Solve a Boundary Value Problem Using a Green's Function

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

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eqn = -u''[x] - u'[x] + 6 u[x] == f[x];

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bc0 = u[0] == 0;

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bc1 = u[1] == 0;

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

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

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

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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.

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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 .

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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 .

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Plot[Table[sol, {a, {1/4, 1, 2, 4}}] // Evaluate, {x, 0, 1}]

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