Regularizing properties of difference schemes for singular integral-differential equations (Q450890)

The author studies in the space \( C^{'}[0,1] \) a system of integral-differential equations of the form NEWLINE\[NEWLINE A(t) x'(t) + B(t) x(t) = F( t,\int^{t}_{0}K(t,s,x(s))ds),\qquad t\in [0,1], NEWLINE\]NEWLINE with the initial condition \( x(0)= x_{0}\). It is assumed that the matrix \(A(t)\) satisfies the condition NEWLINE\[NEWLINE det A= 0,\qquad \forall t\in [0,1].NEWLINE\]NEWLINE The approximate system for this problem is constructed using the backward differentiation formula of order \(k\) for the derivative \( x'(t)\) and the Adams quadrature formula to approximate the integral with the variable upper limit. The author proves that this difference scheme can be stable using the step size \(h\) as a regularization parameter.