Miscellaneous Exercises 253 d an analytic (integral) solution of this wave problem 6 au 1 a2u -00
2. Consider the following problem au au at2 = 2,2 -00<< ,> 0. 1- C for - 1<x<1 u(a,0) = 1 0 for x > 1 (3,0) = sin(x), -o0 < x < 00. Write the solution of the problem as a sum of a forward and backward wave.
(1) Consider the following BVP for the wave equation, which models a string that is free at both ends: (r, t) E (0, L) x (0, 0o) (0, t) ur(L, t) u(r, 0) f() u(r, 0) g() 0 t0 E [0, L E [0, L The total energy of the solution at time t is 1 E(t) 2 0 (au (r, t) +(u(T, t)? ds. Show that the total energy is constant, i.c., E'(t) 0. [Hint: Start by differentiating under...
(1) Consider the following BVP for the wave equation, which models a string that is free at both ends: (r, t) E (0, L) x (0, 0o) (0, t) ur(L, t) u(r, 0) f() u(r, 0) g() 0 t0 E [0, L E [0, L The total energy of the solution at time t is 1 E(t) 2 0 (au (r, t) +(u(T, t)? ds. Show that the total energy is constant, i.c., E'(t) 0. [Hint: Start by differentiating under...
1. Let u be a solution of the wave equation u 0. Let the points A, B, C, D be the vertices of the paralleogram formed by the two pairs of characteristic lines r-ctC1,x- ct-2,+ ct- di,r +ct- d2 Show that u (A)+u (C)-u (B) + u (D Use this to find u satisfying For which (x, t) can you determine u (x, t) uniquely this way? 2. Suppose u satisfies the wave equation utt -curr0 in the strip 0...
Problem 3 (5 pts). Find solution to = 4uxx -00 <1 00 - <t<oo, cos(x), [17 ut-o = 19191919 Remark. Solution should be represented in the form of the appropriate Fourier integral
d1=8
d2=9
lu for Find the solution u(x,t) for the l-D wave equation-=- Qx2 25 at2 (a) oo < x < oo with initial conditions u(x,0)-A(x) , where A(x) Is presented in the diagram below, and zero initial velocity. For full marks u(x,t) needs to be expressed as an equation involving x and t, somewhat similar to f(x) on page 85 of the Notes Part 2. d2+5 di+10 di+15dı+20 (b) Check for the wave equation in (a) that if (x...
1. We solved heat/diffusion equation on the entire line -00
d1= 3 & d2= 2
Question 2 Find the solution 11(x, 1) for the 1-D wave equation aT = (a) 25-for -o <x < oo with initial conditions it (x,0) = A (x) , where A(x) is presented in the diagram below, and zero initial velocity. For full marks u(x,t) needs to be expressed as an equation involving x and 1, somewhat similar to fex) on page 8s of the Notes Part 2. 2 d2+5 r-0 di+10 di+15 di+20 3...
PROBLEM 1 IS SUPPOSED TO BE A WAVE EQUATION NOT HEAT
EQUATION
1. Find the solution to the following boundary value initial value problem for the Heat Equation au 22u 22 = 22+ 2 0<x<1, c=1 <3 <1, C u(0,t) = 0 u(1,t) = 0 (L = 1) u(x,0) = f(x) = 3 sin(7x) + 2 sin (3x) (initial conditions) (2,0) = g(x) = sin(2x) 2. Find the solution to the following boundary value problem on the rectangle 0 <...
Question 2 ul lu (a) Find the solution u(x,t) for the 1-D wave equationfor -oo < x < oo with initial conditions u (x,0)-A(x) , where A(x) s presented in the diagram below, and zero initial velocity. For full marks u(x,t) needs to be expressed as an equation involving x and t, somewhat similar to f(x) on page 85 of the Notes Part 2. di+10 dı+15di+20 (b) Check for the wave equation in (a) that if f(xtct) (use appropriate value...