Message Boards Message Boards

Numerical solution of fourth order differential equation

sergei pronkevich, Univercity
Posted 11 years ago

Could you tell me to solve numerically the equation with the next boundary conditions

\[Pi]^4 Y[y] - b^4 N0 (Y^\[Prime]\[Prime])[y] - 
  2 b^2 \[Pi]^2 (Y^\[Prime]\[Prime])[y] + 
  b^3 N0 y \[Alpha] (Y^\[Prime]\[Prime])[y] + b^4 
\!\(\*SuperscriptBox[\(Y\), 
TagBox[
RowBox[{"(", "4", ")"}],
Derivative],
MultilineFunction->None]\)[y] == 0
Y[0] == 0;
Y[1] == 0;
Y''[0] == 0;
Y''[1] == 0;

equation

Attachments:
POSTED BY: sergei pronkevich
6 Replies
Sort By:

Thank you very much. I did not know about "AccuracyGoal". I'm solving an equation of elastic buckling of rectangular plate with linearly varying in-plane load and I expected the solution similar to Y[y] = Sin[[Pi]*y/b]. This is the solution of my equation if [Alpha] = 0 and load is constant. I expected solution similar to the last picture.
POSTED BY: sergei pronkevich

If you modify the boundary conditions, the solver gives the reasonable solution. I guess in your case the solution is very likely Y[x] == 0.

code
POSTED BY: Shenghui Yang

But all the deviations from zero that you see, are only numerical artefacts, aren't they?

sols = ParametricNDSolve[{\[Pi]^4 Y[y] - 
     b^4 N0 (Y^\[Prime]\[Prime])[y] - 
     2 b^2 \[Pi]^2 (Y^\[Prime]\[Prime])[y] + 
     b^3 N0 y \[Alpha] (Y^\[Prime]\[Prime])[y] + 
     b^4 D[Y[y], {y, 4}] == 0,
   Y[0] == 0, Y[1] == 0, Y''[0] == 0, Y''[1] == 0}, 
  Y, {y, 0, 1}, {N0, b, \[Alpha]}, AccuracyGoal -> 100]

gives

enter image description here

I suppose that the existence and uniqueness theorem would suggest that there is only one solution, which is the "trivial" solution?

Cheers,

M.

PS: Another clue that the solutions that deviate from zero cannot be accurate is that they do not fulfil one of the boundary conditions, i.e. Y[1]==0.

POSTED BY: Marco Thiel

You are right. These deviations are just either numerical error or internal values.

POSTED BY: Shenghui Yang

Here is what you can try. I believe this time Mathematica hasn't found a non-trivial solution yet. You may try give this equation a reasonable choice of parameters:

enter image description here

Attachments:
POSTED BY: Shenghui Yang

Hi,

there are quite some issues with that. First of all there are many free parameters such $b, N0, \alpha$. To integrate numerically you need to substitute values for them (or use ParametricNDSolve). Syntax that produces an output is:

sols = NDSolve[{\[Pi]^4 Y[y] - b^4 N0 (Y^\[Prime]\[Prime])[y] - 
      2 b^2 \[Pi]^2 (Y^\[Prime]\[Prime])[y] + 
      b^3 N0 y \[Alpha] (Y^\[Prime]\[Prime])[y] + 
      b^4 D[Y[y], {y, 4}] == 0 /. {b -> 1., N0 -> 1, \[Alpha] -> 1},
   Y[0] == 0, Y[1] == 0, Y''[0] == 0, Y''[1] == 0}, Y[y], {y, 0, 1}]

There might be some numerical issues though because 

Plot[Y[y] /. sols, {y, 0, 1}]

gives

enter image description here

which is numerically zero, where it should be plainly zero. You can fix this by setting a higher Accuracy Goal:

sols = NDSolve[{\[Pi]^4 Y[y] - b^4 N0 (Y^\[Prime]\[Prime])[y] - 
      2 b^2 \[Pi]^2 (Y^\[Prime]\[Prime])[y] + 
      b^3 N0 y \[Alpha] (Y^\[Prime]\[Prime])[y] + 
      b^4 D[Y[y], {y, 4}] == 0 /. {b -> 1., N0 -> 1, \[Alpha] -> 1},
   Y[0] == 0, Y[1] == 0, Y''[0] == 0, Y''[1] == 0}, Y[y], {y, 0, 1}, 
  AccuracyGoal -> 100]

That gives a flatline zero.

Cheers,

Marco
POSTED BY: Marco Thiel
Reply to this discussion
Community posts can be styled and formatted using the Markdown syntax.
Tag limit exceeded
Note: Only the first five people you tag will receive an email notification; the other tagged names will appear as links to their profiles.
Reply Preview
Attachments
Remove
or Discard

Group Abstract Group Abstract