Notes-PDE pt1

Notes-PDE pt1 - Second Order Linear Partial Differential...

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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: α 2 u xx = u t (one-dimensional heat conduction equation) a 2 u xx = u tt (one-dimensional wave equation) u xx + u yy = 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.
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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 : α 2 u xx = u t Where the constant coefficient α 2 is the thermo diffusivity of the bar, given by α 2 = 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 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. 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 ).
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Hence, what we have is a problem given by: (Heat conduction eq.) α 2 u xx = u t , 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
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Notes-PDE pt1 - Second Order Linear Partial Differential...

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