A Numerical Calculation of Arbitrary Integrals of Functions

Abstract

This paper presents a numerical technique for solving fractional integrals of functions by employing the trapezoidal rule in conjunction with the finite difference scheme. The proposed scheme is only a simple modification of the trapezoidal rule, in which it is treated as an algorithm in a sequence of small intervals for finding accurate approximate solutions to the corresponding problems. This method was applied to solve fractional integral of arbitrary order α > 0 for various values of alpha. The fractional integrals are described in the Riemann-Liouville sense. Figurative comparisons and error analysis between the exact value, two-point and three-point central difference formulae reveal that this modified method is active and convenient.

Computational Algorithm for Approximating Fractional Derivatives of Functions

“…Therefore, we can observe that when ℎ (step size) is reduced by a factor of 1/2 the successive errors are diminished by approximately 1/4 this confirms the order is O(ℎ 2 ) and consistent with the error analysis presented above. This method is effective and consistent especially when compared with other methods and solvers [38]- [42].…”

“…Trapezoidal rule is an effective tool for approximation of derivatives and integral of arbitrary order particularly when combined with the finite difference scheme [38]. Engineers and scientist find it useful especially in dealing with problems that are either difficult or cannot be solved analytically, this approach is not only unique but limited in literature and efficient in practice especially when numerical solution is sought, [38], [42], [45]. .…”

“…This section presents some Mathematical basics which will be necessary for further evaluation in this paper, some of which include definitions, properties and theorems and can be found in [5], [6], [42]- [44].…”

“…This article shares the author's perspective on the major, and rapidly developing topic of fractional calculus and propose and algorithm for easy computation of functions with the aid of finite difference scheme and the trapezoidal rule, with this method the problems are resolved due to the methods high adaptability and applications are made easier while maintaining efficiency. Engineers and scientists use numerical integration which is fundamental to obtain approximations of definite integrals that are difficult to solve analytically [42], [43]. One method among others that can be used for approximation of definite integrals of a specific function value at particular points is the Trapezoidal rule which is based on dividing the area between the curve of f(x) and the horizontal axis into strips and then interpolating the function f(x) by a sequence of (straight) lines [5].…”