Math 250A (Fall 2008) - Lab II (SOLUTIONS)

Transcription

1 Math 25A (Fall 28) - Lab II (SOLUTIONS) Part I Consider the differential equation d d = 2 + () Solve this equation analticall to obtain an epression for (). Your answer should contain an arbitrar constant. Answer: A general solution can be found b first separating variables and then integrating both sides. One can then make a trigonometric substitution = tan θ and upon solving for, the solution is () = tan ( + C) where C is a constant dependent upon the initial condition. (2) Solve the differential equation numericall using Euler s method on the interval [, ] for the initial condition () =. On a single figure, plot our estimated solution curve using the following step sizes for :.5,.2,.,.5, and.. Make clear which curve corresponds to each step-size (Hint: use different line stles/colors). Answer: An important component of the assignment was in this specific question. The main challenge stemmed from learning how to write a Matlab script that would do the numeric integration when eecuted. An eample code is given below as well as the figure (Fig. ) that was asked for. Make sure ou understand how each line of the code works and how the ultimatel all tie together to form the program. % ### odesolvepti.m ###..8 % Matlab code to use Euler s method to solve the differential equation % = ˆ2 + % NOTE: there are a lot of different was (e.g., structure, snta, naming % conventions) one could use for this code that would still ultimatel % produce the same result clear clf % - % User Input Parameters % nsteps= ; % # of steps MIN= ; % starting -value MAX= ; % ending -value

2 2.6 Numerical Solution to ODE d/d = 2 + using Eulers Method step size = FIGURE. Comparison of numerical solutions for the differential equation = 2 + using Euler s method for different step-sizes. deltax=.; % step-size % initial conditions (i.e. what is (MIN)?) = ; % - % [user shouldn t have to change parameters below here] = MIN; % rename initial -value % determine # of steps needed based upon interval bounds and specified step-size nsteps= (MAX-MIN)/deltaX; % create the arras S and S that will contain all the and -values % b setting first value equal to initial condition S()= ; S()= ; % use a for loop to go through each step for nn=2:nsteps+ end % use Euler s method to solve for net -value % (this is where the form of the differential equation is input) S(nn) = S(nn-) + deltax*(s(nn-)ˆ2 + ); % update -arra S(nn) = S(nn-)+ deltax; % plot the numerical solution

3 3.6 Numerical Solution to ODE d/d = 2 + using Eulers Method Eulers method Eact FIGURE 2. Comparison of eact solution for the differential equation = 2 + with solution estimated numericall using Euler s method. plot(s,s) hold on; label( ) label( ) title( Numerical Solution to ODE d/d =ˆ2+ using Eulers Method ) % plot eact (i.e., analticall-derived) solution (onl true if ()=) A= tan(s); plot(s,a, r ) grid on legend( Eulers method, Eact, Location, SouthEast ) (3) On a different figure, plot our numerical solution above for =.5 together with the eact solution ou solved for analticall (given the specified initial condition). Make sure that it is clear which of the two different curves is which. Discuss an differences ou see between the two plots and tr to eplain how those differences arise. How does this difference change as gets smaller? Answer: See Fig. 2. We can see that the numerical solution (solid blue line) under-estimates the eact solution (() = tan ( + C)). From Fig., we can sa that the difference between the eact and numerical solutions decreases as gets smaller.

4 Eulers method Eact Numerical Solution to ODE d/d = 2 + using Eulers Method FIGURE 3. Comparison of eact solution for the differential equation = 2 + with solution estimated numericall using Euler s method with a step-size of =.5. (4) Etend the range of comparison between the numerical solution and the eact solution to [, π/2]. How does our numerical solution compare to the analtical solution as gets close to π/2? Eplain wh one is larger than the other. Answer: From Fig. 3, we can see that the numerical solution gets worse and worse (as it under-estimates) the eact solution diverges as π/2 and that its derivative goes to. Remember that Euler s method assumes that the derivative is constant over the interval, using the value at the left-hand side. Since the derivative becomes increasingl larger as π/2, it makes sense that the total error in the numerical approimation will increase and that the numerical solution will be an under-esitmate (since the assumed value of the derivative is alwas the smallest of anwhere on the actual interval).

5 5 FIGURE 4. Phase-line portrait for d/d = 2 c. Note that the stabilit of the equilibria can be determined b whether the quadratic function is above or below the -ais. Part II Consider the differential equation d d = 2 c where c is a constant greater than zero. () Determine all equilibrium solutions and their stabilit. Answer: Factoring the right-hand side of the differential equation, we can see that there will be equilibria solutions at = ± c. There are man was one could determine their stabilit (e.g. use DFIELD, linearize about the fied point and find the eigenvalues, etc..). One approach is to plot the phase-line portrait (Fig.4). The right-hand side is just a concave-up parabola and the points where it crosses the horizontal-ais indicates the equilibrium locations. Whether the curve is above or below the ais indicates the directionalit of the phase-line. We can see that = c is unstable and = c is a stable equilibrium. (2) Solve this equation analticall to obtain an epression for (). Your answer should depend upon c and contain an arbitrar constant. Answer: Similar to Part I, we would use separation of variables, then integrate both sides. For this case, we could use a partial fraction epansion to simplif the

6 6.2.4 Numerical Solution to ODE d/d = 2 c using Eulers Method step size = Numerical Solution to ODE d/d = 2 c using Eulers Method step size = log(+.) FIGURE 5. Comparison of eact solution for the differential equation = 2 c with solution estimated numericall using Euler s method for different step-sizes. Also shown using a logarithmic -ais (where the values were slightl offset so to visualize the initial point). denominator, ielding [ d 2 c = 2 c ( + c) + ] 2 c ( d = + C c) The resulting integral is easil solved (make a substitution and use du u = ln u + c) and solving for, we have the final solution c () = c + Ae2 Ae 2 c where A is an arbitrar constant. (3) Write a code to solve the equation numericall using Euler s method on the interval [, 5] for the initial condition () = and with c = 4. On a single figure, plot our estimated solution curve using the following step sizes for :.5,.2,.,.5, and.. Make clear which curve corresponds to each stepsize. How does the solution depend upon? Answer: An eample code is given below as well as the figure (Fig. 5) that was asked for. The solution approaches the equilibrium value faster for smaller values of. A sample code for solving the problem is shown below. % ### odesolveptii.m ###..8 % Matlab code to use Euler s method to solve the differential equation % = ˆ2 - c % (where c is a positive const.)

7 7 clear clf % - % User Input Parameters MIN= ; % starting -value MAX= 5; % ending -value deltax=.; % step-size % initial conditions = ; % - = MIN; nsteps= (MAX-MIN)/deltaX; S()= ; S()= ; for nn=2:nsteps+ end % note the difference here from the code for Part I S(nn) = S(nn-) + deltax*(s(nn-)ˆ2 - c); % update -arra S(nn) = S(nn-)+ deltax; % plot the numerical solution plot(s,s) hold on; label( ) label( ) title( Numerical Solution to ODE d/d =ˆ2-c using Eulers Method ) % plot eact (i.e., analticall-derived) solution (onl true if ()=) A= sqrt(c)*(-ep(2*sqrt(c)*s))./((+ep(2*sqrt(c)*s))); % could also have used -sqrt(c)*tanh(sqrt(c)*s) here too plot(s,a, r ) grid on legend( Eulers method, Eact, Location, SouthEast ) (4) Using =., find solution curves for different initial conditions () = o. How do the solutions depend upon o? Answer: See Fig.6. (5) Eplain our answer to the last part in terms of our analtic solution. Are the two results consistent? Answer: The constant A will depend upon the initial condition such that A = o c o + c

8 8 2 Numerical Solution to ODE d/d =^2 -c using Eulers Method.5.5 = FIGURE 6. Comparison of numerical solutions for the differential equation = 2 4 with solution estimated numericall using Euler s method with a step-size of =. for different initial values at =. Though not shown on this figure, initial conditions where o > 2 will diverge towards + (as ) and o < 2 will converge towards = 2. where o = (). Solutions will diverge towards + (for increasing ) if o > c. Solutions will be sigmoidal when o < c. The shape will be a reverse- S since solutions will move awa from the unstable equilibrium (i.e., c) and towards the stable one ( c) as increases. Decreasing o from around 2 towards -2 will move the center of the S-shape to the left. For o < c, solution curves will asmptoticall approach the stable equilibrium. When o = c, the solutions will be constant (i.e., equilibrium solutions). Both the numerical ( =.) and analtical solutions are consistent with one another, though the numerical solution will underestimate for the case o > c as the solution diverges. Note that for the initial condition () =, we have the solution. c () = c e2 + e = c tanh ( c) 2 c (6) What is the effect of varing c? Eplain in the contets of both our analtical answer and numerical simulations. Do both agree?

9 Answer: The effect of changing c is two-fold. First, it changes the equilibrium values and thus what the asmptotic limits are. Second, because c appears in the argument of the eponent, increasing c will increase the rate at which solution asmptoticall approach (or move awa from) the equilibria. For eample, if o < c, then a larger value of c means a sharper transition in the S-shape that the solution takes. One could also introduce the variables u = / c and τ = c. This change of variables would reformulate the differential equation as du dτ = u2 and thereb removing the parameter-dependence in terms of understanding the underling dnamics. 9