5. (15 %) (a) Solve the partial differential equation u,-ku +cu,, u(x,0)-f(x) k.c are constants a...
Fourier transform: 3. Consider the equation a(x, 0) = f(x) u(x,t) lim 0 Using a Fourier transform, solve this equation. Evaluate your solution in the case when f(x)-δ(x). 3. Consider the equation a(x, 0) = f(x) u(x,t) lim 0 Using a Fourier transform, solve this equation. Evaluate your solution in the case when f(x)-δ(x).
2. Solve the following partial differential equation using Laplace transform. Express the solution of u in terms of t&x. alu at2 02u c2 2x2 u(x,0) = 0 u(0,t) = f(t) ou = 0 == Ot=0 lim u(x, t) = 0
where M=7 322-M2 4) Find the inverse - transform of F(z) = (2-1)(2-2M)' (15 marks) 0 t<-M/2 M <t< - 5) Show that the Fourier transform of function f(t) sin 7 s (10 marks) au 6) Show that u = ln(x2 + xy + y2) satisfies the partial differential equation x x ди +y 2. (7 marks) au 7) Solve the partial differential equation = e-cos(x) where at du x = 0, at =tet ax at and t = 0,...
solve k2 Solve the following partial differential equation by Laplace transform: д?у ду dx2 at , with the initial and boundary conditions: t = 0, y = A x = 0, y = B[u(t) – uſt - to)] x = 0, y = 1 5 Where, k, A, B and to are constants
The function u(x, t) satisfies the partial differential equation with the boundary conditions u(0,t) = 0 , u(1,t) = 0 and the initial condition u(x,0) = f(x) = 2x if 0<x<} 2(1 – x) if}<x< 1 . The initial velocity is zero. Answer the following questions. (1) Obtain two ODES (Ordinary Differential Equations) by the method of separation of variables and separating variable -k? (2) Find u(x, t) as an infinite series satisfying the boundary condition and the initial condition.
Use the Fourier transform to find a solution of the ordinary differential equation u´´-u+2g(x) =0 where g∈L1. (The solution obtained this way is the one that vanishes at ±∞. What is the general solution?) 1. Use the Fourier transform to find a solution of the ordinary differential equation u" - u + 2g(x) = 0 where g E L. (The solution obtained this way is the one that vanishes at £oo. What is the general solution?) eg(y)dy eg(y)dy e Answer:...
(4) (a) Compute the Fourier series for the function f(s)-- interval [-T, on the (b) Compute the solution u(t,a) for the partial differential equation on the interval [o, ) luWith u(t, 0) u(t,1)-0 for t>0 (boundary conditions) u(o,z)-3 sin(2x)-5 sin(5z) + sin(6z), for O < < 1 (initial conditions) (20 points) (4) (a) Compute the Fourier series for the function f(s)-- interval [-T, on the (b) Compute the solution u(t,a) for the partial differential equation on the interval [o, )...
9) Solve the following partial differential equation au a2u ax2 n(0, t) = u(2, t)-0 t > 0 (x, 0) = 0 au at It=0 =x(2-x) 9) Solve the following partial differential equation au a2u ax2 n(0, t) = u(2, t)-0 t > 0 (x, 0) = 0 au at It=0 =x(2-x)
1. Consider the Partial Differential Equation ot u(0,t) = u(r, t) = 0 a(x, 0)-x (Y), sin (! We know the general solution to the Basic Heat Equation is u(z,t)-Σ b e ). n= 1 (b) Find the unique solution that satisfies the given initial condition ur, 0) -2. (Hint: bn is given by the Fourier Coefficients-f(z),sin(Y- UsefulFormulas/Facts for PDEs/Fourier Series 1)2 (TiT) » x sin aL(1)1 a24(부) (TiT) 1)+1 0 1. Consider the Partial Differential Equation ot u(0,t) =...
(4) (a) Compute the Fourier series for the function f(x) interval [-π, π]. 1-z on the (b) Compute the solution u(t, z) for the partial differential equation on the interval [0, T): 16ut = uzz with u(t, 0)-u(t, 1) 0 for t>0 (boundary conditions) (0,) 3 sin(2a) 5 sin(5x) +sin(6x). for 0 K <1 (initial conditions) (20 points) Remember to show your work. Good luck. (4) (a) Compute the Fourier series for the function f(x) interval [-π, π]. 1-z on...