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One end of the pipe is closed, which corresponds to the boundary conditionu(0, t) = 0, for t > 0. The other end of the pipe is open, which correspondsto the boundary condition ux(L, t) = 0, for t > 0.(a) Suppose that µ < 0, so µ = −k^2 for some k > 0. Find the non-trivial solution X(x) that satisfies equations (3), stating clearly what values k is allowed to take.(b) Write down the general solution of equation...
QUESTION 22.1 Assume you have a 2D concrete slab that has dimensions of 1 x 1. The boundaries x=0 and y=0 both have a temperature of 00C, while the boundaries x=1 and y=1 both have a temperature of 150x and 120y respectively. (i) Plot/draw the computational domain taking step sizes in both the x-direction and the y-direction to be 0.1 units. Show all the boundary conditions. (10)(ii) How many computational nodes are in this problem? (5)2.2 Assuming the problem is a steady state heat...
WAVE EQUATION (Applied Differential Equations) Write out the solution of with , and . Then graph the shape of the wave at different t-values and describe the motion. Graph the velocity at different times and discuss its values. 41111-utt ці0. t) = u(6, t) = 0 a(2, 0-0 1 12 41111-utt ці0. t) = u(6, t) = 0 a(2, 0-0 1 12
Solve the following wave partial differential equation of the vibration of string for ?(? ,?). yxx=16ytt y(0,t)=y(1,t)=0 y(x,0)=2sin(x)+5sin(3x) yt(x,0)=6sin(4x)+10sin(8x) We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
3. (20 points) Consider a modified wave equation partial differential equation -2+ utt = ur- (a) Take the Fourier transform of uu u ug t u, e expressing ün in terms of ú where Flu). u (b) Pind a solution for u(w, t) to the equation derived in part a. (c) Describe how your solution in part b compares to the solution of the original wave equation (ull = urr), what is the longterm behavior of your solution to the...
The temperature distribution Θ(x, t) along an insulated metal rod oflength L is described by the differential equation.The rod is held at a fixed temperature of 0◦ C atone end and is insulated at the other end, which gives rise to the boundaryconditions Θ(0, t) = 0 and Θx(L, t) = 0, for t > 0.Show that function Xn(x) satisfies the boundary conditions that you found. Show that Xn(x) satisfies differential equation (1) for some constant µ (which you should...
Solved examples of second shifting property of Laplace transform...?