Hot Water Spherical Ball Introduction to Partial Differential Equations 10013

# Hot water spherical ball introduction to partial

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Hot Water Spherical Ball
Introduction to Partial Differential Equations 10.01.3 In spherical co-ordinates, the location is given by r , , co-ordinates. Figure 1 Spherical Coordinate System. The differential equation would now be a partial differential equation and is given as a t t C r k r k r r r r k ) 0 ( , 0 , sin sin sin 2 2 2 2 2 2 2 0 a h r , at the surface (6) where k = thermal conductivity of material, ) /( K m W = density of material, 3 / m kg As an introduction to solve PDEs, most textbooks concentrate on linear second order PDEs with two independent variables and one dependent variable. The general form of such an equation is 0 2 2 2 2 2 D y u C y x u B x u A (7) Where C B A and , , are functions of y x and and D is a function of y u x u u y x , and , , . Depending on the value of AC B 4 2 , a 2 nd order linear PDE can be classified into three categories. x y z r P
10.01.4 Chapter 10.01 1. if 0 4 2 AC B , it is called elliptic 2. if 0 4 2 AC B , it is called parabolic 3. if 0 4 2 AC B , it is called hyperbolic Elliptic Equation The Laplace equation for steady state temperature in a plate is an example of an elliptic second order linear partial differential equation. The Laplace equation for steady state temperature in a plate is given by 0 2 2 2 2 y T x T (8) Using the general form of second order linear PDEs with one dependent variable and two independent variables, 0 2 2 2 2 2 D y u C y x u B x u A 0 , 1 , 0 , 1 D C B A , gives AC B 4 2 ) 1 )( 1 ( 4 0 4 0 4 This classifies Equation (8) as elliptic.

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• Spring '16
• Partial differential equation, Autar Kaw, Sri Harsha Garapati

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