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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 $$u'(x)=a(x)u(x)+ b(x) + \int_{-1}^{x}K(x,s)u(s)ds$$ 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 {\it T. Tang, X. Xu} and {\it J. Cheng} [J. Comput. Math. 26, No. 6, 825--837 (2008; Zbl 1174.65058)]. A theoretical $L_{\infty}$-estimation of error of the required solution (via it’s different norms) is derived. A numerical example illustrates the theoretical results.

##### MSC:
 65R20 Integral equations (numerical methods) 45J05 Integro-ordinary differential equations 65M70 Spectral, collocation and related methods (IVP of PDE)
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##### References:
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