Lecture10

# Lecture10 - We consider Dirichlet boundary value problems...

This preview shows pages 1–3. Sign up to view the full content.

We consider Dirichlet boundary value problems which are posed either inside or outside of a sphere with radius R > 0: our domain Ω is either Ω := { x R 3 | | x | < R } ( squiggleright “interior problem”), or (361) Ω + := { x R 3 | | x | > R } ( squiggleright “exterior problem”). (362) Notice that Ω is a bounded domain, whereas Ω + is not. In both cases, the boundary value problem has the form Δ u = 0 in Ω , (363) u ( R, ϕ, ϑ ) = f ( ϕ, ϑ ) , ϕ [0 , 2 π ) , ϑ [0 , π ] , (364) lim r →∞ u ( r, ϕ, ϑ ) = 0 , ϕ [0 , 2 π ) , ϑ [0 , π ] . (365) Separation of Variables We write the unknown function in the form u ( r, ϕ, ϑ ) = F ( r ) G ( ϕ, ϑ ). We separate the radial variable from the angular variables by r 2 F ′′ F + 2 r F F = - 1 G Δ S 2 G = k R , (366) and the separate differential equations are given by r 2 F ′′ + 2 rF - kF = 0 , r < R or r > R, (367) Δ S 2 G + kG = 0 , ϕ [0 , 2 π ) , ϑ [0 , π ] . (368) We solve for G first. For that purpose, we separate again, G ( ϕ, ϑ ) = H ( ϕ ) Q ( ϑ ), and we obtain - H ′′ H = sin ϑ Q (sin ϑQ ) + k sin 2 ϑ = μ 2 R . (369) The separate ODEs are now given by H ′′ + μ 2 H = 0 , ϕ [0 , 2 π ) , (370) sin 2 ϑQ ′′ + sin ϑ cos ϑQ + ( k sin 2 ϑ - μ 2 ) Q = 0 , ϑ [0 , π ] . (371) 64

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Auxiliary Conditions and Solution of Separate ODEs 2 π -periodic solutions of ( 370 ) are found for μ = m Z , and they are given by H m ( ϕ ) = e imϕ . (372) Remark: We prefer to use the complex exponentials here, instead of cos( ) and sin( ), m Z .
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern