A survey of singularly perturbed Volterra equations
The paper contains a survey of results concerning the second kind Volterra integral equations of the form \[ \varepsilon y(t)=g(t)+ \int_0^t k(t,s,y(s))ds, \qquad 0\leq t\leq T,\tag{1} \] and the Volterra integro-differential equation of the form \[ \varepsilon y'(t)= f\Biggl( t,y(t), \int_0^t k(t,s,y(s))ds \Biggr), \quad 0<t\leq T, \qquad y(0)=y_0.\tag{2} \] Both in (1) and (2) \(\varepsilon\) is a parameter from the interval \((0,1]\). Several results concerning the existence, uniqueness, qualitative properties and asymptotic expansion of solutions of the problems (1) and (2) are presented and described. A review of numerical methods for singularly perturbed Volterra equations is also added. Moreover, the paper contains two original results associated with asymptotic expansions of solutions of linear convolution equations and with a numerical method for the solution of weakly singular Volterra integral equations.
scientific article; zbMATH DE number 1036647
scientific article