Consider the symmetric

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Partial differential equations and boundary-value problems · Partial differential equations and boundary-value problems

Conditions of Validity, Ch Integral equations, kernels and the Fredholm alternative for the last time

Problem

Consider the symmetric, two-term separable kernel K(x,ξ)=x+ξK(x,\xi)=x+\xi on [0,1][0,1] in the Fredholm equation of the second kind ϕ(x)=f(x)+μ∫01(x+ξ) ϕ(ξ) dξ,\phi(x)=f(x)+\mu\int_0^1(x+\xi)\,\phi(\xi)\,\mathrm{d}\xi, where μ\mu is the Fredholm parameter and the interval is treated as dimensionless. ((a)) Using a1(x)=xa_1(x)=x, b1(ξ)=1b_1(\xi)=1, a2(x)=1a_2(x)=1, b2(ξ)=ξb_2(\xi)=\xi, reduce the equation to a 2×22\times2 linear system for c1=∫01ϕ dξc_1=\int_0^1\phi\,\mathrm{d}\xi and c2=∫01ξϕ dξc_2=\int_0^1\xi\phi\,\mathrm{d}\xi, and solve it exactly for μ=1/2\mu=1/2 and f(x)=xf(x)=x to give ϕ(x)\phi(x). ((b)) Find the two characteristic values of μ\mu, the roots of the 2×22\times2 system's determinant. ((c)) At the positive characteristic value, μ=43−6\mu=4\sqrt3-6, find (up to an overall scale) the homogeneous solution ψ(x)\psi(x) of ψ=μKψ\psi=\mu K\psi, and use the Fredholm alternative for kernels to decide whether ϕ=f+μKϕ\phi=f+\mu K\phi has a solution there for (i) the same f(x)=xf(x)=x used in part (a), and (ii) f(x)=x−3/3f(x)=x-\sqrt3/3.

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