By Michel Hervé

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**Extra info for Analyticity in infinite dimensional spaces**

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4 Consider the two-point boundary-value problem −u = cosh x, u(0) = u(1) = 0, where we seek an approximate solution of the form u ˜4 = x(1 − x)(c0 + c1 x + c2 x2 + c3 x3 + c4 x4 ), with the exact solution u = 1 + (cosh 1 − 1)x − cosh x. 2, . . 9 to ﬁnd u ˜4 . Compare the results with the exact solution. 5 For the two-point boundary-value problem u = ex , 0 < x < 1, u(0) = u(1) = 0, use the Rayleigh–Ritz method with the two trial functions u ˜A = x(1 − x)c0 + c1 x and u ˜B = x(ex − e). Compare the results with the exact solution.

E. 29) uLu dx dy − 2 I [u] = D uf dx dy. 29). Suppose that u0 is the exact solution; then Lu0 = f. 27 Weighted residual and variational methods Thus uLu dx dy − 2 I [u] = D uLu0 dx dy D uL (u − u0 ) dx dy − = uLu0 dx dy D D uL (u − u0 ) dx dy − = (u − u0 ) Lu0 dx dy − D D u0 Lu0 dx dy. D Since L is self-adjoint and the boundary conditions are homogeneous, (u − u0 ) Lu dx dy − I [u] = (u − u0 ) Lu0 dx dy − D D (u − u0 ) L (u − u0 ) dx dy − = D u0 Lu0 dx dy D u0 Lu0 dx dy. D Since L is positive deﬁnite and u0 is non-trivial, u0 Lu0 dx dy > 0 D and (u − u0 ) L (u − u0 ) dx dy ≥ 0, D equality occurring if and only if u ≡ u0 .

1), and examples are given in Appendix A. These equations are often equivalent to the problem of the minimization of a functional, which itself may be interpreted in terms of the total energy of the system under consideration. In any physical situation, an expression for the total energy could be obtained and then minimized to ﬁnd the equilibrium solution. However, instead of ﬁnding the energy explicitly, it would be useful to be able to start with the governing partial diﬀerential equation and develop the corresponding functional.