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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.

65R20Integral equations (numerical methods)
45J05Integro-ordinary differential equations
65M70Spectral, collocation and related methods (IVP of PDE)
Full Text: DOI
[1] Brunner, H.: Collocation methods for Volterra integral and related functional equations methods, (2004) · Zbl 1059.65122
[2] Guo, Ben-Yu; Shen, J.: Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval, Numer. math. 86 (2000) · Zbl 0969.65094 · doi:10.1007/s002110000168
[3] Mastroianni, G.; Monegato, G.: Nyström interpolants based on zeros of Laguerre polynomials for someweiner-Hopf equations, IMA J. Numer. anal. 17 (1997) · Zbl 0884.65133 · doi:10.1093/imanum/17.4.621
[4] Shen, J.: Stable and efficient spectral methods in unbounded domains using Laguerre functions, SIAM J. Numer. anal. 38 (2000) · Zbl 0979.65105 · doi:10.1137/S0036142999362936
[5] Xu, Cheng-Long; Guo, Ben-Yu: Laguerre pseudospectral method for nonlinear partial differential equations, J. comp. Math. 20, 413-428 (2002) · Zbl 1005.65115
[6] Elnagar, G. N.; Kazemi, M.: Chebyshev spectral solution of nonlinear Volterra--Hammerstein integral equations, J. comput. Appl. math. 76, 147C158 (1996) · Zbl 0873.65122 · doi:10.1016/S0377-0427(96)00098-2
[7] H. Fujiwara, High-accurate numerical method for integral equations of the first kind under multiple-precision arithmetic, Preprint, RIMS, Kyoto University, 2006
[8] Tang, T.; Xu, X.; Cheng, J.: On spectral methods for Volterra type integral equations and the convergence analysis, J. comput. Math. 26, No. 6, 825-837 (2008) · Zbl 1174.65058
[9] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A.: Spectral methods fundamentals in single domains, (2006) · Zbl 1093.76002
[10] Qu, C. K.; Wong, R.: Szegos conjecture on Lebesgue constants for Legendre series, Pacific J. Math. 135, 157C188 (1988) · Zbl 0664.42012