# Wolfram Mathematica

## Solve a Boundary Value Problem Using a Green's Function

Solve the following second-order differential equation subject to the given homogeneous boundary conditions.

In[1]:=
`eqn = -u''[x] - u'[x] + 6 u[x] == f[x];`
In[2]:=
`bc0 = u[0] == 0;`
In[3]:=
`bc1 = u[1] == 0;`

The forcing term is given by the following function f[x].

In[4]:=
`f[x_] := E^(-a x)`

Compute the Green's function for the corresponding differential operator.

In[5]:=
`gf[y_, x_] = GreenFunction[{eqn[[1]], bc0, bc1}, u[x], {x, 0, 1}, y]`
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Plot the Green's function for different values of lying between 0 and 1.

In[6]:=
`Plot[Table[gf[y, x], {y, 0, 1, 0.2}] // Evaluate, {x, 0, 1}]`
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The solution of the original differential equation with the given forcing term can now be computed using a convolution integral on the interval .

In[7]:=
```sol = Integrate[gf[y, x] f[y], {y, 0, 1}, Assumptions -> 0 < x < 1] // Simplify```
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Plot the solution for different values of the parameter .

In[8]:=
`Plot[Table[sol, {a, {1/4, 1, 2, 4}}] // Evaluate, {x, 0, 1}]`
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