025 -0.Jt r the posih'on tolevance Pind virtua condiion and Boundary
3. (a) Solve the boundary value problem on the wedge u(r, 0) = 0 0<r<p, a(r, g) = 0 0<r<p, u(p, 0)-/(0), 0 < θ < θο. (b) State the mathematical and physical boundary conditions for this problem. (c) Suppose ρ-1.00-π/3, and f(9)-66ere. Plot the solution surface and polar contour plot for N -10
3. (a) Solve the boundary value problem on the wedge u(r, 0) = 0 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.
4. Consider the boundary-value problem on the region given by {(r, 0, 6)|1 < r < 2}: vu= 0, 1 <r< 2, u(r = 1)= 1, ur(r = 2) = -u(r = 2). Using our work with the Laplace equation in class, find the solution to this problem. [Hint: it depends only on r, not on 0 or ø.
4. Consider the boundary-value problem on the region given by {(r, 0, 6)|1
30] Find th e solution of the following boundary value problem. 1<r<2, u(r, θ = 0) = 0, u(r, θ = π) =0, 1,0-0, u(r-2,0)-sin(20), 0 < θ < π. u(r Please also draw the sketch associated with this problem. You may assume that An -n2, Hn(s)sin(ns), n 1,2,3,. are the eigenpairs for the eigenvalue problem H(0) 0, H(T)0.
30] Find th e solution of the following boundary value problem. 1
7. Find the interior and boundary of each of the sets(VR:nEN) and r EQ:0<< 2
35 Given a boundary-value problem defined by =i+1, 0<r <1 subject to (0)= 0 and 0(1)= 1, use the finite difference method to find (0.5). You may take A = 0.25 and perform 5 iterations. Compare your result with the exact solution.
Assume u E C2 (B (0,r)) solves the boundary-value problem where g E C(OB(0,r)). Show that gry.ndS(y) (хев"(0. т.)) which is called Poisson's formula with Poisson's kernel
Assume u E C2 (B (0,r)) solves the boundary-value problem where g E C(OB(0,r)). Show that gry.ndS(y) (хев"(0. т.)) which is called Poisson's formula with Poisson's kernel
Find the solution to ∇2 Ψ(r, θ, φ) = 0 inside a sphere with the
following boundary conditions:
∂Ψ (1,θ,φ)=sin2θcosφ.∂r
Find the solution to V2(r, e, p) =0 inside a sphere with the following boundary conditions: ay (1, e, p) sin20 cosp. ar
Find the solution to V2(r, e, p) =0 inside a sphere with the following boundary conditions: ay (1, e, p) sin20 cosp. ar
4. Solve the initial, boundary value problem by the Fourier integral method. u (0,t)0, u(r,t) bounded as-00
4. Solve the initial, boundary value problem by the Fourier integral method. u (0,t)0, u(r,t) bounded as-00
QUESTION 2 Consmder the problem ди 2k, 0<r< 1, t>O оt and the boundary conditions u(0,t)= 1, u (1,t) = 3, t > 0 (a) Find the equiltbrium solutiou ug (r) (b) Find the solution u (z.t) of the PDE and the boundary condition which also satisfies the mitial condition (,0)-1+++sin (3wx), 0<o< 1 [25]