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Unformatted text preview: Definitions and examples The wave equation The heat equation Chapter 12: Partial Differential Equations Chapter 12: Partial Differential Equations Definitions and examples The wave equation The heat equation Definitions Examples 1. Partial differential equations A partial differential equation (PDE) is an equation giving a relation between a function of two or more variables, u , and its partial derivatives. The order of the PDE is the order of the highest partial derivative of u that appears in the PDE. A PDE is linear if it is linear in u and in its partial derivatives. A linear PDE is homogeneous if all of its terms involve either u or one of its partial derivatives. A solution to a PDE is a function u that satisfies the PDE. Finding a specific solution to a PDE typically requires an initial condition as well as boundary conditions . Chapter 12: Partial Differential Equations Definitions and examples The wave equation The heat equation Definitions Examples Examples Check that u = f ( x + ct ) + g ( x ct ), where f and g are two smooth functions, is a solution (called dAlemberts solution ) to the onedimensional wave equation , 2 u t 2 = c 2 2 u x 2 . Is the twodimensional wave equation (given below) linear? 2 u t 2 = c 2 2 u x 2 + 2 u y 2 What is the order of the heat equation u t = 2 u x 2 ? The Laplace equation reads u = 0, where is the two or threedimensional Laplacian. Is this equation homogeneous? Chapter 12: Partial Differential Equations Definitions and examples The wave equation The heat equation The onedimensional wave equation Separation of variables The twodimensional wave equation 2. The onedimensional wave equation The onedimensional wave equation models the 2dimensional dynamics of a vibrating string which is stretched and clamped at its end points (say at x = 0 and x = L ). The function u ( x , t ) measures the deflection of the string and satisfies 2 u t 2 = c 2 2 u x 2 , c 2 T , T tension of the string with Dirichlet boundary conditions u (0 , t ) = u ( L , t ) = 0 , for all t . In what follows, we assume that the initial conditions are u ( x , 0) = f ( x ) , u t ( x , 0) u t ( x , 0) = g ( x ) , for x [0 , L ] . Chapter 12: Partial Differential Equations Definitions and examples The wave equation The heat equation The onedimensional wave equation Separation of variables The twodimensional wave equation Solution by separation of variables We look for a solution u ( x , t ) in the form u ( x , t ) = F ( x ) G ( t ). Substitution into the onedimensional wave equation gives 1 c 2 G ( t ) d 2 G dt 2 = 1 F d 2 F dx 2 ....
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 Spring '08
 STAFF
 Math, Equations, Partial Differential Equations

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