Asymptotic solution of some singularly perturbed Fredholm integral equations
The authors consider certain physical problems reducing to a singularly perturbed Fredholm integral equation of the type \(K_{\epsilon}u(x)+c(x)u(x)=f(x,\epsilon)\) with \[ K_{\epsilon}u(x)=1/\epsilon \int^{b}_{a}k((x- s)/\epsilon,x,s)u(s)ds,\quad a\leq x\leq b, \] where \(\epsilon >0\) is a small parameter. The kernel k(z,x,s) is continuous in x and s as well as in z except possibly at \(z=0\). The authors demonstrate a formal approach to construct the leading order boundary layer (at either or both of the end points \(x=a\), \(x=b)\) corrections to \(u_ 0(x)\) which is the solution of the reduced (as \(\epsilon\) \(\to 0)\) problem. They consider an asymptotic expansion of the form \(u(x)=u_ 0(x)V(\eta)W(\zeta),\) \(\eta =(x-a)/\epsilon,\) \(\zeta =(b-x)/\epsilon.\) Building the asymptotic expansion is reduced to solving two Wiener-Hopf problems for W and V. The authors consider concrete problems in heat transfer, diffraction theory, crack mechanics, Markov processes and low order eigenvalue problems.
singularly perturbed Fredholm integral equation
scientific article; zbMATH DE number 4107426
scientific article