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A novel iterative numerical method is constructed for solving second kind Volterra fractional integral equations. The method uses at each iterative step a Bernstein spline interpolation procedure combined with the corresponding quadrature formula. In this way, based on the nice approximation and shape preserving properties of the Bernstein polynomials, we propose an alternative to the classical product integration technique that uses trapezoidal, Simpson, Gauss type and other well-known quadrature formulas. The convergence of the method is proved with the error estimate expressed in terms of the Lipschitz constants and the accuracy is illustrated on some numerical experiments.

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<p style='text-indent:20px;'>In this work we will discuss about an approximation method for initial value problems associated to fractional order differential equations. For this method we will use Bernstein spline approximation in combination with the Banach's Fixed Point Theorem. In order to illustrate our results, some numerical examples will be presented at the end of this article.</p>

In this paper an improved error bound is obtained for the complete quartic spline with deficiency 2, in the less smooth class of continuous functions. In the case of Lipschitzian functions, the obtained estimate improves the constant from Theorem 3, in J. Approx. Theory 58 (1989) 58-67. Some applications of the complete quartic spline in the numerical integration and in the construction of an iterative numerical method for fourth order two-point boundary value problems with pantograph type delay are presented.