Linear differential-algebraic equations perturbed by Volterra integral operators
In the present work, the authors study the initial problem (IP) for linear inhomogeneous vector ordinary differential equations of arbitrary order in the form \[ {{\Lambda }_{k}}x:={{A}_{k}}(t){{x}^{(k)}}(t)+{{A}_{k-1}}(t){{x}^{(k-1)}}(t)+\dots+{{A}_{0}}(t)x(t)=f(t),\tag{1} \] \[ {{x}^{(j)}}(\alpha )={{\alpha }_{j}},\;j=1,\dots,k-1,\tag{2} \] where \(t\in T=[\alpha ,\beta ]\), \(k,n\in \mathbb{N}\), \({{\alpha }_{j}}\in {{\mathbb{R}}^{n}}\) are given vectors, \({{A}_{j}}:T\to {{\mathbb{R}}^{n\times n}}\) and \(\det {{A}_{k}}(t)=0,\;t\in T\). Along with IP (1), (2) is studied the an auxiliary IP (3), (4) obtained by perturbations with a Volterra operator (not necessarily small) of equation (1) in the form \[ ({{\Lambda }_{k}}+\lambda V)z:={{A}_{k}}(t){{z}^{(k)}}(t)+\dots+{{A}_{0}}(t)x(t)+\lambda \int\limits_{\alpha }^{t}{K(t,s)z(s)ds}=f(t),\tag{3} \] \[ {{z}^{(j)}}(\alpha )={{\alpha }_{j}},\;j=1,\dots,k-1.\tag{4} \] In the work under natural conditions are obtained sufficient conditions for the existence and uniqueness of the solutions of the IP (1), (2) and IP (3), (4) respectively, as well as and representations of their general solutions. The influence of small perturbations of the free term and the initial data on the solutions of the IP (1), (2) and IP (3),(4) are considered. A numerical method is suggested and single out a class of differential-algebraic equations for which one, when computing the index does not need to find derivatives of the input data and hence can be applyed a broad class of finite-difference methods. The results are illustrated with numerical experiments.
scientific article; zbMATH DE number 6844046