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Numerical methods for a class of singular integro-differential equations based on semigroup approximation. (English) Zbl 0744.65103

The (scalar) integro-differential equation of neutral type,

(d/dt)( -r 0 g(θ)x(t+θ)dθ)=a 0 x(t)+ -r 0 a(θ)x(t+θ)dθ+a 1 x(t-r)+f(t),

t>0, x(θ)=φ(θ) (-rθ<0), where g is positive, nondecreasing, and weakly singular at θ=0 (e.g. g(θ)=|θ| -p , 0<p<1), is considered in the weighted Lebesgue space L g 2 .

Using approximation techniques introduced by H. T. Banks and J. A. Burns [SIAM J. Control Optim., 18, 169-208 (1978; Zbl 0379.49025)], the convergence of the spline-based semi-discrete and fully- discrete numerical schemes is analyzed within the framework of semigroup theory (where the given integro-differential equation is formulated as a first-order hyperbolic partial differential equation with nonlocal boundary condition).

Two examples (a singular neutral functional differential equation and an Abel-Volterra integral equation of the first kind) are employed to illustrate the feasibility of these numerical schemes.


MSC:
65R20Integral equations (numerical methods)
45J05Integro-ordinary differential equations