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Partial Differential Equations PARTIAL DIFFERENTIAL EQUATIONS Math 124A { Fall 2010 «Viktor Grigoryan [email protected] Department of Mathematics University of California, Santa Barbara These lecture notes arose from the course \Partial Differential Equations" { Math 124A taught by the author in the Department of Mathematics at UCSB in the fall quarters of 2009 and 2010. The selection of topics and the order in which they are introduced is based on [Str]. Most of the problems appearing in this text are also borrowed from Strauss. A list of other references that were consulted while teaching this course appears in the bibliography at the end. These notes are copylefted, and may be freely used for noncommercial educational purposes. I will appreciate any and all feedback directed to the e-mail address listed above. The most up-to date version of these notes can be downloaded from the URL given below. Version 0.1 - December 2010 http://www.math.ucsb.edu/~grigoryan/124A.pdf Contents 1 Introduction 1 1.1 Classification of PDEs . 1 1.2 Examples . 2 1.3 Conclusion . 3 Problem Set 1 4 2 First-order linear equations 5 2.1 The method of characteristics . 6 2.2 General constant coefficient equations . 7 2.3 Variable coefficient equations . 8 2.4 Conclusion . 10 3 Method of characteristics revisited 11 3.1 Transport equation . 11 3.2 Quasilinear equations . 12 3.3 Rarefaction . 13 3.4 Shock waves . 14 3.5 Conclusion . 14 Problem Set 2 15 4 Vibrations and heat flow 16 4.1 Vibrating string . 16 4.2 Vibrating drumhead . 17 4.3 Heat flow . 18 4.4 Stationary waves and heat distribution . 18 4.5 Boundary conditions . 19 4.6 Examples of physical boundary conditions . 19 4.7 Conclusion . 20 5 Classification of second order linear PDEs 21 5.1 Hyperbolic equations . 23 5.2 Parabolic equations . 24 5.3 Elliptic equations . 24 5.4 Conclusion . 25 Problem Set 3 26 6 Wave equation: solution 27 6.1 Initial value problem . 28 6.2 The Box wave . 29 6.3 Causality . 31 6.4 Conclusion . 33 7 The energy method 34 7.1 Energy for the wave equation . 34 7.2 Energy for the heat equation . 36 7.3 Conclusion . 37 Problem Set 4 38 i 8 Heat equation: properties 39 8.1 The maximum principle . 39 8.2 Uniqueness . 41 8.3 Stability . 42 8.4 Conclusion . 42 9 Heat equation: solution 43 9.1 Invariance properties of the heat equation . 43 9.2 Solving a particular IVP . 44 9.3 Solving the general IVP . 45 9.4 Conclusion . 46 Problem Set 5 47 10 Heat equation: interpretation of the solution 48 10.1 Dirac delta function . 48 10.2 Interpretation of the solution . 50 10.3 Conclusion . 52 11 Comparison of wave and heat equations 53 11.1 Comparison of wave to heat . 55 11.2 Conclusion . 56 Problem Set 6 57 12 Heat conduction on the half-line 58 12.1 Neumann boundary condition . 60 12.2 Conclusion . 61 13 Waves on the half-line 62 13.1 Neumann boundary condition . 64 13.2 Conclusion . 65 14 Waves on the finite interval 66 14.1 The parallelogram rule . 69 14.2 Conclusion . 69 Problem Set 7 70 15 Heat with a source 71 15.1 Source on the half-line . 73 15.2 Conclusion . 74 16 Waves with a source 75 16.1 Source on the half-line . 77 16.2 Conclusion . 78 17 Waves with a source: the operator method 79 17.1 The operator method . 80 17.2 Conclusion . 82 Problem Set 8 83 ii 18 Separation of variables: Dirichlet conditions 84 18.1 Wave equation . 84 18.2 Heat equation . 85 18.3 Eigenvalues . 86 18.4 Conclusion . 87 19 Separation of variables: Neumann conditions 88 19.1 Wave equation . 88 19.2 Heat equation . 90 19.3 Mixed boundary conditions . 90 19.4 Conclusion . 90 Problem Set 9 91 Bibliography 92 iii 1 Introduction Recall that an ordinary differential equation (ODE) contains an independent variable x and a dependent variable u, which is the unknown in the equation. The defining property of an ODE is that derivatives 0 du of the unknown function u = dx enter the equation. Thus, an equation that relates the independent variable x, the dependent variable u and derivatives of u is called an ordinary differential equation. Some examples of ODEs are: u0(x) = u u00 + 2xu = ex u00 + x(u0)2 + sin u = ln x In general, and ODE can be written as F (x; u; u0; u00;::: ) = 0. In contrast to ODEs, a partial differential equation (PDE) contains partial derivatives of the depen- dent variable, which is an unknown function in more than one variable x; y; : : : . Denoting the partial @u @u derivative of @x = ux, and @y = uy, we can write the general first order PDE for u(x; y) as F (x; y; u(x; y); ux(x; y); uy(x; y)) = F (x; y; u; ux; uy) = 0: (1.1) Although one can study PDEs with as many independent variables as one wishes, we will be primar- ily concerned with PDEs in two independent variables. A solution to the PDE (1.1) is a function u(x; y) which satisfies (1.1) for all values of the variables x and y. Some examples of PDEs (of physical significance) are: ux + uy = 0 transport equation (1.2) ut + uux = 0 inviscid Burger's equation (1.3) uxx + uyy = 0 Laplace's equation (1.4) utt − uxx = 0 wave equation (1.5) ut − uxx = 0 heat equation (1.6) ut + uux + uxxx = 0 KdV equation (1.7) iut − uxx = 0 Shr¨odinger'sequation (1.8) It is generally nontrivial to find the solution of a PDE, but once the solution is found, it is easy to verify whether the function is indeed a solution. For example to see that u(t; x) = et−x solves the wave equation (1.5), simply substitute this function into the equation: t−x t−x t−x t−x (e )tt − (e )xx = e − e = 0: 1.1 Classification of PDEs There are a number of properties by which PDEs can be separated into families of similar equations. The two main properties are order and linearity. Order. The order of a partial differential equation is the order of the highest derivative entering the equation. In examples above (1.2), (1.3) are of first order; (1.4), (1.5), (1.6) and (1.8) are of second order; (1.7) is of third order. Linearity. Linearity means that all instances of the unknown and its derivatives enter the equation linearly. To define this property, rewrite the equation as Lu = 0; (1.9) @2 where L is an operator, which assigns u a new function Lu. For example L = @x2 +1, then Lu = uxx +u. The operator L is called linear if L(u + v) = Lu + Lv; and L(cu) = cLu (1.10) 1 for any functions u; v and constant c. The equation (1.9) is called linear, if L is a linear operator. In our examples above (1.2),.
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