3
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While trying to solve

With[{h = h[x]},
{eq, bc} = {D[h, x, x] == Sin[Pi x], {h == 0 /. x -> 0, 
 D[h, x, x] == -D[h, x] + h /. x -> 1}}];
g = NDSolveValue[{eq, bc}, h, {x, 0, 1}]

I got the error

NDSolve::nlnum1: The function value {0.,0.31831 +(NDSolve`h$9$1^[Prime])1} is not a list of numbers with dimensions {2} when the arguments are {0.,0.,0.31831,0.63662}.

Then I found this trick so I tried

With[{h = h[x]},
{eq, bc} = {D[h, x, x, x] == 
 Pi Cos[Pi x], {D[h, x, x] == Sin[Pi x] /. x -> 0, 
 h == 0 /. x -> 0, D[h, x, x] == -D[h, x] + h /. x -> 1}}];
 g = NDSolveValue[{eq, bc}, h, {x, 0, 1}]
 Plot[g[x], {x, 0, 1}]

I obtained enter image description here

I guess this solution is not correct due to high values (maybe instability).

Is there a better way to solve such an equation?

Improve this question
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2
  • 1
    $\begingroup$ for second order ode, IC/BC should not really have derivative terms in it of equal of higher than the 2. And why do you write things this strange way? why not write ode = h''[x] == Sin[Pi x] and bc = {h[0] == 0, h''[1] == -h'[1] + h[1]} which is much more readable and clear. $\endgroup$
    – Nasser
    Commented Mar 18 at 17:51
  • $\begingroup$ @Nasser yes I realized that, but still looking for good solutions.For the form I wrote, it's because I simplified complicated eqs. $\endgroup$
    – S. Euler
    Commented Mar 18 at 17:58

1 Answer 1

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Your ode-system has no solution!

general solution depends on two parameters

H = DSolveValue[h''[x] == Sin[Pi x], h, x]
(*Function[{x}, C[1] + x C[2] - Sin[\[Pi] x]/\[Pi]^2]*)

Solve[{H[0] == 0, (H^\[Prime]\[Prime])[1] ==H[1] - Derivative[1][H][1] }, {C[1], C[2]}]
(* {} *)

Only bc h[0]==1/Pi gives a solution

Solve[{H[0] == 1/Pi, (H^\[Prime]\[Prime])[1] ==H[1] - Derivative[1][H][1] }, {C[1], C[2]}][[1]]
(* {C[1] -> 1/\[Pi]} *)
Improve this answer
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4
  • $\begingroup$ thanks for pointing this out. This is because I simplified the eq. I can take h[0]==1/Pi to focus on bc issue. $\endgroup$
    – S. Euler
    Commented Mar 18 at 18:04
  • $\begingroup$ @S.Euler As Nasser mentioned in hist comment NDSolve doesn't allow boundary conditions with order of ode! $\endgroup$
    – Ulrich Neumann
    Commented Mar 18 at 18:09
  • $\begingroup$ what about approximating h'' by first oder derivative? $\endgroup$
    – S. Euler
    Commented Mar 19 at 1:49
  • $\begingroup$ @S.Euler Try DSolveValue[{h''[x] == Sin[Pi x], h[0] == 1/Pi, Sin[Pi] == h[1] - h'[1]}, h, x] , but Mathematica gives an error message. $\endgroup$
    – Ulrich Neumann
    Commented Mar 19 at 9:15

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