Differential Equations Lecture Work Solutions 91

Differential Equations Lecture Work Solutions 91 -...

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4 PDEs in Higher Dimensions 4.1 Introduction 4.2 Heat Flow in a Rectangular Domain Problems 1. Solve the heat equation u t ( x, y, t )= k ( u xx ( x, y, t )+ u yy ( x, y, t )) , on the rectangle 0 <x<L , 0 <y<H subject to the initial condition u ( x, y, 0) = f ( x, y ) , and the boundary conditions a. u (0 ,y,t )= u x ( L, y, t )=0 , u ( x, 0 ,t )= u ( x, H, t )=0 . b. u x (0 ,y,t )= u ( L, y, t )=0 , u y ( x, 0 ,t )= u y ( x, H, t )=0 . c. u (0 ,y,t )= u ( L, y, t )=0 , u ( x, 0 ,t )= u y ( x, H, t )=0 .
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Unformatted text preview: &lt; x &lt; L, &lt; y &lt; H, &lt; z &lt; W, u t ( x, y, z, t ) = k ( u xx + u yy + u zz ) , subject to the boundary conditions u (0 , y, z, t ) = u ( L, y, z, t ) = 0 , u ( x, , z, t ) = u ( x, H, z, t ) = 0 , u ( x, y, , t ) = u ( x, y, W, t ) = 0 , and the initial condition u ( x, y, z, 0) = f ( x, y, z ) . 91...
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