it is distinct from the Paley-Walsh system: its step functions use the M¨obius function in their definition. Functions have almost-everywhere convergent Fourier-series expan- sions if and only if they have almost-everywhere convergent step-function-series expansions (in terms of the members of the new orthonormal system). Thus, for instance, the new system and the Fourier system are both complete for Lp(0; 1); 1 <p2 R: 1. Introduction Analytical desiderata and applications, ever and anon, motivate the elaboration of systems of orthogonal step functions|as exemplified by the Haar system, the Paley-Walsh system and the Rademacher system. Our motivation is the example- based classification of digitized images, represented by rational points of the real interval [0,1], the domain of interest in the sequel. It is imperative to establish the \completeness" of any orthogonal system. Much is known, in this regard, for the classical systems, as this has been the subject of numerous investigations, and we use the latter results to establish analogous properties for the orthogonal system developed herein. h i RDefinition 1. Let the inner product f(x);g(x) denote the Lebesgue integral 1 0 f(x)g(x)dx: Definition 2.