Coordinate Transformation

# Coordinate Transformation - Case A y Coordinate...

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Coordinate Transformation x y T=f(x) T=0 T=0 T=0 x y T=0 T=0 T=f(x) T=0 For the obvious reason, the solution of case A will be different from that of case B. However, the question is whether do we need to derive the solution B from scratch if solution A is available Case A Case B x=a y=b

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The problem can be simplified by doing a simple coordinate transformation: by assigning a new variable such that ' - Therefore 0 ' ; ' 0 In other words, the case A will transform to y y b y y y b y b y b y b y = = = - = = = - = case B if the new variable ' (shown as the red axis) is used and vice versa. y x T=f(x) T=0 T=0 T=0 x=a y=b y’=0 y=0 y’=b y y’
2-D Steady Heat Equation 2 2 2 2 2-D Heat Equation: 0 Bounday Conditions: ( 0, ') 0, ( , ') 0 ( , ' ) 0, ( , ' 0) ( ) as shown below. Note that I have changed into ', the new variable T T x y T x y T x a y T x y b T x y f x y y + = = = = = = = = = x y’ y’=b y=0 x=0 x=a T(x,b)=f(x)

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New Example ( 29 1 1 1 If 3, 6 ( ) 100(3 )sin( ) between 0 3 ( ,6) ( ) 100(3 )sin( ) sin sinh 2 3 100(3 )sin( ) sinh sin sin where sinh i n n n n n n n n a b f x x x x n x T x f x x x
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