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...
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
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
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
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
#6
6. What is the solution to the following interior Dirichlet problem with radius R 2 u (2,0) sin θ 0 < θ < 2π BC
6. What is the solution to the following interior Dirichlet problem with radius R 2 u (2,0) sin θ 0
Q3. [22 marks] The Dirichlet's problem for a disc of radius a is stated as follows: r(a, θ)-/(0) for osas2m, where the function f (0) is integrable [10 marks] Find the general solution of u(r, θ) (i) (7 marks] if f (θ)-sin|-θ | , find the specific solution u (r,0) (ii) [ (ii) [5 marks] Use the solution in (ii) to deduce that 4n1-9) 18 Q4. [24marks] Consider the second order linear partial differential equation
Q3. [22 marks] The Dirichlet's...
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
nonhomogeneous vibrating string problem for u(x with homogeneous boundary conditions t > 0 u(0, t) u(r,t) = 0, 0, = and the initial conditions 0stst tr(z,0)=0, u(z, 0) sin(2x), = Find the solution u(x,t) to the IBVP using an eigenfunction expansion: u(z, t) = Σ an(t) sin(nz) n-1
nonhomogeneous vibrating string problem for u(x with homogeneous boundary conditions t > 0 u(0, t) u(r,t) = 0, 0, = and the initial conditions 0stst tr(z,0)=0, u(z, 0) sin(2x), = Find the...
1. Find the solution to the following boundary value problem on Ω (0,2) × (0,00): (102 -) u(x,t)-0 (, t) E S2 () 0, I] r E1,2 u(0, t) = u(2, t) = 2 , where t > 0 a [0,2
1. Find the solution to the following boundary value problem on Ω (0,2) × (0,00): (102 -) u(x,t)-0 (, t) E S2 () 0, I] r E1,2 u(0, t) = u(2, t) = 2 , where t > 0...
4.[10] Find the solution to given initial-boundary value problem: 4uxx = U, 0 < x <TT, t> 0 u(0,t) = 5, u(t, t) = 10, t> 0 u(x,0) = = sin 3x - sin 5x, 0<x<