BVP examples of the AnalyticalApproximations`LdeApprox` package

Copyright © 2003-2017 DigiArea, Inc.. All rights reserved.

This notebook illustrates AnalyticalApproximations`LdeApprox` package capability of working with BVPs.

This loads the package.


In[31]:=

bvp_1.gif

Example 1

Boundary value problem.

In[32]:=

bvp_2.gif

Out[32]=

bvp_3.gif

Finding polynomial approximation for solution of the BVP.

In[33]:=

bvp_4.gif

Out[33]=

bvp_5.gif

Using Mathematica function DSolve to get exact solution of the BVP.

In[34]:=

bvp_6.gif

Out[34]=

bvp_7.gif

Comparing exact and approximate results for bvp_8.gif using Mathematica function Plot.

In[35]:=

bvp_9.gif

Out[35]=

bvp_10.gif

Example 2

Boundary value problem.

In[36]:=

bvp_11.gif

Out[36]=

bvp_12.gif

Finding polynomial approximation for solution of the BVP.

In[37]:=

bvp_13.gif

Out[37]=

bvp_14.gif

Using Mathematica function DSolve to get exact solution of the BVP.

In[38]:=

bvp_15.gif

Out[38]=

bvp_16.gif

Comparing exact and approximate results using Mathematica function Plot3D.

In[39]:=

bvp_17.gif

Out[39]=

bvp_18.gif

Example 3

Boundary value problem.

In[40]:=

bvp_19.gif

Out[40]=

bvp_20.gif

Finding polynomial approximation for solution of the BVP.

In[41]:=

bvp_21.gif

Out[41]=

bvp_22.gif

Using Mathematica function DSolve to get exact solution of the BVP.

In[42]:=

bvp_23.gif

Out[42]=

bvp_24.gif

Comparing exact and approximate results using Mathematica function Plot3D.

In[43]:=

bvp_25.gif

Out[43]=

bvp_26.gif

Example 4

Boundary value problem.

In[44]:=

bvp_27.gif

Out[44]=

bvp_28.gif

Trying to find polynomial approximation for solution of the BVP.

In[45]:=

bvp_29.gif

bvp_30.gif

Out[45]=

bvp_31.gif

Using ToRatCoeffs function to get rational coefficients of the LDE on interval [0,1].

In[46]:=

bvp_32.gif

Out[46]=

bvp_33.gif

Finding polynomial approximation for solution of the BVP.

In[47]:=

bvp_34.gif

Out[47]=

bvp_35.gif

Using Mathematica function DSolve to get exact solution of the BVP.

In[48]:=

bvp_36.gif

Out[48]=

bvp_37.gif

Comparing exact and approximate results using Mathematica function Plot.

In[49]:=

bvp_38.gif

Out[49]=

bvp_39.gif

Example 5

Boundary value problem.

In[50]:=

bvp_40.gif

Out[50]=

bvp_41.gif

Finding polynomial approximation for solution of the BVP.

In[51]:=

bvp_42.gif

Out[51]=

bvp_43.gif

Using Mathematica function DSolve to get exact solution of the BVP.

In[52]:=

bvp_44.gif

Out[52]=

bvp_45.gif

Comparing exact and approximate results using Mathematica function Plot.

In[53]:=

bvp_46.gif

Out[53]=

bvp_47.gif

Example 6

Boundary value problem.

In[54]:=

bvp_48.gif

Out[54]=

bvp_49.gif

Finding polynomial approximation for solution of the BVP.

In[55]:=

bvp_50.gif

Out[55]=

bvp_51.gif

Using Mathematica function DSolve to get exact solution of the BVP.

In[56]:=

bvp_52.gif

Out[56]=

bvp_53.gif

Comparing exact and approximate results using Mathematica function Plot.

In[57]:=

bvp_54.gif

Out[57]=

bvp_55.gif

Example 7

Boundary value problem.

In[58]:=

bvp_56.gif

Out[58]=

bvp_57.gif

Finding polynomial approximation for solution of the BVP.

In[59]:=

bvp_58.gif

Out[59]=

bvp_59.gif

Using Mathematica function DSolve to get exact solution of the BVP.

In[60]:=

bvp_60.gif

Out[60]=

bvp_61.gif

Comparing exact and approximate results using Mathematica function Plot3D.

In[61]:=

bvp_62.gif

Out[61]=

bvp_63.gif

Example 8

Boundary value problem.

In[62]:=

bvp_64.gif

Out[62]=

bvp_65.gif

Finding polynomial approximation for solution of the BVP.

In[63]:=

bvp_66.gif

Out[63]=

bvp_67.gif

Using Mathematica function DSolve to get exact solution of the BVP.

In[64]:=

bvp_68.gif

Out[64]=

bvp_69.gif

Comparing exact and approximate results.

In[65]:=

bvp_70.gif

Out[65]=

bvp_71.gif

Example 9

Polynomial approximation of solution of boundary value problem bvp_72.gif.

The boundary value problem.

In[66]:=

bvp_73.gif

Out[66]=

bvp_74.gif

Finding polynomial approximation for solution of the BVP.

In[67]:=

bvp_75.gif

bvp_76.gif

bvp_77.gif

Out[67]=

bvp_78.gif

Finding normalized polynomial approximation for solution of the BVP on interval [0,1].

In[68]:=

bvp_79.gif

Out[68]=

bvp_80.gif

Unfortunately DSolve can not find exact solution of the BVP because it contains parameter α.

In[69]:=

bvp_81.gif

Out[69]=

bvp_82.gif

As the BVP is homogeneous one then to get the exact solution we can use the following technic. First of all we find exact solution for the following problem bvp_83.gif. After that we find constant α. And finally we normalize the result to get exact solution of the initial BVP.

In[70]:=

bvp_84.gif

Out[70]=

bvp_85.gif

Finding constant α using the exact solution.

In[71]:=

bvp_86.gif

Out[71]=

bvp_87.gif

In[72]:=

bvp_88.gif

Out[72]=

bvp_89.gif

Comparing exact and approximate results for constant α.

In[73]:=

bvp_90.gif

Out[73]=

bvp_91.gif

Finding normalized exact solution of the BVP on interval [0,1].

In[74]:=

bvp_92.gif

Out[74]=

bvp_93.gif

Comparing normalized exact and approximate results using Mathematica function Plot.

In[75]:=

bvp_94.gif

Out[75]=

bvp_95.gif