On a canonical standard form of second order linear ordinary differential equations with a small parameter
Recall that the function \(v(\gamma,\epsilon)=a+b-be^{-\gamma /\epsilon}\), \(\gamma >0\), \(\epsilon >0\), a,b being constants, serves as a most basic illustration of the initial layer phenomena at \(\gamma =0\) as \(\epsilon\downarrow 0\), and satisfies the equation (1) \(\epsilon d^ 2v/d\gamma^ 2+dv/d\gamma =0,\gamma >0\), \(\epsilon >0\), with the initial data \(v(0,\epsilon)=a\), \(\epsilon dv(0,\epsilon)/d\gamma =b\). Consider a slightly more general initial value problem: \[ (2)\quad \epsilon^ 2d^ 2u/dt^ 2+\epsilon p(t,\epsilon)du/dt+q(t,\epsilon)u=0,\quad t>0,\quad \epsilon >0, \] with the initial data \(u(0,\epsilon)=f\) 0(\(\epsilon)\), \(\epsilon du(0,\epsilon)/dt=f\) 1(\(\epsilon)\). The author shows, in the present paper, that the problem (2) can be reduced to the problem (1) by a change of the dependent and independent variables which incorporates the initial layer at \(t=0\) as \(\epsilon\downarrow 0\) provided the coefficients p(t,\(\epsilon)\) and q(t,\(\epsilon)\) are taken from the appropriate asymptotic class, p(t,\(\epsilon)\) positive and q(t,\(\epsilon)\) small in some sense.
scientific article; zbMATH DE number 4032170
scientific article