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Q1:Problem 6 (15 points) Let u(x, t) be a solution to the wave equation describing a vibrating string UttUrr = 0, x € R, t > 0. (i) Show that the energy density satisfies a certain conservation law E(x, t) = ½ (u²(x, t) + u²(x, t)) Et(x, t) + z(x, t) = 0. Find a flux function. There are many flux functions. You are asked to find one of them, not all. Remarks. • For the explanation of why & is the energy density of a vibrating string with a small amplitude, see the file "energy of a string" in Canvas → Files → Readings → Wave equation. (1) • This mathematical result shows that the energy density can be interpreted as a nonnegative physical quantity that moves in space as the string vibrates. (ii) Show that the total energy of the (infinite) string = [ε (x, t) da -∞ E(t) = is a conserved quantity, i.e., does not change in time. Hint: integrate the conservation law. Alternatively, you can directly differentiate E and then integrate by parts. The second approach might be a bit harder than the first one. (iii) Verify that the 'wave train' function u(x, t) = sin(xt) satisfies the wave equation (1). Compute the energy density E(x, t) and explain why it propagates with the same velocity as the wave train.See Answer
Q2:Problem 5 (15 points) Solve the initial boundary value problem on the first quadrant: Ut + Ux = 0, x > 0,t>0, u(x,0) = 0, x ≥ 0, u(0, t) = t², t ≥ 0. Hint: Use the geometric approach to the method of characteristics by sketching the characteristics on the x-t plane. The answer should be a piecewise function depending on where the backward characteristic emanating from (x, t) 'lands' first.See Answer
Q3:Problem 2 (10 points) Let u(x, t) be the solution to the transport equation UtUx=0, x = R, t > 0 with the triangular 'pulse' initial value u(x,0) = X, -x, 2 x ≤ 0, x = (0, 1], 1<x≤ 2, x > 2. Sketch the graph of the solution u(x, 3) u(x, 4) in the x – y plane. In what direction does the triangular pulse move? Hint: Denote g(x) = u(x, 0) and write down the solution u(x, t) in terms of g. After that, sketch the graphs.See Answer
Q4:Problem 2 (10 points) Solve the dam break problem, that is, the wave equation Utt Uxx = 0, x ¤ R, t > 0, u(x,0) = f(x), x ≤ R, ut(x, 0) = 0, x ≤ R. where ƒ(x) = = [1, x ≥ 0, 0, x < 0. 1/2, 1. Include the graphs in your work. Sketch the solution u(x, t) as a function of x for t =See Answer
Q5: 7 2 J File Preview Problem 5 (15 points): Fix a positive real number L. Suppose that u R× [0,T] → R is a positive solution of the heat equation Ut(x,t) = Uxx(x,t) which is 2L-periodic in the first variable. In other words, u(x+2L,t) = u(x,t) for all x and t. We define and = L J(t) J-L u(x, t) dx S(t) = − √ u(x, t) log u(x, t) dr. -L (i) Please show that the derivatives of J(t) and S(t) are given by J'(t) = 0 2 and S'(t) = [[ u(x,t) =¹ u₂(x, t)² dx -L for 0≤t≤T. (Hint: Use integration by parts.) (ii) Please show that the second derivative of S(t) is given by L S" (t) = -2 - 2 [ ", u(x, t)~^¹ [uzz(x, t) — u(x, t)¯¹ uz(x,t)²]² dx - -L for 0<t<T. (Hint: Use (i) and integration by parts.) (iii) We next define - log t 2 W(t) = t S' (t) + S(t) − J(t). Please show that the derivative of W(t) can be written in the form W'(t) = −2t L 1 2 dx ·L u(x,t)¯¹ [uxx(x,t)—u(x,t)¯¹uz(x, t)² + 1 = u(x, t)]². -L 2t for 0 <t<T. Deduce from this that the function W(t) is monotone decreasing for 0 <t<T.See Answer
Q6:Problem 3 (15 points) a) (5 points) Find all the standing wave solutions to the wave equation -Z²uxx Utt = 0, x = R, t > 0, that are periodic in x (for fixed t) with the period 1 > 0. b) (10 points) Find all the standing wave solutions for a finite string that is free at both ends: Utt − c² Uxx = 0, x ≤ (0, π), t > 0, ux (0, t) = 0, t≥ 0, ux (π, t) = 0, t≥ 0.See Answer
Q7:3. Find the solution to Utt = 36UTT u(x,0) = u(x) = e-¹² sin(x), u₁(x, 0) = u₁(x) = 0, uz (0, t) = 0, on the domain 0 ≤ x <∞, 0≤ t <∞. Marks: 3See Answer
Q8:Question 2 Consider a water tank that is being filled with rainfall at the top and is drained through a small hole at the bottom. The change in volume can be modelled by considering the amount of water entering and leaving the tank (per unit of time) as follows dV dt = fin - fout. Here volume of water in the tank V is in litres, flow rate in fin is in litres per hour, flow rate out fout is in litres per hour and time t is in hours. Initially, the tank contains 50L of water: V(0) = 50. (a) (2 marks) Assume the rainfall throughout the day is getting heavier, such that, fin (t) = 10+t and that the tank is losing 10% of its volume of water per hour, such that, fout (V) = 0.1V. Check by direct substitution that the function V(t) = 10(t+5e-0.1t) satisfies the ODE and the initial condition. (b) (4 marks) Now assume that the hole is slowly growing in size, so that the flow rate out increases with time: fout (V, t) = 0.1(1 + 0.1t)V. Solve the ordinary differential equation for V(t) with this flow rate out, using either separation of variables, or the integrating factor method. Make sure to also include the initial condition. (c) (2 marks) use MATLAB to create a plot of both solutions from Q2a and Q2b (in the same figure). You can either use the explicit solutions that come from solving the ODEs, or use ode45 to generate numerical solutions to the ODEs. Make sure you have chosen an interval to plot over so that the behaviour of the solutions can be clearly seen.See Answer
Q9:5. Use Laplace transforms to solve - 7x+6x=et + 8(t-2) + 6(t-4), (0) = 0, (0) = 0.See Answer
Q10:4. A 32-pound weight stretches a spring 2 feet. If the weight is released from rest at the equilibrium position, find the equation of motion z(t) if an impressed force f(t) = sint acts on the system for 0 < t < 27 and is then removed. Ignore any damping and assume g = 32 ft/sec².See Answer
Q11:3. Consider the variable coefficient ODE ty" - y'= 2t2 with y(0) = y(0) = 0. Using Laplace transforms find, but do not solve, the ODE that Y(s) = L{y(t)} must satisfy.See Answer
Q12:2. Solve the IVP y" - y" -y'+y=6et, y(0) = y'(0) = y" (0) = 0 using Laplace Transforms.See Answer
Q13:6) Solve the given differential equation by using an appropriate substitution. d x + 3y 3x + ySee Answer
Q14:2) Verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval I of definition for each solution. y"-6y + 13y=0; y = ³ cos 2xSee Answer
Q15:5. Convert the following initial value problems into a system of first order initial value problems. Write your answer in the form u' Au + f, u(0) = uo. (a) (b) +ty"+y=1, y(0)=0, g'(0)=1, /'(0) = 2 y+3y + 2z=e, y(0)=0, (0)=1 z"+y+22=1, 2(0)=1, '(0)=0 CopySee Answer
Q16:4. Consider the nonconservative mass-spring system governed by +2 +26x = 0, z(0) = 1, ż(0) = 4 (a) Find the solution z(t) and its derivative (t), and evaluate z(7/5) and a(w/5). (b) Calculate the total energy E(t) of the system when t = x/5. (c) Calculate the energy loss in the system due to friction in the time interval from t = 0 tot = x/5. QuiSee Answer
Q17:2. Consider the following differential equations. Determine the form of the particular solution, g,. for use in the method of undeter- mined coefficients. Simply find the form of the particular solution without solving for the coefficients. Remember to check for duplication with solutions to the homogeneous equation. (a) 4y"+y=t-008 () (b) "5y+6y=cost-te (c) "" "t²te^ (d) y(4)ytet + sint 23See Answer
Q18:2. Determine solution of the following partial differential equation a²U/əx² = a² au/at subjected to: U (x, 0) = 150 = U(o, t) = 0 U (1,t) = 0See Answer
Q19:An infinite solid circular cylinder is initially at a uniform temperature of 150° C. At time t = 0 the temperature around the entire boundary is suddenly reduced to 0°C, and maintained thereafter. Determine the temperature at any point of the region at any subsequent time.See Answer
Q20:1. For a cylindrical coordinate, write the partial differential equation for the followings: a. the Heat Equation b. the Steady State Equation, c. the Wave Equation. 2. Determine solution of the following partial differential equation 8²U/ax² = a² du/dt = 0 subjected to: U (x,0) = 0 U (o, t) = 50 U (1,t) = 50See Answer

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