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Coordinate Transformation for 2-D Heat Equation: Case A vs. Case B Comparison

Explore solutions for 2-D steady heat equation in Case A and Case B scenarios using coordinate transformations. Investigate if deriving solution B is necessary when solution A is known. Example: Temperature distribution with shifted constant temperature contours.

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Coordinate Transformation for 2-D Heat Equation: Case A vs. Case B Comparison

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  1. Coordinate Transformation Case A Case B y y T=f(x) T=0 y=b T=0 T=0 T=0 T=0 x x T=0 x=a T=f(x) 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

  2. y T=f(x) y=b y’=0 T=0 T=0 y=0 y’=b x T=0 x=a y’

  3. 2-D Steady Heat Equation y’ y’=b x=a x y=0 x=0 T(x,b)=f(x)

  4. New Example

  5. New variable

  6. Temperature Distribution in x at different y stations f(x)=100(3-x)sin(x) y’=6-y

  7. Constant temperature contour plot has shifted so that the maximum temperature occurs at the bottom as it should be in case B.

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