Second Order Linear Partial Differential （二阶线性偏微分）.pdfVIP

Second Order Linear Partial Differential Equations Part I Second linear partial differential equations; Separation of Variables; 2 point boundary value problems; Eigenvalues and Eige nfunctions 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: 2 α u = u (onedimensional heat conduction equation) xx t 2 a u = u (onedimensional wave equation) xx tt uxx + uyy = 0 (twodimensional 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 E1 1 (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 u + b u + c u + d u + e u + f u = g(x ,y ). xx xy yy 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