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 g ( y ) ( L, y u y ( x, 0) = u ( x, H )=0 . c. u (0 u ( L, y ( x, 0) u y ( x, 0) = 0 ( x, H f ( x ) . 6. Solve Laplace’s equation outside a circular disk of radius a , subject to a. u ( a, θ )=ln2+4cos3 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 (1 f ( θ ) . b. u θ ( 0) = u θ ( r, π/ 2) = 0 r (1 g ( θ ) . c. u ( 0) = u ( r, π/ 2) = 0 r (1 )=1 . 8. Solve Laplace’s equation inside a circular annulus ( a<r<b ), subject to a. u ( a, θ f ( θ ) ( b, θ g ( θ ) . b. u r ( a, θ f ( θ ) r ( b, θ g ( θ ) . 9. Solve Laplace’s equation inside a semi-inFnite strip (0 <x< , 0 <y<H ) subject to u y ( x, 0) = 0 y ( x, H (0 f ( y ) . 10. Consider the heat equation u t = u xx + q ( x, t ) , 0 <x<L , subject to the boundary conditions
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