5. First-Order ODEs
The initial value problem for the constant coefficients Riccati differential equation:
Vt > 0 : y' t, =
appears in a variety of modelling contexts, from population dynamics to the electrostatics
of dispersed media.
(a) Integrate the ODE above to find the general solution y(t;yo) parameterized by Yo-
For what values of Yo is there only a trivial (i.e. zero or constant) solution?
Are there values of yo for which this IVP is NOT well-posed (in the sense of Hadamard:
that a solution exists, is unique, and varies smoothly with the initial condition)? Does
your answer depend on the domain of t for which a solution y(t) is required?
(b) The method of Picard iterations permits construction of the solution to an IVP in the
neighborhood of the initial point, if such a solution exists. One begins by setting:
yo(t) = yo,
substitutes this initial approximation for y(t) on the right-hand side of the ODE,
integrates from to (0 in this case) to t, and adds yo to obtain the next approximation:
y1(t) = yo + dt Yo
and then repeats this procedure to obtain each successive approximation by substi-
tuting the previous explicit result for y(t) on the rhs of the ODE:
y2(t) = yo +
yn (t) = yo + I. dt f(t,yn-1(t))
Follow this procedure to obtain 1(t) and 12(t) for our Riccati equation.
Compare your formulae for Yo, Y1, and y to the Taylor series for the exact y(t) around
t =0. At which order in t does the first disagreement occur for each approximation?
Does this hold true for all values of yo? Is there a general formula for this leading-order
error, i.e. can you write: yn (t) - yt ~ O(tf(r)) for some simple f(n)?
(c) An alternative approach is numerical approximation, for example a finite-difference
approach such as the forward Euler method:
(t+h) 22 Yh (t+h) = yh (t) + h y' (t,yn(t)), for some choice of the step size h.
Assuming yo = 0, implement Euler's method. Plot the total absolute error Eh (t) =
(t) - y(t) for a reasonable range of t and several choices of h. By doing so, estimate
the leading order of the total error in terms of h, i.e. (t) ~ O(h?) as h t 0.
Next, plot the local absolute error Des(t) = (t)-eh(t-h) Estimate the leading
order of the local error in terms of h, i.e. Aen(t) ~ O(h?) as h 0.
Hints: try 0 < t < 2 and h = 0.1 2-" for some n = 0,1, Note that y(t) approaches
an asymptote as t 00, SO Eh (t) and AEh (t) may not necessarily be monotonic in t.
Logarithmic scaling of co-ordinates may be helpful in elucidating the scaling behavior.
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