System of equations

Systems of equations can be handled by creating a matrix of operators and functionals. For example, we can solve the system

\[\begin{gathered} \mathop{u}'' - \mathop{u} + 2 \mathop{v} = \mathop{e}^x, \\ \mathop{u} + \mathop{v}' + \mathop{v} = \cos{x}, \\ \mathop{u}(-1) = \mathop{u}'(-1) = \mathop{v}(-1) = 0 \end{gathered}\]

using the following code:

using ApproxFun
using LinearAlgebra

x = Fun();
B = Evaluation(Chebyshev(),-1);
A = [B      0;
     B*𝒟    0;
     0      B;
     𝒟^2-I  2I;
     I      𝒟+I];
u,v = A \ [0;0;0;exp(x);cos(x)];


import Plots
Plots.plot(u, label="u", xlabel="x", legend=:topleft)
Plots.plot!(v, label="v")

In this example, the automatic space detection failed and so we needed to specify explicitly that the domain space for B is Chebyshev().

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