Olve the following heat problem using the method of separation of variables: lxx 6. S
Please write clearly!
(a) Using the method of separation of variables, find a (formal) solution of the problem describing the heat evolution of an insulated one-dimensional rod (Neumann problem)
Solve the heat equation by the method of separation of variables 3π u(x,0)--2cos( x)
Solve the heat equation by the method of separation of variables 1(1, t) = 0 Эт u,(0, t) = 0, u(x,0) =-2cos( 12.
Solve the heat equation by the method of separation of variables 1(1, t) = 0 Эт u,(0, t) = 0, u(x,0) =-2cos( 12.
Page 6 of 6 5. (20 pts) a) Solve the heat equation by the method of separation of variables t>0, 0Sxs1 11T b) Once solution is obtained, find u(0.6,0.002) c) (Bonus) What physical phenomenon is described by this equation? d) (bonus) What physical laws (name at least two) are the used in deriving of this equation?
Page 6 of 6 5. (20 pts) a) Solve the heat equation by the method of separation of variables t>0, 0Sxs1 11T b) Once...
Q2.PNGA sphere of radius R has a specified potential at it’s surface that is given by: V (R, θ) = kR /epsilon0
(3 cos^2
θ − 1)
. a) Using the method of separation of variables in spherical coordinate, solve
Laplace’s equation to find the potential inside and outside.of the sphere. Refer to Griffith’s examples 3.6 and 3.7 for the method and on how to ”eye-ball”
the coefficients in the general solution. (10 points)Using the continuity equation, find the surface charge density...
2. Use the method of separation of variables to solve the boundary value problem ( au = karu 0<x<L t > 0 (0,t) = 0, > 0 (1.1) -0. > 0 (u(a,0) - (x) 0<x<L. Be sure to detail exactly how f(x) enters your solution E-
Solve the following partial differential equation using separation of variables method to determine the function 0 (x,t). Simplify the solution using Fourier series method. 2²0 2²0 (30 marks) Ox² at is Where: (x,0) = 0 0(0,t) = 0.21 0<t< 20 Q(x,20) = 0 do(0,1)= (1 - 2t) dx = (1-21)
part A
PART IV. 4. Use the Vibration problem. method of Separation of Variables to find the solution of a String A. Ue (x, t)-0.16us (x, ) 0,0x<8 u(0, t)u(8, t)-0,t0 u(x, 0) = 0 , 0
3. Using separation of variables to solve the heat equation, u -kuxx on the interval 0x<1 with boundary conditions u(0 and ur(1, t)-0, yields the general solution, u(x, t) =A0 + Σ Ane-k,t cos(nm) (with A, = ㎡π2) 0<x<l/2 0〈x〈1,2 u(x,0)=f(x)-.., , . . .) when u(x,0) = f(x)- Determine the coefficients An (n - 0, 1,2,
Problem 2 Use the method of separation of variables to compute the solution of SEE NEXT PAGE (WE) con una nova = o in (0, 1)*(0,10), (0,1) = (1,0) = 0,1% 0, M(3:0) = cos(27), (1,0) = 1,0<x<1.