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9-1 CHAPTER 9 NUMERICAL PARTIAL DIFFERENTIAL EQUATIONS (PDE) A classic example of partial differential equations (PDE) is the heat equation that governs the temperature distribution as time evolves. For a uniform one-dimensional object, the equation takes the form of: 2 2 x Q D t Q = , ------(*) where D is called the diffusion coefficient and Q ( x , t ) is the unknown temperature at location x and time t . x 0 L /2 - L /2 t =0 Insulator Hot spot t = Uniform t >0 Diffusion 8 EE 3108 Semster B 2007/2008 S C Chan 9-2 The differentiations are said to be partial because they treat the two independent variables separately. The equation is referred to as homogeneous because it has no dependence on x nor t (except for the derivatives). Similar to ODEs, the initial conditions at t =0 (for all x ) are typically given for an PDE problem. In addition, the boundary conditions at x = ± L /2(for all t ) should also be given. Therefore, the problem is an initial value problem (IVP) and also a boundary problem (BVP). Two approaches are considered in this chapter, which are based on the approximation of the first-order time-derivative. The explicit method uses the forward-difference approximation whilst the implicit method uses the backward- difference approximation. These methods can be modified for solving other types of PDE.
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9-3 I. Physical background A. Heat equation The heat equation belongs to the class of parabolic PDE that are frequently used in electrical engineering such as semiconductor carrier transport and paraxial electromagnetic wave propagation. The heat equation for a one-dimensional rod can be obtained by considering the heat flow J. x Hot Cold Heat flow, J J Absorbs heat On the left, we can deduce that the heat flow is proportional to the heat gradient: x Q J On the right, we can deduce that the change of heat flow has to be absorbed: x J t Q Combine the two equations, we have: 2 2 x Q D t Q = , where D represents the proportionality constant. For copper, D = 1.16 cm 2 /s. 9-4 B. Boundary conditions Suppose we use a copper rod of length L = 10 cm and put it in between two insulators. No heat is permitted to flow at the boundaries: J = 0 at x = ± L /2. Equivalently, in terms of Q , we have the boundary conditions: 0 2 / = ± = L x x Q . Boundary conditions are usually expressed in terms of Q or its derivatives. C. Initial conditions The center portion of the copper rod is first heated before t = 0. The PDE can be used to find the temperature distribution for t > 0. We shall be interested in the time window from t =0 to T , where T = 10 s. The initial temperature distribution is given as: < < = otherwise 25 2 / 2 / for 100 ) 0 , ( a x a x Q , where a = 1 cm is the length of the heated region. The unit for the temperature is chosen to be Celsius here.
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