Partial Differential Equations
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Elliptic Pdes
viii CHAPTER 4 Elliptic PDEs One of the main advantages of extending the class of solutions of a PDE from classical solutions with continuous derivatives to weak solutions with weak deriva- tives is that it is easier to prove the existence of weak solutions. Having estab- lished the existence of weak solutions, one may then study their properties, such as uniqueness and regularity, and perhaps prove under appropriate assumptions that the weak solutions are, in fact, classical solutions. There is often considerable freedom in how one defines a weak solution of a PDE; for example, the function space to which a solution is required to belong is not given a priori by the PDE itself. Typically, we look for a weak formulation that reduces to the classical formulation under appropriate smoothness assumptions and which is amenable to a mathematical analysis; the notion of solution and the spaces to which solutions belong are dictated by the available estimates and analysis. 4.1. Weak formulation of the Dirichlet problem Let us consider the Dirichlet problem for the Laplacian with homogeneous boundary conditions on a bounded domain Ω in Rn, (4.1) −∆u = f in Ω, (4.2) u =0 on ∂Ω. First, suppose that the boundary of Ω is smooth and u,f : Ω → R are smooth functions. Multiplying (4.1) by a test function φ, integrating the result over Ω, and using the divergence theorem, we get ∞ (4.3) Du · Dφ dx = fφ dx for all φ ∈ Cc (Ω). ZΩ ZΩ The boundary terms vanish because φ = 0 on the boundary. -
Elliptic Pdes
CHAPTER 4 Elliptic PDEs One of the main advantages of extending the class of solutions of a PDE from classical solutions with continuous derivatives to weak solutions with weak deriva- tives is that it is easier to prove the existence of weak solutions. Having estab- lished the existence of weak solutions, one may then study their properties, such as uniqueness and regularity, and perhaps prove under appropriate assumptions that the weak solutions are, in fact, classical solutions. There is often considerable freedom in how one defines a weak solution of a PDE; for example, the function space to which a solution is required to belong is not given a priori by the PDE itself. Typically, we look for a weak formulation that reduces to the classical formulation under appropriate smoothness assumptions and which is amenable to a mathematical analysis; the notion of solution and the spaces to which solutions belong are dictated by the available estimates and analysis. 4.1. Weak formulation of the Dirichlet problem Let us consider the Dirichlet problem for the Laplacian with homogeneous boundary conditions on a bounded domain Ω in Rn, (4.1) −∆u = f in Ω; (4.2) u = 0 on @Ω: First, suppose that the boundary of Ω is smooth and u; f : Ω ! R are smooth functions. Multiplying (4.1) by a test function φ, integrating the result over Ω, and using the divergence theorem, we get Z Z 1 (4.3) Du · Dφ dx = fφ dx for all φ 2 Cc (Ω): Ω Ω The boundary terms vanish because φ = 0 on the boundary. -
Advanced Partial Differential Equations Prof. Dr. Thomas
Advanced Partial Differential Equations Prof. Dr. Thomas Sørensen summer term 2015 Marcel Schaub July 2, 2015 1 Contents 0 Recall PDE 1 & Motivation 3 0.1 Recall PDE 1 . .3 1 Weak derivatives and Sobolev spaces 7 1.1 Sobolev spaces . .8 1.2 Approximation by smooth functions . 11 1.3 Extension of Sobolev functions . 13 1.4 Traces . 15 1.5 Sobolev inequalities . 17 2 Linear 2nd order elliptic PDE 25 2.1 Linear 2nd order elliptic partial differential operators . 25 2.2 Weak solutions . 26 2.3 Existence via Lax-Milgram . 28 2.4 Inhomogeneous bounday value problems . 35 2.5 The space H−1(U) ................................ 36 2.6 Regularity of weak solutions . 39 A Tutorials 58 A.1 Tutorial 1: Review of Integration . 58 A.2 Tutorial 2 . 59 A.3 Tutorial 3: Norms . 61 A.4 Tutorial 4 . 62 A.5 Tutorial 6 (Sheet 7) . 65 A.6 Tutorial 7 . 65 A.7 Tutorial 9 . 67 A.8 Tutorium 11 . 67 B Solutions of the problem sheets 70 B.1 Solution to Sheet 1 . 70 B.2 Solution to Sheet 2 . 71 B.3 Solution to Problem Sheet 3 . 73 B.4 Solution to Problem Sheet 4 . 76 B.5 Solution to Exercise Sheet 5 . 77 B.6 Solution to Exercise Sheet 7 . 81 B.7 Solution to problem sheet 8 . 84 B.8 Solution to Exercise Sheet 9 . 87 2 0 Recall PDE 1 & Motivation 0.1 Recall PDE 1 We mainly studied linear 2nd order equations – specifically, elliptic, parabolic and hyper- bolic equations. Concretely: • The Laplace equation ∆u = 0 (elliptic) • The Poisson equation −∆u = f (elliptic) • The Heat equation ut − ∆u = 0, ut − ∆u = f (parabolic) • The Wave equation utt − ∆u = 0, utt − ∆u = f (hyperbolic) We studied (“main motivation; goal”) well-posedness (à la Hadamard) 1. -
Bessel's Differential Equation Mathematical Physics
R. I. Badran Bessel's Differential Equation Mathematical Physics Bessel's Differential Equation: 2 Recalling the DE y n y 0 which has a sinusoidal solution (i.e. sin nx and cos nx) and knowing that these solutions can be treated as power series, we can find a solution to the Bessel's DE which is written as: 2 2 2 x y xy (x p )y 0 . The solution is represented by a series. This series very much look like a damped sine or cosine. It is called a Bessel function. To solve the Bessel' DE, we apply the Frobenius method by mn assuming a series solution of the form y an x . n0 Substitute this solution into the DE equation and after some mathematical steps you may find that the indicial equation is 2 2 m p 0 where m= p. Also you may get a1=0 and hence all odd a's are zero as well. The recursion relation can be found as a a n n2 (n m 2)2 p 2 with (n=0, 1, 2, 3, ……etc.). Case (1): m= p (seeking the first solution of Bessel DE) We get the solution 2 4 p x x y a x 1 2(2p 2) 2 4(2p 2)(2p 4) This can be rewritten as: x p2n J (x) y a (1)n p 2n n0 2 n!( p n)! 1 Put a 2 p p! The Bessel function has the factorial form x p2n J (x) (1)n p 2n p , n0 n!( p n)!2 R. -
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). -
The Standard Form of a Differential Equation
Fall 06 The Standard Form of a Differential Equation Format required to solve a differential equation or a system of differential equations using one of the command-line differential equation solvers such as rkfixed, Rkadapt, Radau, Stiffb, Stiffr or Bulstoer. For a numerical routine to solve a differential equation (DE), we must somehow pass the differential equation as an argument to the solver routine. A standard form for all DEs will allow us to do this. Basic idea: get rid of any second, third, fourth, etc. derivatives that appear, leaving only first derivatives. Example 1 2 d d 5 yx()+3 ⋅ yx()−5yx ⋅ () 4x⋅ 2 dx dx This DE contains a second derivative. How do we write a second derivative as a first derivative? A second derivative is a first derivative of a first derivative. d2 d d yx() yx() 2 dx dxxd Step 1: STANDARDIZATION d Let's define two functions y0(x) and y1(x) as y0()x yx() and y1()x y0()x dx Then this differential equation can be written as d 5 y1()x +3y ⋅ 1()x −5y ⋅ 0()x 4x dx This DE contains 2 functions instead of one, but there is a strong relationship between these two functions d y1()x y0()x dx So, the original DE is now a system of two DEs, d d 5 y1()x y0()x and y1()x +3y ⋅ 1()x −5y ⋅ 0()x 4x⋅ dx dx The convention is to write these equations with the derivatives alone on the left-hand side. d y0()x y1()x dx This is the first step in the standardization process. -
Delta Functions and Distributions
When functions have no value(s): Delta functions and distributions Steven G. Johnson, MIT course 18.303 notes Created October 2010, updated March 8, 2017. Abstract x = 0. That is, one would like the function δ(x) = 0 for all x 6= 0, but with R δ(x)dx = 1 for any in- These notes give a brief introduction to the mo- tegration region that includes x = 0; this concept tivations, concepts, and properties of distributions, is called a “Dirac delta function” or simply a “delta which generalize the notion of functions f(x) to al- function.” δ(x) is usually the simplest right-hand- low derivatives of discontinuities, “delta” functions, side for which to solve differential equations, yielding and other nice things. This generalization is in- a Green’s function. It is also the simplest way to creasingly important the more you work with linear consider physical effects that are concentrated within PDEs, as we do in 18.303. For example, Green’s func- very small volumes or times, for which you don’t ac- tions are extremely cumbersome if one does not al- tually want to worry about the microscopic details low delta functions. Moreover, solving PDEs with in this volume—for example, think of the concepts of functions that are not classically differentiable is of a “point charge,” a “point mass,” a force plucking a great practical importance (e.g. a plucked string with string at “one point,” a “kick” that “suddenly” imparts a triangle shape is not twice differentiable, making some momentum to an object, and so on. -
Analytic Solutions of Partial Differential Equations
Analytic Solutions of Partial Differential Equations MATH3414 School of Mathematics, University of Leeds 15 credits • Taught Semester 1, • Year running 2003/04 • Pre-requisites MATH2360 or MATH2420 or equivalent. • Co-requisites None. • Objectives: To provide an understanding of, and methods of solution for, the most important • types of partial differential equations that arise in Mathematical Physics. On completion of this module, students should be able to: a) use the method of characteristics to solve first-order hyperbolic equations; b) classify a second order PDE as elliptic, parabolic or hyperbolic; c) use Green's functions to solve elliptic equations; d) have a basic understanding of diffusion; e) obtain a priori bounds for reaction-diffusion equations. Syllabus: The majority of physical phenomena can be described by partial differential equa- • tions (e.g. the Navier-Stokes equation of fluid dynamics, Maxwell's equations of electro- magnetism). This module considers the properties of, and analytical methods of solution for some of the most common first and second order PDEs of Mathematical Physics. In particu- lar, we shall look in detail at elliptic equations (Laplace?s equation), describing steady-state phenomena and the diffusion / heat conduction equation describing the slow spread of con- centration or heat. The topics covered are: First order PDEs. Semilinear and quasilinear PDEs; method of characteristics. Characteristics crossing. Second order PDEs. Classifi- cation and standard forms. Elliptic equations: weak and strong minimum and maximum principles; Green's functions. Parabolic equations: exemplified by solutions of the diffusion equation. Bounds on solutions of reaction-diffusion equations. Form of teaching • Lectures: 26 hours. 7 examples classes. -
CORSO ESTIVO DI MATEMATICA Differential Equations Of
CORSO ESTIVO DI MATEMATICA Differential Equations of Mathematical Physics G. Sweers http://go.to/sweers or http://fa.its.tudelft.nl/ sweers ∼ Perugia, July 28 - August 29, 2003 It is better to have failed and tried, To kick the groom and kiss the bride, Than not to try and stand aside, Sparing the coal as well as the guide. John O’Mill ii Contents 1Frommodelstodifferential equations 1 1.1Laundryonaline........................... 1 1.1.1 Alinearmodel........................ 1 1.1.2 Anonlinearmodel...................... 3 1.1.3 Comparingbothmodels................... 4 1.2Flowthroughareaandmore2d................... 5 1.3Problemsinvolvingtime....................... 11 1.3.1 Waveequation........................ 11 1.3.2 Heatequation......................... 12 1.4 Differentialequationsfromcalculusofvariations......... 15 1.5 Mathematical solutions are not always physically relevant . 19 2 Spaces, Traces and Imbeddings 23 2.1Functionspaces............................ 23 2.1.1 Hölderspaces......................... 23 2.1.2 Sobolevspaces........................ 24 2.2Restrictingandextending...................... 29 2.3Traces................................. 34 1,p 2.4 Zero trace and W0 (Ω) ....................... 36 2.5Gagliardo,Nirenberg,SobolevandMorrey............. 38 3 Some new and old solution methods I 43 3.1Directmethodsinthecalculusofvariations............ 43 3.2 Solutions in flavours......................... 48 3.3PreliminariesforCauchy-Kowalevski................ 52 3.3.1 Ordinary differentialequations............... 52 3.3.2 Partial differentialequations............... -
Differential Equations and Linear Algebra
Chapter 1 First Order Equations 1.1 Four Examples : Linear versus Nonlinear A first order differential equation connects a function y.t/ to its derivative dy=dt. That rate of change in y is decided by y itself (and possibly also by the time t). Here are four examples. Example 1 is the most important differential equation of all. dy dy dy dy 1/ y 2/ y 3/ 2ty 4/ y2 dt D dt D dt D dt D Those examples illustrate three linear differential equations (1, 2, and 3) and a nonlinear differential equation. The unknown function y.t/ is squared in Example 4. The derivative y or y or 2ty is proportional to the function y in Examples 1, 2, 3. The graph of dy=dt versus y becomes a parabola in Example 4, because of y2. It is true that t multiplies y in Example 3. That equation is still linear in y and dy=dt. It has a variable coefficient 2t, changing with time. Examples 1 and 2 have constant coefficient (the coefficients of y are 1 and 1). Solutions to the Four Examples We can write down a solution to each example. This will be one solution but it is not the complete solution, because each equation has a family of solutions. Eventually there will be a constant C in the complete solution. This number C is decided by the starting value of y at t 0, exactly as in ordinary integration. The integral of f.t/ solves the simplest differentialD equation of all, with y.0/ C : D dy t 5/ f.t/ The complete solution is y.t/ f.s/ds C : dt D D C Z0 1 2 Chapter 1. -
The Calculus of Trigonometric Functions the Calculus of Trigonometric Functions – a Guide for Teachers (Years 11–12)
1 Supporting Australian Mathematics Project 2 3 4 5 6 12 11 7 10 8 9 A guide for teachers – Years 11 and 12 Calculus: Module 15 The calculus of trigonometric functions The calculus of trigonometric functions – A guide for teachers (Years 11–12) Principal author: Peter Brown, University of NSW Dr Michael Evans, AMSI Associate Professor David Hunt, University of NSW Dr Daniel Mathews, Monash University Editor: Dr Jane Pitkethly, La Trobe University Illustrations and web design: Catherine Tan, Michael Shaw Full bibliographic details are available from Education Services Australia. Published by Education Services Australia PO Box 177 Carlton South Vic 3053 Australia Tel: (03) 9207 9600 Fax: (03) 9910 9800 Email: [email protected] Website: www.esa.edu.au © 2013 Education Services Australia Ltd, except where indicated otherwise. You may copy, distribute and adapt this material free of charge for non-commercial educational purposes, provided you retain all copyright notices and acknowledgements. This publication is funded by the Australian Government Department of Education, Employment and Workplace Relations. Supporting Australian Mathematics Project Australian Mathematical Sciences Institute Building 161 The University of Melbourne VIC 3010 Email: [email protected] Website: www.amsi.org.au Assumed knowledge ..................................... 4 Motivation ........................................... 4 Content ............................................. 5 Review of radian measure ................................. 5 An important limit ....................................