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solving DE numerically....

  • To: mathgroup at smc.vnet.net
  • Subject: [mg126690] solving DE numerically....
  • From: raj kumar <rajesh7796gm at gmail.com>
  • Date: Thu, 31 May 2012 02:48:36 -0400 (EDT)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com

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]?



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