Skip to main content

bugs - Solving integro-differential equation with DSolve in MMA 11.0


In this question DSolve(symbolic solution) gives wrong answer.


I analyzed the 13 equations in which DSolve gives the same solution.


 ClearAll["Global`*"];
 Remove["Global`*"];

 eqn1 = y''[x] == 1 + y[x] - Integrate[y'[t], {t, 0, x}];

 eqn2 = y''[x] == 1 + y[x] - Integrate[y'[x - t], {t, 0, x}];
 eqn3 = y''[x] == 1 + y[x] - Integrate[y[x - t], {t, 0, x}];
 eqn4 = y''[x] == 1 + y[x] - Integrate[y'[x + t], {t, 0, x}];
 eqn5 = y''[x] == 1 + y[x] - Integrate[y[x + t], {t, 0, x}];
 eqn6 = y''[x] == 1 + y[x] - Integrate[2*y'[t], {t, 0, x}];
 eqn7 = y''[x] == 1 + y[x] - Integrate[t*y'[t], {t, 0, x}];
 eqn8 = y''[x] == 1 + y[x] - Integrate[x*y'[t], {t, 0, x}];
 eqn9 = y''[x] == 1 + y[x] - Integrate[y[t]/(1 + x), {t, 0, x}];
 eqn10 = y''[x] == 1 + y[x] - Integrate[y'[t]/(1 + x), {t, 0, x}];
 eqn11 = y''[x] == 1 + y[x] - Integrate[y'[t]/(1 + t), {t, 0, x}];

 eqn12 = y''[x] == 1 + y[x] - Integrate[y'[t]*Sin[t], {t, 0, x}];
 eqn13 = y''[x] == 1 + y[x] - Integrate[y'[t]*Sin[x], {t, 0, x}];

ic = {y[0] == 0, y'[0] == 0};

sol1 = y[x] /. First@DSolve[{eqn1, ic}, y@x, x](*Wrong solution.It should be: y[x]=x^2/2 *)
sol2 = y[x] /. First@DSolve[{eqn2, ic}, y@x, x](*Wrong solution*)
sol3 = y[x] /. First@DSolve[{eqn3, ic}, y@x, x](*Wrong solution*)
sol4 = y[x] /. First@DSolve[{eqn4, ic}, y@x, x](*Wrong solution*)
sol5 = y[x] /. First@DSolve[{eqn5, ic}, y@x, x](*Wrong solution*)

sol6 = y[x] /. First@DSolve[{eqn6, ic}, y@x, x](*Wrong solution*)
sol7 = y[x] /. First@DSolve[{eqn7, ic}, y@x, x](*Wrong solution*)
sol8 = y[x] /. First@DSolve[{eqn8, ic}, y@x, x](*Wrong solution*)
sol9 = y[x] /. First@DSolve[{eqn9, ic}, y@x, x](*Wrong solution*)
sol10 = y[x] /. First@DSolve[{eqn10, ic}, y@x, x](*Wrong solution*)
sol11 = y[x] /. First@DSolve[{eqn11, ic}, y@x, x](*Wrong solution*)
sol12 = y[x] /. First@DSolve[{eqn12, ic}, y@x, x](*Wrong solution*)
sol13 = y[x] /. First@DSolve[{eqn13, ic}, y@x, x](*Wrong solution*)

Gives (*1/2 E^-x (-1 + E^x)^2*) the same answer for 13 equations?



The question is: is a bug?


Other equations that DSolve give wrong answer:


eqn14 = y''[x] == 1 + y[x] - Integrate[1 + y'[t], {t, 0, x}];
eqn15 = y''[x] == 1 + y[x] - Integrate[x + y'[t], {t, 0, x}];
eqn16 = y''[x] == 1 + y[x] - Integrate[t + y'[t], {t, 0, x}];
eqn17 = y''[x] == 1 + y[x] - Integrate[x + y[t], {t, 0, x}];
eqn18 = y''[x] == 1 + y[x] - Integrate[t + y[t], {t, 0, x}];
eqn19 = y''[x] == 1 + y[x] - Integrate[1 + y[t], {t, 0, x}];

Checks eqn19 equation:



sol19 = y[x] /. First@DSolve[{eqn19, ic}, y@x, x]
(D[#, {x, 2}] == 1 + # - Integrate[(1 + (# /. x -> t)), {t, 0, x}]) & /@ {sol19} // FullSimplify
 (* {2 + x^2 == 2 (E^-x + x)} *)

not True.


Edited:


From Wolfram Technical Support:


Thank you for contacting Wolfram Technical Support. I have confirmed that DSolve is giving the wrong result for the examples you provided by direct substitution. I have filed a report with our developers regarding this issue, and included the example equations you provided. Thank you for bringing this to our attention. RE: [CASE:3808125]



Answer



A clear pattern emerges. (The initial conditions are ignored for generality.)



(y[x] /. First@DSolve[#, y[x], x]) & /@ {eqn1, eqn2, eqn3, eqn4, eqn5,
    eqn6, eqn7, eqn8, eqn9, eqn10, eqn11, eqn12, eqn13}
(* {-1 + E^x C[1] + E^-x C[2], -1 + E^x C[1] + E^-x C[2], -1 + E^x C[1] + E^-x C[2], ... *)

indeed gives the same answer in each case, and the answer is the same as that from


(y[x] /. First@DSolve[#, y[x], x]) &[y''[x] == 1 + y[x]]
(* -1 + E^x C[1] + E^-x C[2] *)

Evidently, the integral is set to zero by DSolve. Consider next the second set of equations


(y[x] /. First@DSolve[#, y[x], x]) & /@ {eqn14, eqn15, eqn16, eqn17, eqn18, eqn19}

(* {-1 + x + E^x C[1] + E^-x C[2], 1 + x^2 + E^x C[1] + E^-x C[2], 
    x^2/2 + E^x C[1] + E^-x C[2], 1 + x^2 + E^x C[1] + E^-x C[2], 
    x^2/2 + E^x C[1] + E^-x C[2], -1 + x + E^x C[1] + E^-x C[2]} *)

Now, compare these with the following, obtained by explicitly integrating the portions of the integrands independent of y and discarding the portions dependent on y.


(y[x] /. First@DSolve[y''[x] == 1 + y[x] - #, y[x], x]) & /@ {x, x^2, x^2/2, x^2, x^2/2, x}

The outputs are identical to those just above. DSolve simply ignores y-dependent integrands. Incidentally, this same behavior occurs in the equation of the earlier question cited in the first line of this question.


Surely, this is a bug.


Version 11.1 update (Still true for 11.3)



DSolve now returns unevaluated for all thirteen equations. Although perhaps unsatisfying, this certainly is better than returning the wrong answer.


Comments

Post a Comment

Popular posts from this blog

front end - keyboard shortcut to invoke Insert new matrix

I frequently need to type in some matrices, and the menu command Insert > Table/Matrix > New... allows matrices with lines drawn between columns and rows, which is very helpful. I would like to make a keyboard shortcut for it, but cannot find the relevant frontend token command (4209405) for it. Since the FullForm[] and InputForm[] of matrices with lines drawn between rows and columns is the same as those without lines, it's hard to do this via 3rd party system-wide text expanders (e.g. autohotkey or atext on mac). How does one assign a keyboard shortcut for the menu item Insert > Table/Matrix > New... , preferably using only mathematica? Thanks! Answer In the MenuSetup.tr (for linux located in the $InstallationDirectory/SystemFiles/FrontEnd/TextResources/X/ directory), I changed the line MenuItem["&New...", "CreateGridBoxDialog"] to read MenuItem["&New...", "CreateGridBoxDialog", MenuKey["m", Modifiers-...
Post a Comment
Read more

How to thread a list

I have data in format data = {{a1, a2}, {b1, b2}, {c1, c2}, {d1, d2}} Tableform: I want to thread it to : tdata = {{{a1, b1}, {a2, b2}}, {{a1, c1}, {a2, c2}}, {{a1, d1}, {a2, d2}}} Tableform: And I would like to do better then pseudofunction[n_] := Transpose[{data2[[1]], data2[[n]]}]; SetAttributes[pseudofunction, Listable]; Range[2, 4] // pseudofunction Here is my benchmark data, where data3 is normal sample of real data. data3 = Drop[ExcelWorkBook[[Column1 ;; Column4]], None, 1]; data2 = {a #, b #, c #, d #} & /@ Range[1, 10^5]; data = RandomReal[{0, 1}, {10^6, 4}]; Here is my benchmark code kptnw[list_] := Transpose[{Table[First@#, {Length@# - 1}], Rest@#}, {3, 1, 2}] &@list kptnw2[list_] := Transpose[{ConstantArray[First@#, Length@# - 1], Rest@#}, {3, 1, 2}] &@list OleksandrR[list_] := Flatten[Outer[List, List@First[list], Rest[list], 1], {{2}, {1, 4}}] paradox2[list_] := Partition[Riffle[list[[1]], #], 2] & /@ Drop[list, 1] RM[list_] := FoldList[Transpose[{First@li...
Post a Comment
Read more

plotting - How to draw lines between specified dots on ListPlot?

I would like to create a plot where I have unconnected dots and some connected. So far, I have figured out how to draw the dots. My code is the following: ListPlot[{{1, 1}, {2, 2}, {3, 3}, {4, 4}, {1, 4}, {2, 5}, {3, 6}, {4, 7}, {1, 7}, {2, 8}, {3, 9}, {4, 10}, {1, 10}, {2, 11}, {3, 12}, {4,13}, {2.5, 7}}, Ticks -> {{1, 2, 3, 4}, None}, AxesStyle -> Thin, TicksStyle -> Directive[Black, Bold, 12], Mesh -> Full] I have thought using ListLinePlot command, but I don't know how to specify to the command to draw only selected lines between the dots. Do have any suggestions/hints on how to do that? Thank you. Answer One possibility would be to use Epilog with Line : ListPlot[ {{1, 1}, {2, 2}, {3, 3}, {4, 4}, {1, 4}, {2, 5}, {3, 6}, {4, 7}, {1, 7}, {2, 8}, {3, 9}, {4, 10}, {1, 10}, {2, 11}, {3, 12}, {4, 13}, {2.5, 7}}, Ticks -> {{1, 2, 3, 4}, None}, AxesStyle -> Thin, TicksStyle -> Directive[Black, Bold, 12], Mesh -> Full, Epilog -> { Line[ ...
Post a Comment
Read more