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Re: solving DE numerically....

  • To: mathgroup at smc.vnet.net
  • Subject: [mg126702] Re: solving DE numerically....
  • From: danl at wolfram.com
  • Date: Fri, 1 Jun 2012 05:16:38 -0400 (EDT)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com
  • References: <jq74dd$qj1$1@smc.vnet.net>

On Thursday, May 31, 2012 1:51:25 AM UTC-5, raj kumar wrote:
> 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]?

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



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