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Homework 13 Due April 27, 2016 .T D T0 . @T [email protected] D 0. Solve r T D 0 for the hemisphere shown. The boundary conditions are T .

Hi, Please help me solve these questions below. These are related to partial differential equations.

Homework 13 Due April 27, 2016 1. Solve r 2 T D 0 for the hemisphere shown. The boundary conditions are T .R; S/ D T 0 S and @T @D ˇ ˇ ˇ ˇ D D 1 2 E D 0 or equivalently @T @S ˇ ˇ ˇ ˇ S D 0 D 0: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................................................................................................................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T D T 0 S @[email protected] D 0 2. Solve r 2 V.r; D/ D 0 for the region around the sphere r D R . The region of interest therefore is R d r < 1 , and the origin is excluded. The boundary conditions are V j r D R D V 0 cos D; and V j r !1 D 0: [Carry out the full Legendre series expansion. However, in the end, the BC at r D R will leave only one term since S D cos D can be expressed as just P 1 .S/; N 1 d S d 1 .] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........................................ .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............... . . . . . . . . . . . . . . . . . . . . ........... . . . . . . . . . . . . . . . . . . . . . . ......................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........ . . . . . . . . . . ................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................................................. .. . V D V 0 cos D V ! 0 as r ! 1 3. Obtain the solution for 1 ˛ @T @t D @ 2 T @r 2 C 2 r @T @r in the region 0 < r < R . The initial/boundary conditions are T D 0 at r D R T < 1 as r ! 0 T D T 0 r=R at t D 0: 4. Solve 1 ˛ @T @t D @ 2 T @r 2 C 2 r @T @r in the spherical shell region c d r d R . 1
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The initial/boundary conditions are T D 0 at r D R @T @r D 0 at r D c (will lead to transcendental equation for eigenvalues) T D T 0 at t D 0: 5. Find the solution for r 2 T D 0 in the hemispherical shell region. The boundary conditions are given by T D 0 at D D 1 2 E T D T 0 at r D R @T @r D 0 at r D c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................................................................................................................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........................................ @T @r D 0 T D T 0 T D 0 2
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