solving DE numerically....

[raj kumar]

hi all,
greetings from sunny malaysia!
this pertains to solving a DE numerically at 5 differnt points.Should be straigth forward. But i get different results when i try to insert one parameter "by hand".have i overlooked something?

any feed-back will be most appreciated

Please consider the following:

In[137]:= pw3 = Piecewise[{{V0, x < a}, {V0, x == a}}, 0];
k = Sqrt[10 - pw3];
V0 = -1;
a = 2;
Table[{x, k}, {x, 1, 5}]
(*Plot[Piecewise[{{V0, x<a}, {V0,x==a}},0],{x,0,5}]*)


Out[141]= {{1, Sqrt[11]}, {2, Sqrt[11]}, {3, Sqrt[10]}, {4, Sqrt[
10]}, {5, Sqrt[10]}}

Solve the following DE numerically for y1[
1.0], y1[2.0], y1[3], y1[4], y1[5]

In[148]:=
s33 = NDSolve[{(y1^\[Prime]\[Prime])[x] + Sqrt[k] y1[x] == 0,
y1[0.1] == 0, Derivative[1][y1][0.1] == 1/10^6}, y1, {x, 1, 10}];


We obtain

In[149]:= {y1[1.0], y1[2.0], y1[3], y1[4], y1[5]} /. s33

Out[149]= {{6.9408*10^-7,
4.08303*10^-7, -5.09458*10^-7, -6.64851*10^-7, 1.70684*10^-7}}

now check the output above for the values x =
1 to x = 5. For x = 1, 2,
we insert k =
Sqrt[11] by hand in the same DE above and work out the values for \
the same points again. We get

In[144]:=
s11 = NDSolve[{(y11^\[Prime]\[Prime])[x] + Sqrt[ 11] y11[x] == 0,
y11[0.1] == 0, Derivative[1][y11][0.1] == 1/10^6},
y11, {x, 1, 10}];

In[145]:= {y11[1.0], y11[2.0]} /. s11

Out[145]= {{5.47931*10^-7, -1.68202*10^-7}}

Out[105]= {{5.47931*10^-7, -1.68202*10^-7}}

For x = 3, 4, 5, k = Sqrt[10]. We get

In[146]:=
s11 := NDSolve[{(y11^\[Prime]\[Prime])[x] + Sqrt[10] y11[x] == 0,
y11[0.1] == 0, Derivative[1][y11][0.1] == 1/10^6},
y11, {x, 1, 10}];



In[147]:= {y11[3], y11[4], y11[5]} /. s11

Out[147]= {{-5.30318*10^-7, 3.34051*10^-7, 4.16315*10^-7}}

Qn : Should the values for y11[1-5] be any different from that calculated earlier ie y1[1-5]?

[da...@wolfram.com]

As you note, k is sqrt(11) for x<2. You are using sqrt(k) in your DE, not k, so you'd want (11)^(1/4) rather than sqrt(11) in the second DE.

Daniel Lichtblau
Wolfram Research

[Kevin J. McCann]

Greetings from a rather steamy Washington, DC.

I think your problem is that k is a function of x not a fixed parameter
in the DE. Here is my take on your problem. I made both pw3 and k
explicit functions of x. Drop this into a notebook, put the cursor in
the cell and type ctrl-shift-n to make it look nice.

Cheers,

Kevin

Clear[pw3, x, k]
(* note the set delayed (:=) in the next two *)
pw3[x_] := Piecewise[{{V0, x < a}, {V0, x == a}}, 0];
k[x_] := Sqrt[10 - pw3[x]];

V0 = -1;
a = 2;

Table[{x, k[x]}, {x, 1, 5}]
Plot[pw3[x], {x, 0, 5}]
s33 = NDSolve[{(y1^\[Prime]\[Prime])[x] + Sqrt[k[x]] y1[x] == 0,
y1[0.1] == 0, Derivative[1][y1][0.1] == 10^-6}, y1, {x, 0.1, 10}]
Y1 = y1 /. s33[[1]]
(*Check the IC *)
{Y1[0.1], Y1'[0.1]}
Table[{x, Y1[x]}, {x, 1, 5, 1}]
Plot[Y1[x], {x, 0.1, 10}]
(* Check that the de is zero (almost) *)
Plot[(Y1^\[Prime]\[Prime])[x] + Sqrt[k[x]] Y1[x], {x, 0.1, 10}]