Introduction to Partial Differential Equations

Transcription

1 Introduction to Partial Differential Equations Philippe B. Laval KSU Current Semester Philippe B. Laval (KSU) Key Concepts Current Semester 1 / 25

2 Introduction The purpose of this section is to define what partial differential equations are, discuss why such equations are so important, present the various techniques used in solving them. We finish with a brief discussion on classification. These concepts will be studied in great detail throughout the semester. One can think of this section as an outline for the work we will be doing this semester. Philippe B. Laval (KSU) Key Concepts Current Semester 2 / 25

3 Operators The general definition of an operator coincides with the definition of a mapping or a function. Definition (Operator) Let X and Y be two sets. A rule which assigns a uniquely defined element A (x) Y to every element x of a subset D X is called an operator from X into Y. The term operator is mostly used when both X and Y are vector spaces. Operators are usually denotes with upper case letters. Remark In this class we will study operators which apply to functions. They belong to the class of differential operators, that is they involve derivatives or partial derivatives. When we apply an operator L to a function u we write L (u). Some operators which are used often have their own notation as we will see in the examples. Philippe B. Laval (KSU) Key Concepts Current Semester 3 / 25

4 Operators: s The process of differentiating can be written as an operator. In differential d calculus, is an operator. It acts on functions and compute their dx derivative with respect to x. If we define L = d dx Similarly, In multivariable calculus, x to x operator. If we define L = f then L (f ) = x x. then L (f ) = df dx. is the partial derivative with respect In multivariable calculus, another important ( operator is the gradient f denoted. If f : R 2 R, then f = x, f ). If f : R 3 R, then y f = ( f x, f y, f z ). Philippe B. Laval (KSU) Key Concepts Current Semester 4 / 25

5 Operators: s Consider the operator L defined by L (u) = u t C 2 u where C is a x 2 constant. Consider the operator L defined by L (u) = u t 2 u x 2 + u Consider the operator L defined by L (u) = 2 u t 2 C 2 2 u where C is a x 2 constant. Philippe B. Laval (KSU) Key Concepts Current Semester 5 / 25

6 Operators: Linear Definition (Linear Operator) Let L be an operator. L is said to be a linear operator or linear if it satisfies L (c 1 u 1 + c 2 u 2 ) = c 1 L (u 1 ) + c 2 L (u 2 ) for any two functions u 1 and u 2 and any two constants c 1 and c 2. Prove that the operator we defined in the first example, L = d, is a linear dx operator. Is the operator defined by L (u) = ( ) du 2 a linear operator? dx Philippe B. Laval (KSU) Key Concepts Current Semester 6 / 25

7 What are PDEs? Definition A partial differential equation (PDE) is an equation involving an unknown function of several variables and one or more of its partial derivatives. Every PDE can be written as an operator equation of the form L (u) = f where u denotes the unknown function, L is the operator which corresponds to the portion of the PDE which involves the unknown function u and its partial derivatives. f denotes the part of the PDE which does not involve u and any of its partial derivatives. We illustrate this with some examples. Philippe B. Laval (KSU) Key Concepts Current Semester 7 / 25

8 What are PDEs? PDEs are useful because most of the natural laws of physics, such as Maxwell s equations, Newton s law of cooling, Newton s equations of motion, Schroedinger s equation, Navier-Stokes equations, can be stated in terms of PDEs involving space as well as time derivatives. These derivatives occur because they have a physical meaning such as velocity, acceleration, flux, force, current,... The purpose of this class is: 1 Learn how to formulate the PDE as well as the auxiliary conditions (see below) corresponding to a physical system. That is, we will learn how to construct a mathematical model. 2 Learn how to solve and study the PDE. More modern applications of PDEs include image processing, medical imaging, various mathematical models used in biology, medicine, finance, chemistry... Philippe B. Laval (KSU) Key Concepts Current Semester 8 / 25

9 s of PDEs? u t = cu xx This is the heat equation in one dimension. It can be written as L (u) = 0 with L (u) = u t cu xx. u t = cu xx + sin x. This is the heat equation with source. u t = c (u xx + u yy ) This is the heat equation in two dimensions. u t = c (u xx + u yy + u zz ) This is the heat equation in three dimensions. Philippe B. Laval (KSU) Key Concepts Current Semester 9 / 25

10 s of PDEs? u tt = c 2 (u xx + u yy ) This is the wave equation in two dimensions. u tt = c 2 (u xx + u yy + u zz ) This is the wave equation in three dimensions. u xx + u yy = 0 This is the two dimensional Laplace equation after the mathematician Pierre Laplace ( ). Show that u (x, y) = e x sin y is a solution of Laplace equation in two dimensions. Philippe B. Laval (KSU) Key Concepts Current Semester 10 / 25

11 How to Solve a PDE? This is a very complex question in the sense that there is not a simple answer. There exists many different methods, some of which involve very advanced mathematical topics. Here, we will only list the most important techniques used. Some (but not all) of these techniques will be studied in this class. The technique or techniques used will depend on the type of equation we are trying to solve. Often, the techniques used will transform an equation we cannot solve into one we can solve. The most important techniques are: Separation of variables. This technique reduces a PDE in n variables into n ordinary differential equations (ODE s). Eigenfunction expansion. This method attempts to find the solution of a PDE as an infinite sum of eigenfunctions. These eigenfunctions. are found by solving what is known as an eigenvalue problem corresponding to the given problem. Transformation of the dependent variable. This methods transforms the unknown in a PDE into one easier to find. Philippe B. Laval (KSU) Key Concepts Current Semester 11 / 25

12 How to Solve a PDE? Change of coordinates. This method changes a PDE into an ODE or a PDE easier to solve by changing the coordinates of the problems. For example, one may rewrite a problem given in Cartesian coordinates into one in cylindrical or spherical coordinates. Integral transforms. This procedure reduces a PDE in n independent variables to one in n 1 variables. Numerical methods. These methods transform a PDE into equations which solutions can be approximated on a computer. They usually involve a very large number of computations and can most of the time only be carried out with a computer. Often, only these techniques will work. Perturbation methods. This method changes a nonlinear problem into a sequence of linear ones which approximates the nonlinear one. Philippe B. Laval (KSU) Key Concepts Current Semester 12 / 25

13 How to Solve a PDE? Impulse-response techniques. This procedure decomposes initial and boundary conditions of the problem into simple impulses and finds the response to each impulse. The overall response is then found by adding these simple responses. Integral equations. This technique changes a PDE into an equation where the unknown is inside an integral which is easier to solve. Calculus of variations methods. These methods find the solution by reformulating the equation as a minimization problem. It turns out that the minimum of certain expressions is also the solution the PDE. Philippe B. Laval (KSU) Key Concepts Current Semester 13 / 25

14 Boundary Conditions Usually, the PDEs that model physical systems and have solutions tend to have infinitely many solutions. To select the function which corresponds to the solution for a given problem, auxiliary conditions must be imposed. These fall into two categories: Boundary Conditions: These are conditions which must hold on the spatial boundary of our physical system. The three forms most commonly encountered are: Dirichlet Condition: u = g (t) Neumann Condition: u n = g (t) Mixed Condition: αu + β u n = g (t) In which u is the unknown, α, β, and g are functions prescribed on the boundary and n is the normal to the boundary. Philippe B. Laval (KSU) Key Concepts Current Semester 14 / 25

15 Initial Conditions Initial Conditions: These are the conditions which must be satisfied throughout our physical system, at a given time, usually the initial time that is the time when the study of our system begins. Usually, the PDE describing a physical system tells us how the system is evolving. So, to know how the system is at a certain time, we need to know how it was at some time in the past. This is the initial condition. Suppose that we are studying the temperature of a thin rod 1 meter long, one end being maintained at 0 and the other end at 20. The initial temperature of the rod is uniform and equal to 50. Let the temperature of the rod be given by u (x, t). If we position the rod along the x-axis so that its left end is at the origin, what are the initial and boundary conditions? Philippe B. Laval (KSU) Key Concepts Current Semester 15 / 25

16 Well-Posed Problems Definition A PDE and the auxiliary conditions comprise a well-posed problem if: 1 A solution to the problem exists. 2 The solution is unique. 3 The solution depends continuously on the data that is if small changes in the data correspond to small changes in the solution. If any of these conditions is not satisfied, the problem is said to be ill-posed. Philippe B. Laval (KSU) Key Concepts Current Semester 16 / 25

17 Classification of PDEs: Order As we saw above, there are many techniques which can be used to solve a PDE. The technique used depends on the PDE. To help in deciding which technique to use, PDEs are classified according to several criteria. These include order, number of variables, linearity, homogeneity, kinds of coeffi cients. We look at each criterion in more detail. The order of a PDE is the order of the highest partial derivative in the equation. What is the order of the PDEs below? 1 u t = u xx + u yy 2 u t + u xx + (u yy ) 3 = 0 3 u t + u xxx = u yy Philippe B. Laval (KSU) Key Concepts Current Semester 17 / 25

18 Classification of PDEs: Number of Variables This is the number of independent variables which appear in the equation (space and time variables). u t = u xx + u yy has three independent variables: x, y, and t. u tt = u xx + u yy + u zz has four independent variables: x, y, z, and t. Philippe B. Laval (KSU) Key Concepts Current Semester 18 / 25

19 Classification of PDEs: Linearity A PDE L (u) = f is linear if L is a linear operator. The general form of a second order linear PDE in two variables is: L (u) = G where L (u) = A 2 u x 2 + 2B 2 u y x + C 2 u y 2 + D u x + E u y + Fu In other words, the general form of a second order linear PDE in two variables is: Au xx + 2Bu xy + Cu yy + Du x + Eu y + Fu = G where A, B, C, D, E, F, G can either be constants or functions of the independent variables, some of which could be 0. Remark The part of the equation equal to Au xx + 2Bu xy + Cu yy is called the principal part of the second order linear PDE. As we will see below, it plays an important role in the classification of the equation. Philippe B. Laval (KSU) Key Concepts Current Semester 19 / 25

20 Classification of PDEs: Linearity Which of the equations below are linear? 1 u tt = e t u xx + sin t 2 u x = u 2 3 u x = u xx u yy 4 e t u t + e y u x + e x u y = 0 Philippe B. Laval (KSU) Key Concepts Current Semester 20 / 25

21 Classification of PDEs:Homogeneous Recall we said that a PDE could be written as an operator equation of the form L (u) = f where L is the operator which contains u and its derivative and f is an expression not involving u and any of iuts derivatives. A PDE L (u) = f is homogeneous if f = 0. For example, a second order linear PDE in two variables is homogeneous if G = 0. If the equation is not homogeneous, it is said to be nonhomogeneous. Is u y = u xx homogeneous? Is u tt = e t u xx + sin t homogeneous? Philippe B. Laval (KSU) Key Concepts Current Semester 21 / 25

22 Classification of PDEs:Kind of Coeffi cients If the coeffi cients are constants, then the equation is said to have constant coeffi cients. If the coeffi cients are not constant, they are said to be variable coeffi cients. Note that in this case by coeffi cient, we mean whatever is in front of u or its derivatives. e t u t + e y u x + e x u y has variable coeffi cients. u t + 2u xxx = u yy has constant coeffi cients. Philippe B. Laval (KSU) Key Concepts Current Semester 22 / 25

23 Classification of PDEs: Type Second order linear PDEs (Au xx + 2Bu xy + Cu yy + Du x + Eu y + Fu = G) are either 1 parabolic. Parabolic equations describe heat flow and diffusion processes. They satisfy B 2 AC = 0 2 hyperbolic. Hyperbolic equations describe vibrating systems and wave motion. They satisfy B 2 AC > 0 3 elliptic. Elliptic equations describe steady-state phenomena. They satisfy B 2 AC < 0 Philippe B. Laval (KSU) Key Concepts Current Semester 23 / 25

24 Classification of PDEs: Type Find the type of each equation below: 1 u t = u xx 2 u tt = u xx 3 u xx + u yy = 0 4 yu xx + u yy = 0. Philippe B. Laval (KSU) Key Concepts Current Semester 24 / 25

25 Exercises See the problems at the end of my notes on PDEs: key concepts. Philippe B. Laval (KSU) Key Concepts Current Semester 25 / 25