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Mathematica Calculus Equation Solving Wolfram Language

How to solve 2-D Laplace equation in a rectangular domain analytically?

Phil R

Posted 11 years ago

POSTED BY: Phil R

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Phil R

Posted 11 years ago

POSTED BY: Phil R

S M Blinder

S M Blinder, WRI and Univ of Michigan

Posted 11 years ago

Correction: In[3]:= DSolve[{D[f[x], {x, 2}]/f[x] == k, f[0] == 0, f[20] == 1}, f[x], x] Out[3]= {{f[x] -> ( E^(20 Sqrt[k] - Sqrt[k] x) (-1 + E^(2 Sqrt[k] x)))/(-1 + E^( 40 Sqrt[k]))}} In[4]:= DSolve[{D[g[y], {y, 2}]/g[y] == -k, g[0] == 0, g[20] == 2}, g[y], y] Out[4]= {{g[y] -> 2 Csc[20 Sqrt[k]] Sin[Sqrt[k] y]}}

Correction:

In[3]:= DSolve[{D[f[x], {x, 2}]/f[x] == k, f[0] == 0, f[20] == 1}, 
 f[x], x]

Out[3]= {{f[x] -> (
   E^(20 Sqrt[k] - Sqrt[k] x) (-1 + E^(2 Sqrt[k] x)))/(-1 + E^(
    40 Sqrt[k]))}}

In[4]:= DSolve[{D[g[y], {y, 2}]/g[y] == -k, g[0] == 0, g[20] == 2}, 
 g[y], y]

Out[4]= {{g[y] -> 2 Csc[20 Sqrt[k]] Sin[Sqrt[k] y]}}

POSTED BY: S M Blinder

S M Blinder

S M Blinder, WRI and Univ of Michigan

Posted 11 years ago

You can find separable solution with u[x,y]=f[x] g[y] In[1]:= DSolve[{D[f[x], {x, 2}] == k, f[0] == 0, f[20] == 1}, f[x], x] Out[1]= {{f[x] -> 1/20 (x - 200 k x + 10 k x^2)}} In[2]:= DSolve[{D[g[y], {y, 2}] == -k, g[0] == 0, g[20] == 2}, g[y], y] Out[2]= {{g[y] -> 1/10 (y + 100 k y - 5 k y^2)}}

You can find separable solution with u[x,y]=f[x] g[y]

In[1]:= DSolve[{D[f[x], {x, 2}] == k, f[0] == 0, f[20] == 1}, f[x], x]

Out[1]= {{f[x] -> 1/20 (x - 200 k x + 10 k x^2)}}

In[2]:= DSolve[{D[g[y], {y, 2}] == -k, g[0] == 0, g[20] == 2}, g[y], y]

Out[2]= {{g[y] -> 1/10 (y + 100 k y - 5 k y^2)}}

POSTED BY: S M Blinder

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