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5. (20 pts). Solve the following initial-value problem: Ut + 2uuz - 0<x<, 0 <t<oo 0...
4. [10] Find the solution to given initial-boundary value problem: 4uxx = ut 0<x<TI, t> 0 u(0,t) = 5, uit, t) = 10, t> 0 u(x,0) = sin 3x - sin 5x, 0<x<T
1. Solve the initial-boundary value problem one = 4 for () <<3, t> 0, u(0,t) = u(3, 1) = 0 for t> 0, u(x,0) = 3x – 2” for 0 < x < 3. (30 pts.)
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
7. (a) Find the solution of the heat conduction problem: Suxx = ut, 0<x< 5, u(0, 1) = 20, tu(5, 1) = 80, 1>0 u(x,0) = f(x) = 12x + 20 + 13sin(tor) - 5sin(3 tex). (b) Find lim u(2, t). (c) If the initial condition is, instead, u(x,0) = 10x – 20 + 13sin( Tox) - 5sin(3 7ox), will the limit in (b) be different? What would the difference be?
-). Solve the initial and boundary value problem: uUx=0, TE (0,), t > 0, U (0,t) = u(,t) = 0, >0, u(,0) - cos', 1€ (0,7).
7. Find the solution of the heat conduction problem 100uzz = ut, 0 < x < 1, t > 0; u(0,t) 0, u1,t 0, t>0; In Problem 10, consider the conduction of heat in a rod 40 cm in length whose ends are maintained at 0°C for all t0. Find an expression for the temperature u(,t) if the initial temperature distribution in the rod is the given function. Suppose that a
2. Solve the initial-boundary value problem One = 48m2 for 0 < x < 8, t > 0, u(0, t) = u(8,t) = 0 for t > 0, u(2,0) = 2e-4x for 0 < x < 8. (60 pts.)
(1 point) Solve the nonhomogeneous heat problem Ut Uzz + 3 sin(3.c), 0<x<1, u(0,t) = 0, u(T,t) = 0 u(2,0) sin(52) u(x, t) = Steady State Solution lim oo u(a,t) =
Solve the y"+ 4y = initial value problem s 1 if 0<xsa To if x>,T ylo)= 1, g(0)=0
Find a formula for the solution of the initial value problem for for t>0, -oc < x < oo ut = uzz-u a(1:0) = g(z) -x < 1 < x where g is continuous and bounded.( Hint: use v(x, t) = et u(z. t).) Find a formula for the solution of the initial value problem for for t>0, -oc