4. Find the harmonic function in the exterior {r > a} of a sphere that satisfies...
(a) Find the function u which is harmonic outside the circle r - a and satisfies the Neumann condition ou (a,d) =cos(24). (b) State clearly the Weierstrass M-test. Use it to justify mathematically that with solves the equation ou- u=0, 0<z<a, t > to > 0. (a) Find the function u which is harmonic outside the circle r - a and satisfies the Neumann condition ou (a,d) =cos(24). (b) State clearly the Weierstrass M-test. Use it to justify mathematically that...
7. (a) Find the harmonic function in the semi-infinite strip {0 < x < π, 0sy oo) that satisfies the "boundary conditions": u(r, y) (b) What would go awry if we omitted the condition at infinity? 7. (a) Find the harmonic function in the semi-infinite strip {0
9. Find a bounded solution to the exterior boundary value problem Δυ = 0, r>R, u = 1 + 2 sin θ on r-R. 9. Find a bounded solution to the exterior boundary value problem Δυ = 0, r>R, u = 1 + 2 sin θ on r-R.
Find a vector function r that satisfies the following conditions. r"(t) = 7 cos 2ti + 5 sin 9tj + t?k, r(0) = i + k, r'(O) = i +j+k
\((30\) marks) The electric potential in \(V(r, \theta)\); mtside a hollow empty sphere of radius 1 satisfies the Laplace equation. On the surface of the sphere, \(V(1, \theta)=1-\cos 2 \theta\). Given that \(\lim _{r \rightarrow \infty} V(r, \theta)=0\), find \(V(r, \theta) .\)
Question 57.5 from Fourier series and Boundary value problems Brown and Churchill S Find the bounded harmonic function ux, y) in the semi-infinite strip0< 1,y that satisfies the conditions 2 Answer: u(x, y)E sinh ax cos α f(s) cos as ds da. S Find the bounded harmonic function ux, y) in the semi-infinite strip0
1. Draw the given sets. Find their interior, exterior and boundary and also sketch them. Moreover, study whether these are open, closed, bounded and/or compact. (0) A = (3.7]{0,1} E R. (ii) B = {(x,y) ERO SysIn... ISIS 2). (iii) C = {(x,y) E R' + ly <l},
Problem 3 (12 points): Let D be a bounded domain in R" with smooth boundary. Suppose that K(x, y) is a Green's function for the Neumann . For each x E D, the function y H K(x, y) is a smooth harmonic For each x E D, the normal derivative of the function y K(x, y) . For each z e D, the function y K(x,y)-Г(z-y) is smooth near problem. This means the following: function on D(r satisfies (VyK(x, y).v(b))-arefor...
se melod of images to find Gren's function for the exterior 3. Use the method of images to find Green's function for the exterior Dirichlet problem: vu = -f. r>R, -00 < < u(R ) = 0.
6. The electric potential at the surface of a sphere of radius R is constant, i.e., V(R,0) = k, where k + 0. Very far away from the sphere (r >> R) the electric potential is V(r,0) = kr cos(0). Find the electric potential outside the sphere, remember to check that your answer matches the boundary conditions (1 point).