Chem Differential Eq HW Solutions Fall 2011 161

# Chem Differential Eq HW Solutions Fall 2011 161 - u-axis...

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Section 12.6 Solving Dirichlet Problems with Conformal Mappings 161 f1 t_ Re f z . x 1, y t f2 t_ Im f z . x 1, y t ParametricPlot Evaluate f1 t ,f2 t , t,0,Pi 2 , AspectRatio Automatic Cos t Sin t ParametricPlot Evaluate f1 t ,f2 t , t,0,Pi , AspectRatio Automatic 9. We map the region onto the upper half-plane using the mapping f ( z ) = z 2 (see Example 1). The transformed problem in the uv -plane is 2 U = 0 with boundary values on the u -axis given by U ( u, 0) = 100 if 0 < u< 1 and 0 otherwise. The solution in the uv -plane follows from Example 5, Section 12.5. We have U ( u,v ) = 100 π cot - 1 u - 1 v - cot - 1 u v . The solution in the xy -plane is φ ( x,y ) = U f ( z ). To find the formula in terms of ( x,y ), we write z = x + iy , f ( z ) = z 2 = x 2 - y 2 + 2 ixy = ( u,v ). Thus u = x 2 - y 2 and v = 2 xy and so φ ( x,y ) = U f ( z ) = U ( x 2 - y 2 , 2 xy ) = 100 π cot - 1 x 2 - y 2 - 1 2 xy - cot - 1 x 2 - y 2 2 xy . 13. We map the region onto the upper half-plane using the mapping f ( z ) = e z (see Example 2). The points on the x -axis, z = x , are mapped onto the positive u -axis, since e x > 0 for all x , as follows: f ( x ) 1 if x 0 and 0 <f ( x ) < 1 if x< 0. The points on the horizontal line
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Unformatted text preview: u-axis, since e x + i π =-e x < 0, as follows: f ( x + iπ ) =-e x ≤ -1 if x ≥ 0 and-1 < f ( x + iπ ) =-e x < 0 if x < 0. With these observations, we see that the transformed problem in the uv-plane is ∇ 2 U = 0 with boundary values on the u-axis given by U ( u, 0) = 100 if-1 < u < 1 and 0 otherwise. The solution in the uv-plane follows from Example 5, Section 12.5. We have U ( u, v ) = 100 π ± cot-1 ² u-1 v ³-cot-1 ² u + 1 v ³¶ . The solution in the xy-plane is φ ( x, y ) = U ◦ f ( z ). To Fnd the formula in terms of ( x, y ), we write z = x + iy , ( u, v ) = f ( z ) = e z = e x cos y + ie x sin y. Thus u = e x cos y and v = e x sin y and so φ ( x, y ) = U ◦ f ( z ) = U ( e x cos y, e x sin y ) = 100 π ± cot-1 ² e x cos y-1 e x sin y ³-cot-1 ² e x cos y + 1 e x sin y ³¶ ....
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