3
$\begingroup$

I want to solve the following equation numerically:

enter image description here

with initial condition y[0]=0 and y'[0]=0

I tried to do so by using the NDSolve function:

eqn = Derivative[2][y][x] == 1+y[x] - Integrate[Derivative[1][y][x - t]/sqrt[t], {t, 0, x}]; NDSolve[{eqn, y[0] == 0, Derivative[1][y][0] == 0}, y, {x, 0, 1}]

But it seems like Matlab cannot solve it directly. Do you have any suggestions?

Improve this question
$\endgroup$
3
  • 2
    $\begingroup$ You mean Mathematica cannot solve? You wrote Matlab $\endgroup$ Commented Oct 6, 2016 at 11:03
  • $\begingroup$ Convert it to a regular ODE and then feed it to NDSolve[]. $\endgroup$ Commented Oct 6, 2016 at 11:18
  • $\begingroup$ sqrt should be Sqrt. This isn't main issue here of course. $\endgroup$ Commented Oct 6, 2016 at 11:22

1 Answer 1

Reset to default
6
$\begingroup$

As mentioned in the comment above, sqrt should be Sqrt and Matlab should be Mathematica, these aren't the main issue here of course. Currently NDSolve can't handle the problem, but we can solve the equation with the help of LaplaceTransform:

teqn = LaplaceTransform[eqn, x, s] /. Rule @@@ ic tsol = Solve[teqn, LaplaceTransform[y[x], x, s]][[1, 1, -1]] (* 1/(s (-1 + Sqrt[\[Pi]] Sqrt[s] + s^2)) *)

InverseLaplaceTransform can't handle tsol, so I turned to this numeric Laplace inversion package:

Plot[FT[(s \[Function] #) &@tsol, x], {x, 0, 1}]

Mathematica graphics

Finally, DSolve in v11 can solve this equation in principle, but sadly it doesn't give the correct answer in this case, I think there's no doubt it's a bug:

eqn = y''[x] == 1 + y[x] - Integrate[y'[x - t]/Sqrt[t], {t, 0, x}]; ic = {y[0] == 0, y'[0] == 0}; sol[x_] = y[x]/.First@DSolve[{eqn, ic}, y@x, x] (* 1/2 E^-x (-1 + E^x)^2 *) Plot[{sol@x, FT[(s \[Function] #) &@tsol, x]}, {x, 0, 1}, PlotLegends -> {"Output of DSolve", "Numeric Solution"}]

Mathematica graphics

What's more awkward is, if the second argument of DSolve is modified to y i.e. if we write DSolve[{eqn, ic}, y, x], DSolve will return the input... Seems that this new feature is still unstable.

Improve this answer
$\endgroup$
2
  • 1
    $\begingroup$ Dsolve gives wrong answer.Can you check it? $\endgroup$ Commented Dec 29, 2016 at 17:24
  • $\begingroup$ @MariuszIwaniuk You're right. DSolve in v11 is more unstable than I thought. Edited. Thanks for point out. $\endgroup$ Commented Dec 30, 2016 at 3:01

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.