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On spectral methods for Volterra-type integro-differential equations. (English) Zbl 1202.65170

The author considers the problem of numerical solving the equation

u ' (x)=a(x)u(x)+b(x)+ -1 x K(x,s)u(s)ds

with |x|<1 and the initial condition u(-1)=u -1 . The origin integro-differential equation is chained on two integral equations with respect to functions u(x) and z(x)=u ' (x). The numerical method for solving the obtained system is a Legendre-collocation method with Gauss quadrature formulas for integral terems proposed by T. Tang, X. Xu and J. Cheng [J. Comput. Math. 26, No. 6, 825–837 (2008; Zbl 1174.65058)]. A theoretical L -estimation of error of the required solution (via it’s different norms) is derived. A numerical example illustrates the theoretical results.

65R20Integral equations (numerical methods)
45J05Integro-ordinary differential equations
65M70Spectral, collocation and related methods (IVP of PDE)