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Please help me with this ! Thank you 4. Consider the partial differential equation au au...
Consider the following partial differential equation. au, au ax? + = u ay? Identify A, B, and C in the above equation and use them to calculate the following. B2 - 4AC = -1 + u X Classify the given partial differential equation as hyperbolic, parabolic, or elliptic. O hyperbolic parabolic elliptic
QUESTION 4 [8 MARKS TOTAL] Q4 Hand calculations (see Workshops 21 and 22 for relevant content) Consider the 1-D diffusion equation: Eqn 3 да d²q at Ox? Write the centered in space, Crank-Nicolson (CN) discretization of this equation. Consider just ONE Fourier mode of the representation of the error, and write it in the simplified form (defined at time level n, and node j) as Eqn (4) Using this Fourier mode, undertake a von Neumann stability analysis for the CN...
4. (50 pts) Consider the following partial differential equation: 1du au Ət22 Ətər2 (BC) u7,t) = 0 20,t) = 0 0 <t (IC) u(3,0) = 0 0 <r <a Follow the steps below to solve it: (a) (8 pts) Separate variables as u(x,t) = X(2)T(t) to derive the following differential equations for X and T, with an unknown parameter 1: T" - T' + XT = 0, X" + 1X = 0.
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67.3. Suppose initially (t = 0) that the traffic density is p = Do + Esin , where le « po. Determine p(x, t). 67.4. Consider Õpildt + copildx) = 0. Suppose we observe pi in a coordinate system moving at velocity v. Show that др. +(c dr 0. v) dx' Does p stay constant moving at the car velocity? 67.5. Based on a linear analysis, would you say...
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Consider the diffusion equation, au(x,t u(x,t) Here u(x,t) > 0 is the concentration of some diffusing substance, the spatial variable is x, time is t and D is a constant called the diffusivity with dimensions [LT-11. We will consider the diffusion equation on a finite spatial domain (0<x< 1) and an infinite time horizon (t > 0). To solve the diffusion equation we must include...
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Differential Equation The following ordinary differential equation is to be solved using nu- merical methods. d + Bar = Ate - where A, 0,8 > 0 and x = x at t = 0. dt It is to be solved from t = 0 to t = 50.0. It has analytical solution r(t) = A te-al + A le-ale"), where A A B-a and A2 А (8 - a)2 Questions Answer the questions given...
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4. Qualitative Behavior of Autonomous First Order Differential Equations: Consider the graphs of g(N) in the panels (a) - (d) in Figure 1. For each graph, identify all equilibrium points and classify them as either stable or unstable. Then, for each panel, make a graph of N(t) vs. t for 0<1<oo with the given conditions: (a) N(0)-1; N(0)-3. (b) N(O) 0.5; N(O)2 (c) N(O) 1.5; N(0)3 (d) N(0)0; N(O)1.5 Assume that N2...
this is numerical analysis. Please do a and b
4. Consider the ordinary differential equation 1'(x) = f(x, y(x)), y(ro) = Yo. (1) (a) Use numerical integration to derive the trapezoidal method for the above with uniform step size h. (You don't have to give the truncation error.) (b) Given below is a multistep method for solving (1) (with uniform step size h): bo +1 = 34 – 2n=1 + h (362. Yn) = f(n=1, 4n-1)) What is the truncation...
4. (5 marks) Consider the partial differential equation (1) for 1 € (0,2) and t > 0, with boundary conditions u(0, 1) = 0 ur(2, 1) = 0. Which of the following are solutions to the PDE and boundary conditions? In each case explain your answer. Note that initial conditions are not given. (Hint: it is not necessary to solve the problem above. (a) -3)*** e ular, 1) = Žen sin [(---) --] ~[(---) ;-)e-(1-1) e+(1-3)*(/2°1 u(3,t) - Cu COS...
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4. [-/1 Points] DETAILS SCALCCC4 7.1.006. Consider the following differential equation. ry + y = 5 (a) Show that every member of the family of functions y = (51n(x) + C)/x is a solution of the differential equation. (Do this on paper. Your instructor may ask you to turn in this work.) (b) Illustrate part (a) by graphing several members of the family of solutions on a common screen. (Do this on paper. Your instructor...