

Ut = Kuzz-cr(z-L) where u = u(x, t) for 0 L and t 0 a(0,t) = 1 (a(L, t) = 1 where к.с > 0 are con...
Solve the wave equation
a2
∂2u
∂x2
=
∂2u
∂t2
, 0 < x < L, t > 0
(see (1) in Section 12.4) subject to the given conditions.
u(0, t) = 0, u(L, t) = 0
u(x, 0) =
4hx
L
,
0
<
x
<
L
2
4h
1 −
x
L
,
L
2
≤
x
<
L
,
∂u
∂t
t = 0
= 0
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Let Z be a random variable where P(X<0) = 0:
a) If
, what is
?
b) If
, what is P = [Z = E(Z)] ?
c) If
, what is
?
6,(W) = jw We were unable to transcribe this imageD() = *(1 + exp(2jw) We were unable to transcribe this imageWe were unable to transcribe this image
(1) (4 points) Let L 〉 0 be a given constant. Solve ut = uxx u(0,t) = u(L, t) = 0 u(x, 0) = sin2 (2x) for all (x, tje (0,L) × (0,+oo), for all t > 0, for all x e [0, L]
1. Solve the boundary value problem ut =-3uzzzz + 5uzz, u(z, 0) = r(z) (-00 < z < oo, t > 0), using direct and inverse Fourier transforms U(w,t)-홅启u(z, t) ei r dr, u(z,t)-二U( ,t) e ur d . You need to explain where you use linearity of Fourier transform and how you transform derivatives in z and in t 2. Find the Fourier transform F() of the following function f(x) and determine whether F() is a continuous function (a)...
l, t)4u (x, t), 0<x< L, 0 <t Evaluate u(1.1; 0.3) where u(x, t) u(0, 1)= u(L, t)- 0v1> 0 u(x, 0)= f(x), u,(x, 0)- g(x), 0<x< L L=T al f(x) 3sin 2x, g(x)=-2sin 3x b/ For f(x)-xn-x & g(x)-0, approximate numerically u(x, t) by the first term. L-S c/f(x)=-3sin g(x)- 5 2sin d/ f(x)-0, g()= .3 x +1 approximate numerically u(x, t) by the first term c/ f(x)-2(5-xx, g(x) x+1 3 approximate numerically u(x, t) by the first couple...
(1 point) Consider the wave equation 1(1)utt = uzz for-oo < z < oo, t>0 with initial conditions ut (z,0-0 and u(z,0) = /(z), where (2) f(z) = 1 for 0 < z < 1, (3) f(z) =-1 for-1 < z < 0, and (4) f(z) = 0 for all other. The slanting lines in the figure below show the characteristics for this PDE that originate on the z-axis at the points of discontinuity of the initial data f f(x)...
The motion of a string with fixed ends in a viscous medium is described by: together with boundary conditions: u(0,t) 0 and u(2, t)0 and initially: u(z,0-sin(nz/2) and ut(z, 0)--sin(2πχ). (a) If u(x, t) - X(x)T(t) find the ordinary differential equations satisfied by X and T e ordinary different (c) Determine u(x, t).
The motion of a string with fixed ends in a viscous medium is described by: together with boundary conditions: u(0,t) 0 and u(2, t)0 and initially: u(z,0-sin(nz/2)...
A. : Suppose that u(x, t) satisfies Ut = Uzr +1, € (0,2) u(x,0) = 0 u(0,t) = u(2,t) = 0 Solve for u(x,t). What is lim u(x,t)? B. Consider the heat equation in the region 0 < x < 1, but supoose that the system is heated with a source. This is represented by: Ut = Uzz + cos(2), 1 € (0,7) u(x,0) = 1+ cos(2x) U (0,t) = U7(TT,t) = 0 Solve for u(x, t).
1 & 5
Solve the following heat equations using Fourier series ux Ut, 0 <x <1,t>0, u (0,t) = 0 = u(1,t), u(x,0) = x/2 1/ 2/ Ux=Ut, 0<x< m ,t>0 ,u(0,t) = 0 = u( 1, t), u(x, O) = sinx- sin3x 3/ usxut, O <x < 1 ,t>0, u(0,t) = 0 = u,(1, t), u(x,0) = 1 -x2 Ux=Ut,O<x <m ,t>0, u(0, t) = 0 = u,( rt , t) , u(x, 0) = (sinxcosx)2 4/ 5/Solve the...
Let X(t) =
2; if 0 t 1;
3; if 1 t 3;
-5; if 3 t 4:
or in one formula X(t) = 2I[0;1](t) +
3I(1;3](t) -
5I(3;4](t).
Give the Itˆo integral
X(t)dB(t)
as a sum of random variables, give its distribution,
specify the mean and the variance.
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