Use the Rayleigh-Ritz method to find three successive approximate solutions to the extremum problem associated with the functional:
J[y]=Integral from 0 to 1 of (y')^2+xy^2+2x^2y dx,
using the trial functions Y_0(x)=x, Y_1(x)=x+c_1 x(1-x) and Y_2(x)=x+c_1 x(1-x)+c_2 x^2(1-x)^2.
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The trial function Y_0(x)=x satisfies the boundary conditions. We have
From here we conclude
inf J(y)leq 1.75<= sup J(y),
where the infimum and supremum range over all functions y satisfying the boundary conditions.
For Y_1(x)=x+c_1x(1-x) we see that Y_1(0)=0 and Y_1(1)=1, so Y_1 satisfies the boundary conditions for all c_1. We
evaluate the ...
We use the Rayleigh-Ritz method to find three successive approximate solutions to the extremum problem associated with a functional. We use three polynomial trial functions.