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Differential Equations Lecture Work Solutions 48

Differential Equations Lecture Work Solutions 48 - 5 Solve...

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5. Solve Laplace’s equation inside a rectangle 0 x L, 0 y H subject to a. u x (0 , y ) = u x ( L, y ) = u ( x, 0) = 0 , u ( x, H ) = f ( x ) . b. u (0 , y ) = g ( y ) , u ( L, y ) = u y ( x, 0) = u ( x, H ) = 0 . c. u (0 , y ) = u ( L, y ) = 0 , u ( x, 0) u y ( x, 0) = 0 , u ( x, H ) = f ( x ) . 6. Solve Laplace’s equation outside a circular disk of radius a , subject to a. u ( a, θ ) = ln 2 + 4 cos 3 θ. b. u ( a, θ ) = f ( θ ) . 7. Solve Laplace’s equation inside the quarter circle of radius 1, subject to a. u θ ( r, 0) = u ( r, π/ 2) = 0 , u (1 , θ ) = f ( θ ) . b. u θ ( r, 0) = u θ ( r, π/ 2) = 0 , u r (1 , θ ) = g ( θ ) . c. u ( r, 0) = u ( r, π/ 2) = 0 , u r (1 , θ ) = 1 . 8. Solve Laplace’s equation inside a circular annulus ( a < r < b ), subject to a. u ( a, θ ) = f ( θ ) , u ( b, θ ) = g ( θ ) . b. u r ( a, θ ) = f ( θ ) , u r ( b, θ ) = g ( θ ) . 9. Solve Laplace’s equation inside a semi-infinite strip (0 < x < , 0 < y < H ) subject to u y ( x, 0) = 0 , u y ( x, H
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