The Reproducing Kernel Hilbert Space Method for Solving System of Linear Weakly Singular Volterra Integral Equations
The exact solutions of a system of linear weakly singular Volterra integral equations (VIE) have been a difficult to find. The aim of this paper is to apply reproducing kernel Hilbert space (RKHS) method to find the approximate solutions to this type of systems. At first, we used Taylor's expansion to omit the singularity. From an expansion the given system of linear weakly singular VIE is transform into a system of linear ordinary differential equations (LODEs). The approximate solutions are represent in the form of series in the reproducing kernel space . By comparing with the exact solutions of two examples, we saw that RKHS is a powerful, easy to apply and full efficiency in scientific applications to build a solution without linearization and turbulence or discretization.
HEP and Springer, Peking and Berlin, (2011).
 A. Tahmasbi and O.S. Fard, "Numerical solution of linear Volterra integral equations
system of the second kind," Applied Mathematics and Computation, vol. 201. No. 1-2, pp.
 E. Babolian, J. Biazar, and A. R. Vahidi, "On the decomposition method for system of
linear equations and system of linear Volterra integral eguations," Applied Mathematics
and Computation, vol. 147, no. 1, pp. 19-27, (2004).
 J. Biazar and H. Ghazvini, "He's homotopy perturbation method for solving systems of
Volterra integral equations of the second kind," Chaos, Solitons and Fractals, vol. 39, no 2,
pp. 770-777, (2009).
 E. Yusufoglu , "A homotopy perturbation algorithm to solve a system of Fredholm-
Volterra type integral equations," Mathematical and Computer Modelling, vol.47, no. 11-
12, pp. 1099–1107, (2008).
 R. Katani and S. Shahmorad, " Block by block method for the systems of nonlinear
Volterra integral equations," Applied Mathematical Modelling, vol. 34, no. 2, pp. 400–406,
 M. Rabbani, K. Maleknejad, and N. Aghazadeh, "Numerical computational solution of the
Volterra integral equations system of the second kind by using an expansion method,"
Applied Mathematics and Computation, vol. 187, no. 2, pp. 1143–1146, (2007).
 H. Yao, "The research of algorithms for some singular differential equations of higher even
-order," Department of mathematics," Harbin Institute of Technology, 2008.
 M. Gui and Y. Lin, "Nonlinear numerical Analysis in the Reproducing kernel space, "Nova
Science Publisher, New York, NY, USA, 2008.
 A. Berlinet and C. Thomas-Agnan, "Reproducing kernel Hilbert space in probability and
statistics," Kluwer Academic Publishers, 2004.
 Z. Chen and Y. Z. Lin, "The exact solution of a linear integral equation with weakly
Singular kernel," Journal of Mathematical Analysis and Applications, vol. 344, pp.726-
 S. Javadi, E. Babolian and E. Moradi, "New implementation of reproducing kernel
Hilbert space method for solving a class of functional integral equations," International
Scientific Publications and consulting Services, vol. 2014, pp. 1-7, 2014.
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