A Fredholm equation of the second kind is stated as (x) = f(x) + _0^1 (x+1)( +1) , ( )

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

A Fredholm equation of the second kind is stated as ϕ(x)=f(x)+μ∫01(x+1)(ξ+1) ϕ(ξ) dξ,x∈[0,1],\phi(x) = f(x) + \mu\int_0^1 (x+1)(\xi+1)\,\phi(\xi)\,\mathrm{d}\xi, \qquad x\in[0,1], where μ\mu is the Fredholm parameter and the kernel and the interval are treated as dimensionless. ((a)) Starting from ϕ=f+μKϕ\phi=f+\mu K\phi, derive by iteration the Neumann series ϕ=∑n=0∞μnKnf\phi=\sum_{n=0}^\infty \mu^nK^nf, and show using the Cauchy-Schwarz inequality that ∥Kng∥≤∥K∥HSn∥g∥\|K^ng\|\le\|K\|_{\rm HS}^n\|g\| for any square-integrable gg. State the resulting condition on μ\mu for the series to converge. ((b)) Compute ∥K∥HS\|K\|_{\rm HS} for the stated kernel and give the resulting radius of convergence for μ\mu. ((c)) The stated kernel has exactly one nonzero eigenvalue, with eigenfunction x+1x+1. Identify the characteristic value μc\mu_{\rm c} at which the homogeneous equation ϕ=μKϕ\phi=\mu K\phi has a nontrivial solution, and use the Fredholm alternative for kernels to state the necessary and sufficient condition on ff for ϕ=f+μcKϕ\phi=f+\mu_{\rm c}K\phi to have a solution.

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