Abstract. In this paper we study the fractional differential equation, containing both the left and right fractional derivatives, with respect to various types of boundary conditions. We transform the fractional differential equation into equivalent integral form. Next, we develop a discrete form of the composition of the left and right fractional integrals, based on the trapezoidal rule of integration, and we obtain a numerical scheme for a fractional integral equation. The discrete form of integral equation is rewritten in the matrix form. Finally, several examples of computations are presented.
Communicated by: H. M. Srivastava MSC Classification: 26A33; 97N50; 97N40We present numerical algorithms calculating compositions of the left and right fractional integrals. We apply quadratic interpolation and obtain the fractional Simpson's rule. We estimate the local truncation error of the proposed approximations, calculate errors generated by presented algorithms, and determine the experimental rate of convergence. Finally, we show examples of numerical evaluation of these operators.
In this paper we present a numerical scheme to calculations of the left fractional integral. To calculate it we use the fractional Simpson's rule (FSR). The FSR is derived by applying quadratic interpolation. We calculate errors generated by the method for particular functions and compare the obtained results with the fractional trapezoidal rule (FTR).
Abstract. In this paper we present different approaches to the transformation of the second order ordinary differential equation, with respect to adequate boundary conditions, into integral equations. The obtained equations are Fredholm integral equations of the second kind. Next, a numerical method based on quadrature methods has been proposed to get an approximate solution of these equations.
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