1. Solve the boundary value problem ut =-3uzzzz + 5uzz, u(z, 0) = r(z) (-00 < z < oo, t > 0), usi...
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
Solve the following problems: ut(x, t)--uxx (x, t) n(x,t), i, (x, t ) → 0, u(x,0) - e-' u, (x,t) + 2u, (x,)-u(x,t) n(x,t), u(x,0)-f(x), u.(x,0)-g(r) u,, (xs (x,t) 4 a) as|x| → t>0 b) as|x| → 0 u(x,0)-f(x), u.(r,0)-g(x) (Write the answer in the inverse Fourier Transform.) n(x, 0) = f(x) Solve the following problems: ut(x, t)--uxx (x, t) n(x,t), i, (x, t ) → 0, u(x,0) - e-' u, (x,t) + 2u, (x,)-u(x,t) n(x,t), u(x,0)-f(x), u.(x,0)-g(r) u,, (xs...
4. Solve the initial, boundary value problem by the Fourier integral method te = kurz, 0<x<oo, t>0, us(0,t) = 0, u(x, t) bounded as T-100 0S$ 0, >4 f(x)-( 4 u(z,0)=f(x), 4. Solve the initial, boundary value problem by the Fourier integral method te = kurz, 04 f(x)-( 4 u(z,0)=f(x),
1. Use the Fourier Transform to solve the following problem with W1 21 (a) Find the Fourier Transform of u by applying F to the equation and initial condition; denote this function U(w, t). (b) Find u u(z, t) by taking the inverse transform of the U(w, t) you found in part (a). 1. Use the Fourier Transform to solve the following problem with W1 21 (a) Find the Fourier Transform of u by applying F to the equation and...
Problem 1: We are interested in solving a modified form of diffusion equation given below using Fourier transforms au(x,t) The domain of the problem is-oo < x < oo and is 0 < t < oo . At time t = 0, the initial condition is given by u (x,0)-0 a) Take the Fourier transform on x and show that the above PDE can be transformed into the following ODE where G() is the Fourier transform of g(x) and U(w,...
PDE questions. Please show all steps in detail. 2. Consider the initial-boundary value problem 0
show all the work 3. Solve the initial-boundary value problem. ur(0,t) 0, u(4,t) 0, A, 0 2 u(z,0) =
Apply method of images and Fourier transformation to solve the following boundary value problem for the heat equation in the quarter plane, ut = 8uzz, cos 2 u(r,0 Uz (0, t) = 0, (10) lim u(x, t)=0 Apply method of images and Fourier transformation to solve the following boundary value problem for the heat equation in the quarter plane, ut = 8uzz, cos 2 u(r,0 Uz (0, t) = 0, (10) lim u(x, t)=0
(a) Find the Fourier transform of the following function (b) Using Fourier transforms, solve the wave equation , -∞<x<∞ t>0 and bounded as ∞ f(r)e We were unable to transcribe this imageu(r, 0)e 4(r.0) =0 , t ur. We were unable to transcribe this image f(r)e u(r, 0)e 4(r.0) =0 , t ur.
3. Consider the non homogeneous heat equation ut- urr+ 1 with non homogeneous boundary conditions u(0. t) 1, u(1t) (a) Find the equilibrium solution ueqx) to the non homogeneous equation. (b) The solution w(r, t) to the homogenized PDE wt-Wra, with w(0,t,t)0 1S -1 Verify that ugen(x, t)Ue(x) +w(x, t) solves the full PDE and BCs (c) Let u(x,0)- f(x) - 2 - ^2 be the initial condition. Find the particular solution by specifying all Fourier coefficients 3. Consider the...