Question

3. A circular plate of unit radius, whose faces are insulated, has half of its boundary kept at constant temperature u, and the other half at temperature uz (see figure). Find the steady state temperature pf the plate. The steady state heat flow is written as oʻu 1du 1 8²u ar?'r or a dz = 0, with the boundary condition 0<¢ <T, luz, < < 21. u(1,0) = {u, After using the proper separation of variable method, we have u=dy + rm (Am cos mo + Bm sin mo). "(An, cosmẹ + B. sin mẹ). m =1 Using the concept of Fourier series, find the constant Amand Bm Show that the temperature distribution can be written as u(r, t) = 4, +43 + 2(4-22r sino + *r sin 30 +r sin 50+...] [20]

Answer 1

Doubt in any step then comment below.. i will explain you..

.

please thumbs up for this solution..thanks..

.

JA let ५(१, ७।= %3D + A6. c, cos (do) +(2 sin Condo as ArH) finite lu(mie)s E (Anwslnolt fn sinrnos Av ull,ola Bn Sin (noj { An coslno) + 4, do + =)

27 Anc 4, Cos(no) + 42 Coslno) 127 4 Sin(no) Sin(no) An Sin(no) brr 42 4 Sih (no) 2 Cos(no) 4, Cos (no) Br (-n° -)) 42 2 (u, -42) n odd

(4it42) 2 (प, -च) nodd (u, - 42)/r Sinl 4l+sin(34) (4+42)