© 2008, 2012 Zachary S Tseng E-1 - 1 Second Order Linear Partial Differential Equations Part I Second linear partial differential equations; Separation of Variables; 2-point boundary value problems; Eigenvalues and Eigenfunctions Introduction We are about to study a simple type of partial differential equations (PDEs): the second order linear PDEs. Recall that a partial differential equation is any differential equation that contains two or more independent variables. Therefore the derivative(s) in the equation are partial derivatives. We will examine the simplest case of equations with 2 independent variables. A few examples of second order linear PDEs in 2 variables are: α2 uxx = ut (one-dimensional heat conduction equation) a2 uxx = utt (one-dimensional wave equation) uxx + uyy = 0 (two-dimensional Laplace/potential equation) In this class we will develop a method known as the method of Separation of Variables to solve the above types of equations.© 2008, 2012 Zachary S Tseng E-1 - 2 (Optional topic) Classification of Second Order Linear PDEs Consider the generic form of a second order linear partial differential equation in 2 variables with constant coefficients: a uxx + b uxy + c uyy + d ux + e uy + f u = g(x,y). For the equation to be of second order, a, b, and c cannot all be zero. Define its discriminant to be b2 – 4ac. The properties and behavior of its solution are largely dependent of its type, as classified below. If b2 – 4ac > 0, then the equation is called hyperbolic. The wave equation is one such example. If b2 – 4ac = 0, then the equation is called parabolic. The heat conduction equation is one such example. If b2 – 4ac < 0, then the equation is called elliptic. The Laplace equation is one such example. In general, elliptic equations describe processes in equilibrium. While the hyperbolic and parabolic equations model processes which evolve over time. Example: Consider the one-dimensional damped wave equation 9uxx = utt + 6ut. It can be rewritten as: 9uxx − utt − 6ut = 0. It has coefficients a = 9, b = 0, and c = −1. Its discriminant is 9 > 0. Therefore, the equation is hyperbolic.© 2008, 2012 Zachary S Tseng E-1 - 3 The One-Dimensional Heat Conduction Equation Consider a thin bar of length L, of uniform cross-section and constructed of homogeneous material. Suppose that the side of the bar is perfectly insulated so no heat transfer could occur through it (heat could possibly still move into or out of the bar through the two ends of the bar). Thus, the movement of heat inside the bar could occur only in the x-direction. Then, the amount of heat content at any place inside the bar, 0 < x < L, and at any time t > 0, is given by the temperature distribution function u(x, t). It satisfies the homogeneous one-dimensional heat conduction equation: α2 uxx = ut Where the constant coefficient α2 is the thermo diffusivity of the bar, given by α2 = k / ρs. (k = thermal conductivity, ρ = density, s = specific heat, of the material of the bar.) Further, let us assume that both ends of the bar are kept constantly at 0 degree temperature (abstractly, by connecting them both to a heat reservoir© 2008, 2012 Zachary S Tseng E-1 - 4 of the same temperature; more practically, say they are immersed in iced water). This assumption imposes explicit restriction on the bar’s ends, in this case: u(0, t) = 0, and u(L, t) = 0. t > 0 Those two conditions are called the boundary conditions of this problem. They literally specify the conditions present at the boundaries between the bar and the outside. Think them as the “environmental factors” of the given problem. In addition, there is an initial condition: the initial temperature distribution within the bar, u(x, 0). It is a snapshot of the temperature everywhere inside the bar at t = 0. Therefore, it is an (arbitrary) function of the spatial variable x only. That is, the initial condition is u(x, 0) = f (x). Hence, what we have is a problem given by: (Heat conduction eq.) α2 uxx = ut , 0 < x < L, t > 0, (Boundary conditions) u(0, t) = 0, and u(L, t) = 0, (Initial condition) u(x, 0) = f (x). This is an example of what is known, formally, as an initial-boundary value problem. Although it is still true that we will find a general solution first, then apply the initial condition to find the particular solution. A major difference now is that the general solution is dependent not only on the equation, but also on the boundary conditions. In other words, the given partial differential equation will have different general solutions when paired with different sets of boundary conditions.© 2008, 2012 Zachary S Tseng E-1 - 5 If the boundary conditions specify u, e.g. u(0, t) = f (t) and u(L, t) = g(t), then they are often called Dirichlet conditions. If they specify the (spatial) derivative, e.g. ux(0, t) = f (t) and ux(L, t) = g(t), then they are often called Neumann conditions. If the boundary conditions are linear combinations of u and its derivative, e.g. α u(0, t) + β ux(0, t) = f (t), then they are called Robin conditions. Those are the 3 most common classes of boundary conditions. If the specified functions in a set of condition are all equal to zero, then they are homogeneous. Our current example, therefore, is a homogeneous Dirichlet type problem. But before any of those boundary and initial conditions could be applied, we will first need to process the given partial differential equation. What can we do with it? There are other tools (by Laplace transforms, for example), but the most accessible method to us is called the method of Separation of Variables. The idea is to somehow de-couple the independent variables, therefore rewrite the single partial differential equation into 2 ordinary differential equations of one independent variable each (which we already know how to solve). We will solve the 2 equations individually, and then combine their results to find the general solution of the given partial differential equation. For a reason that should become clear very shortly, the method of Separation of Variables is sometimes called the method of Eigenfunction Expansion.© 2008, 2012 Zachary S Tseng E-1 - 6 Separation of Variables Start with the one-dimensional heat conduction equation α2 uxx = ut . Suppose that its solution u(x, t) is
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