Use eigenfunction expansion to solve this IBVP.
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Here we have eigen function in term of r.... Because u(0,theta) is bounded....
2. Use eigenfunction expansion to solve the following IBVP: u,(x, t) ="-(x,t) + (t-1)sin(m), 0
2. Use eigenfunction expansion to solve the following IBVP: please answer v) (fifth one) 2. Use eigenfunction expansion to solve the following IBVP u,(x.t)-u(x.t)+(t-1)sin(a) 0<x<1 t>0 u(0,t)0, u(l,r) 0, t>0 u,(x,t)(x) cos(z), 0 <x<1 t>0 n(x,0) = 2-cos(32t) 0 < x < 1 u(0,0, u(l,t) 0, t>0 n(x,0) = 1 u,(x,0) = 0 0 < x < 1 IV Hm(x,y)+u" (x,y)--r', 0<x<1 0<y<2 u(x,0) = 0, u(x2) =-x 0 < x < 1 v) 7" 11(0,8) bounded , -π<θ<π
Consider the area bounded by θ 0 and θ = π/2 and cos θ + sin θ. Calculate this area
3. Use separation of variables to compute the first five terms of the series solution of the IBVP: urr (r,0) + r-rur (r, θ u (1,0, t) 0, u (r, θ, t) , ur(r, θ, t) bounded as r-+0+,-π < θ < π, t > 0, u (r,0,0) = r sin θ, ut (r.0, 0) = 0, o < r < 1, -π < θ < π. Hint: Follow the example from Lecture 19 and use the fact that with...
4. Use the method of eigenfunction expansion to find the solution of the IBVP ut (x, t) u (0,t) u (x, 0) ura' (a, t) + 2t sin (2na:) , 0 < x < 1, 0, u(1,t)=0, t > 0, sin(2π.r)-5 sin (4π.r) , 0 < x < 1. t > 0, = = = 4. Use the method of eigenfunction expansion to find the solution of the IBVP ut (x, t) u (0,t) u (x, 0) ura' (a, t)...
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...
Solve 2cos0+4 = -6cOs 6 on the interval 0<θ< Σπ. π π. 2π 0, π. 2π π3Τ 2 4
Starting with an expression for U(S,V) , show that π(v) = (dU/dV)T is given by π(v)= (dp/dT)V - p .
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
41. Find the distribution of R-A sin θ, where A is a fixed constant and θ is uniformly distributed on (-π/2, π/2). Such a random variable R arises in the theory of ballistics. If a projectile is fired from the origin at an angle α from the earth with a speed v, then the point R at which it returns to the earth can be expressed as R--(W/g) sin 2α, where g is the gravitational constant, equal to 980 centimeters...