PDE_Handout_2x2

PDE_Handout_2x2 - Definitions and examples The wave...

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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 d’Alembert’s solution ) to the one-dimensional wave equation , ∂ 2 u ∂ t 2 = c 2 ∂ 2 u ∂ x 2 . Is the two-dimensional 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 Δ u = 0, where Δ is the two- or three-dimensional Laplacian. Is this equation homogeneous? Chapter 12: Partial Differential Equations Definitions and examples The wave equation The heat equation The one-dimensional wave equation Separation of variables The two-dimensional wave equation 2. The one-dimensional wave equation The one-dimensional wave equation models the 2-dimensional 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 one-dimensional wave equation Separation of variables The two-dimensional 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 one-dimensional wave equation gives 1 c 2 G ( t ) d 2 G dt 2 = 1 F d 2 F dx 2 ....
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This note was uploaded on 12/01/2009 for the course MATH 322 taught by Professor Staff during the Spring '08 term at Arizona.

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PDE_Handout_2x2 - Definitions and examples The wave...

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