A Numerical Calculation of Arbitrary Integrals of Functions


  • John Ojima Mamman College of Science, Federal University of Agriculture, Makurdi
  • Terhemen Aboiyar College of Science, Federal University of Agriculture, Makurdi




Finite difference, Integrals functions, fractional calculus, fractional integral, modified trapezoidal rule, Riemann-Liouville


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.


Download data is not yet available.


<li>Podlubny, “Geometric and physical interpretation of Fractional Integration and Fractional Differentiation,” <em>Fractional Calculus and Applied Analysis</em>, vol. 5, no. 4, pp367-386. (2002) Math.CA/0110241</li>
<li>Podlubny, &amp; R. Magin, &amp; I. Trymorush, “Niels Henrik Abel and the birth of fractional calculus,” <em>Fractional Calculus and Applied Analysis</em>. (2017). 20.</li>
<li>Podlubny, “What Euler could further write, or the unnoticed big bang of the fractional calculus,” <em>Fractional Calculus and Applied Analysis</em>. (2013). 16.</li>
<li>Podlubny, &amp; M. Tavazoei, &amp; B. Vinagre, &amp; D. Xue, &amp; Y. Chen, &amp; M. Haeri,&nbsp; “A Special Issue in ISA Transactions Fractional Order Signals, Systems, and Controls: Theory and Application”. <em>ISA Transactions</em> (2018). 82. 1.</li>
<li>S. Chow, “Fractional dynamics of interfaces between soft-nanoparticles and rough substrates,” <em>Physics Letter A,</em> 342(1-2):148–155. July (2005).</li>
<li>T. Baillie, “Long memory processes and fractional integration in econometrics,” <em>J Econometrics</em>, 73:5–59 (1996).</li>
<li>L. Bagley, P. J. Torvik, “A theoretical basis for the application of fractional calculus to viscoelasticity,” <em>Journal Rheol</em>. 27(3):201–210 (1983).</li>
<li>Panda, M. Dash, “Fractional generalized splines and signal processing,”<em> Signal Process</em>, 86:2340–2350 (2006).</li>
<li>Mainardi, “Fractional calculus: some basic problems in continuum and statistical mechanics. In:Fractals and fractional calculus in continuum mechanics,” <em>New York</em>: Springer, Verlag. p. 291–348 (1997).</li>
<li>Feng, A. Liu “Oscillation for a Class of Fractional Differential Equation,” <em>Journal of Applied Mathematics and Physics</em> 7, 1429-1439. (2019).</li>
<li>Wang, &amp; S. Liu, “He’s fractional derivative and its application for fractional Fornberg-Whitham equation,” <em>Thermal Science</em>. 2016. 54-54. (2016).</li>
<li>Shilpi and A. Praveen “On New Applications of Fractional Calculus,” <em>Boletim da Sociedade Paranaense de Matematica</em> 37(3):113-118 (2019).</li>
<li>Tarasov, (2019). “On History of Mathematical Economics: Application of Fractional Calculus,” <em>Mathematics</em>. 7. 509.</li>
<li>Luo, &amp; J. Wang, &amp; M. Feckan, “Applying Fractional Calculus to Analyze Economic Growth Modelling,” <em>Journal of Applied Mathematics, Statistics and Informatics,</em> (2018). 14. 25-36.</li>
<li>Li, &amp; Y. Chen, &amp; J. Kurths, “Fractional calculus and its applications” <em>Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences</em>. (2013). &nbsp;371. 20130037.</li>
<li>Hilfer “Applications of Fractional Calculus in Physics” <em>Universität Mainz &amp; Universität Stuttgart, Germany</em>. (2000) .</li>
<li>Kochubei, &amp; &nbsp;Y. Kondratiev, “Growth Equation of the General Fractional Calculus” <em>Mathematics</em>, (2019). 7. 615.</li>
<li>E. Tarasov, &amp; V. V. Tarasova, "Macroeconomic models with long dynamic memory: Fractional calculus approach," <em>Applied Mathematics and Computation, Elsevier,</em> vol. 338(C), pages 466-486. 2018.</li>
<li>Sun, &amp; Y. Zhang, &amp; &nbsp;D. Baleanu, &amp; W.&nbsp; Chen, &amp; Y. Chen, &nbsp;“A new collection of real world applications of fractional calculus in science and engineering,” <em>Communications in Nonlinear Science and Numerical Simulation.</em> (2018). &nbsp;64.</li>
<li>Datsko, &amp; I. Podlubny, &amp; Y. Povstenko, “Time-Fractional Diffusion-Wave Equation with Mass Absorption in a Sphere under Harmonic Impact,” <em>Mathematics</em>. (2019). 7. 433.</li>
<li>Magin, &amp; B. Vinagre, &amp; I. Podlubny, “Can Cybernetics and Fractional Calculus Be Partners?: Searching for New Ways to Solve Complex Problems,” <em>IEEE Systems, Man, and Cybernetics Magazine</em>. 4. 23-28. (2018).</li>
<li>Machado, &amp; V. &nbsp;Kiryakova, “The Chronicles of Fractional Calculus”. <em>Fractional Calculus and Applied Analysis. </em>&nbsp;20(2), pp. 307-336. &nbsp;(2017).</li>
<li>Sabatier, &amp; C. Ionescu, &amp; J. Tar, &amp; M. J. Tenreiro, “New Challenges in Fractional Systems,” <em>Mathematical Problems in Engineering</em>. (2013).</li>
<li>Hilfer, and Y.&nbsp; Luchko, “Desiderata for fractional Derivatives and Integrals,” <em>Mathematics</em> (2019) 7(2), 149.</li>
<li>K. RobertoGarrapa, and P. Marina “Evaluation of fractional Integrals and Derivatives of elementary functions: Overview and Tutorial,” <em>Mathematics</em> (2019)&nbsp; 7. 407.</li>
<li>Kiryakova, “Use of fractional calculus to evaluate some improper integrals of special functions.” <em>AIP Conference Proceedings</em>. (2017). 1910. 050012.</li>
<li>Agarwal, “Fractional Integration of the Product of Two Multivariables H-Function and a General Class of Polynomials Praveen Agarwal,” <em>Springer Proceedings in Mathematics &amp; Statistics</em> Volume 41, 2013, pp 359-374. (2013).</li>
<li>Yuri Luchko (Eds.),” <em>Basic Theory Berlin, Boston: De Gruyter. </em>(pp. 111–126). (2019).</li>
<li>&nbsp; J. Tenreiro &amp; V. Kiryakova, &amp; F. Mainardi, &amp; S. Momani, “FCAA-Round Table-ICFDA18,” <em>Fractional Calculus and Applied Analysis</em>. 21. 1151-1155. (2018).</li>
<li>Liu, &amp; M. Meerschaert, &amp; S. Momani, &amp; N. Leonenko, &amp; W. Chen, &amp; O. Agrawal, “Fractional Differential Equations” <em>International Journal of Differential Equations.</em> 2013.</li>
<li>S. Uttam Ghosh, , D. Shantanu “Solution of System of Linear Fractional Differential Equations with Modified Derivative of Jumarie Type,” <em>American Journal of Mathematical Analysis</em>. 2015; 3(3):72-84.</li>
<li>Razzaghi, “A numerical scheme for problems in fractional calculus” <em>ITM Web of Conferences</em>. 20. 02001.</li>
<li>Tarasov, &amp; S. Tarasova, “Probabilistic Interpretation of Kober Fractional Integral of Non-Integer Order,” <em>Progress in Fractional Differentiation and Applications</em>. (2019). &nbsp;5. 1-5.</li>
<li>Diethelm, N. Ford, A. Freed, “Detailed error analysis for a fractional Adams method,”<em> Numerical Algorithms</em> 36:31–52 May (2004).</li>
<li>Odibat, “Approximations of fractional integrals and Caputo fractional derivative,” <em>J Applied Mathematics and Computation</em>, 178:527-533 (2006).</li>
<li>H. Mathews, K. D. Fink, “Numerical Methods Using Mathlab”, <em>Prentice-Hall,</em> (2004).</li>
<li>D. Daniel “Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach,” <em>Wiley Online Books</em>.</li>
<li>W. George Hornberger “Introduction to Finite Difference Methods for Partial Differential Equations,” (2005) .</li>
<li>M.Tai “A Mathematical model for the determination of total area under glucose tolerance and other metaboliccurves,” <em>Diabetes Care </em>17(2): 152-154 (1994).</li>




How to Cite

J. O. Mamman and T. Aboiyar, “A Numerical Calculation of Arbitrary Integrals of Functions”, Adv. J. Grad. Res., vol. 7, no. 1, pp. 11-17, Oct. 2019.



Graduate Research Articles