In this paper we describe numerical methods for solution integral-algebraic equations wiht weakly singular kernels. Methods are based on explicit Adam’s methods, product integration methods for the integral part and on extrapolation formulas for the main part of the equation are proposed.We got weights of quadrature formulas. Presented results of numerical experiments.

1. Apartsyn A.S. Nonclassical Linear Volterra Equations of the First Kind. Nauka, Novosibirsk, 1999 VSP, Utrecht, 2003.

2. Bakhvalov N.S. Numerical Methods: Analysis, Algebra, Ordinary Differential Equations. Moscow, Nauka,1975 Moscow, Mir, 1977.

3. Budnikova O.S., Bulatov M.V. Numerical solution of integral-algebraic equations for multistep methods. Comput. Math. Math. Phys., 2012, vol. 52, no 5 , pp. 691-701.

4. Bulatov M.V., Budnikova O.S. An analysis of multistep methods for solving integral-algebraic equations: Construction of stability domains. Comput. Math. Math. Phys., 2013, vol. 53, no 9, pp. 1260-1271.

5. Bulatov M.V., Budnikova O.S. On stable algorithms of numerical solution of integral-algebraic equations. Bulletin of the South Ural State University, Series "Mathematical Modelling, Programming & Computer;Software", 2013, vol. 6, no 4,pp. 5-14.

6. Verlan’ A.F., Sizikov V.S. Integral equations: methods, algoritms, solutions [Integral’nye uravnenija: metody, algoritmy, reshenija]. Kiev, Naukova dumka, 1986.

7. Gluhkov V.M., Ivanov V.V., Yanenko V.M. Simulation of Evolving Systems (in Russian). Moscow, Nauka, 1983.

8. Samko S.G., Kilbas A.A., Marichev O.I. Integrals and derivatives of fractional order, and some applications(in Russian). Minsk, Nauka and Tehnika, 1987.

9. Sizikov V.S., Smirnov A.V., Fedorov B.A.Numerical solution of the singular Abel integral equation by the generalized quadrature method. Russian Mathematics (Izvestiya VUZ. Matematika), 2004, vol. 48, no 8, pp. 59-66.

10. Ten Men YanApproximate Solution of Linear Volterra Integral Equations of the First Kind(in Russian). Candidate’s Dissertation in Mathematics and Physics. Irkutsk, 1985.

11. Chistyakov V.F. On Singular Systems of Ordinary Differential Equations and Their Integrals Analogues. Lyapunov Functions and Applications. Novosibirsk, Nauka, 1987, pp. 231-239.

12. Bulatov M.V., Lima P.M., Weinm¨uller E.B. Existence and Uniqueness of Solutions to Weakly Singular Integral-Algebraic and Integro-Differential Equations. Central European Journal of Mathematics, 2014, vol. 12, no 2, pp. 308-321.

13. Brunner H. Collocation Methods for Volterra Integral and Related Functioal Equations. Unversity Press, Cambridge, 2004.

14. Brunner H., Bulatov M.V. On singular systems of integral equations with weakly singular kernels. Proceeding of the 11-th Baikal International School Seminar: Optimization Methods and their Applications, 1998, pp. 64-67.

15. Hadizadeh M., Ghoreishi F., Pishbin S. Jacobi spectral solution for integral algebraic equations of index-2. Appl. Numer. Math., 2011, vol. 61, no 1, p. 131-148.

16. Kauthen J.P. The numerical solution of integral-algebraic equations of index-1 by pollinomial spline collocation methods. Math. Comp., 2000, no 236, pp. 1503-1514.

17. Linz P. Analytical and Numerical Methods for Volterra Equations. SIAM, Philadelphia, 1985.

18. Pishbin S. On the numerical solution of integral equations of the fourh kind with higher index: differentiability and tractability index-3 . Journal of Mathematical Modeling, 2015, vol. 2, no 2, pp. 156-169.

19. Pishbin S., Hadizadeh M., Ghoreishi F. The semi-explicit Volterra integral algebraic equations with weakly singular kernel: The numerical treatments. Journal of Computational and Applied Mathematics, 2013, vol. 245, no 1, pp. 121–132.

20. Weiss R.A., Anderssen R.S. Product Integration Method for a Class of Singular First Kind Volterra Equations. Numer. Math., 1972, vol. 18, no 2, pp. 442-456.