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




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.


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


Download data is not yet available.


I. Podlubny, “Geometric and physical interpretation of Fractional Integration and Fractional Differentiation,” Fractional Calculus and Applied Analysis, vol. 5, no. 4, pp367-386. (2002) Math.CA/0110241

I. Podlubny, & R. Magin, & I. Trymorush, “Niels Henrik Abel and the birth of fractional calculus,” Fractional Calculus and Applied Analysis. (2017). 20.

I. Podlubny, “What Euler could further write, or the unnoticed big bang of the fractional calculus,” Fractional Calculus and Applied Analysis. (2013). 16.

I. Podlubny, & M. Tavazoei, & B. Vinagre, & D. Xue, & Y. Chen, & M. Haeri, “A Special Issue in ISA Transactions Fractional Order Signals, Systems, and Controls: Theory and Application”. ISA Transactions (2018). 82. 1.

T. S. Chow, “Fractional dynamics of interfaces between soft-nanoparticles and rough substrates,” Physics Letter A, 342(1-2):148–155. July (2005).

R.T. Baillie, “Long memory processes and fractional integration in econometrics,” J Econometrics, 73:5–59 (1996).

R. L. Bagley, P. J. Torvik, “A theoretical basis for the application of fractional calculus to viscoelasticity,” Journal Rheol. 27(3):201–210 (1983).

R. Panda, M. Dash, “Fractional generalized splines and signal processing,” Signal Process, 86:2340–2350 (2006).

F. Mainardi, “Fractional calculus: some basic problems in continuum and statistical mechanics. In:Fractals and fractional calculus in continuum mechanics,” New York: Springer, Verlag. p. 291–348 (1997).

Q. Feng, A. Liu “Oscillation for a Class of Fractional Differential Equation,” Journal of Applied Mathematics and Physics 7, 1429-1439. (2019).

K. Wang, & S. Liu, “He’s fractional derivative and its application for fractional Fornberg-Whitham equation,” Thermal Science. 2016. 54-54. (2016).

J. Shilpi and A. Praveen “On New Applications of Fractional Calculus,” Boletim da Sociedade Paranaense de Matematica 37(3):113-118 (2019).

V. Tarasov, (2019). “On History of Mathematical Economics: Application of Fractional Calculus,” Mathematics. 7. 509.

D. Luo, & J. Wang, & M. Feckan, “Applying Fractional Calculus to Analyze Economic Growth Modelling,” Journal of Applied Mathematics, Statistics and Informatics, (2018). 14. 25-36.

C. Li, & Y. Chen, & J. Kurths, “Fractional calculus and its applications” Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences. (2013). 371. 20130037.

R. Hilfer “Applications of Fractional Calculus in Physics” Universität Mainz & Universität Stuttgart, Germany. (2000) .

A. Kochubei, & Y. Kondratiev, “Growth Equation of the General Fractional Calculus” Mathematics, (2019). 7. 615.

V. E. Tarasov, & V. V. Tarasova, "Macroeconomic models with long dynamic memory: Fractional calculus approach," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 466-486. 2018.

H. Sun, & Y. Zhang, & D. Baleanu, & W. Chen, & Y. Chen, “A new collection of real world applications of fractional calculus in science and engineering,” Communications in Nonlinear Science and Numerical Simulation. (2018). 64.

B. Datsko, & I. Podlubny, & Y. Povstenko, “Time-Fractional Diffusion-Wave Equation with Mass Absorption in a Sphere under Harmonic Impact,” Mathematics. (2019). 7. 433.

R. Magin, & B. Vinagre, & I. Podlubny, “Can Cybernetics and Fractional Calculus Be Partners?: Searching for New Ways to Solve Complex Problems,” IEEE Systems, Man, and Cybernetics Magazine. 4. 23-28. (2018).

J. Machado, & V. Kiryakova, “The Chronicles of Fractional Calculus”. Fractional Calculus and Applied Analysis. 20(2), pp. 307-336. (2017).

J. Sabatier, & C. Ionescu, & J. Tar, & M. J. Tenreiro, “New Challenges in Fractional Systems,” Mathematical Problems in Engineering. (2013).

R. Hilfer, and Y. Luchko, “Desiderata for fractional Derivatives and Integrals,” Mathematics (2019) 7(2), 149.

E. K. RobertoGarrapa, and P. Marina “Evaluation of fractional Integrals and Derivatives of elementary functions: Overview and Tutorial,” Mathematics (2019) 7. 407.

V. Kiryakova, “Use of fractional calculus to evaluate some improper integrals of special functions.” AIP Conference Proceedings. (2017). 1910. 050012.

P. Agarwal, “Fractional Integration of the Product of Two Multivariables H-Function and a General Class of Polynomials Praveen Agarwal,” Springer Proceedings in Mathematics & Statistics Volume 41, 2013, pp 359-374. (2013).

Yuri Luchko (Eds.),” Basic Theory Berlin, Boston: De Gruyter. (pp. 111–126). (2019).

M. J. Tenreiro & V. Kiryakova, & F. Mainardi, & S. Momani, “FCAA-Round Table-ICFDA18,” Fractional Calculus and Applied Analysis. 21. 1151-1155. (2018).

F. Liu, & M. Meerschaert, & S. Momani, & N. Leonenko, & W. Chen, & O. Agrawal, “Fractional Differential Equations” International Journal of Differential Equations. 2013.

S. S. Uttam Ghosh, , D. Shantanu “Solution of System of Linear Fractional Differential Equations with Modified Derivative of Jumarie Type,” American Journal of Mathematical Analysis. 2015; 3(3):72-84.

M. Razzaghi, “A numerical scheme for problems in fractional calculus” ITM Web of Conferences. 20. 02001.

V. Tarasov, & S. Tarasova, “Probabilistic Interpretation of Kober Fractional Integral of Non-Integer Order,” Progress in Fractional Differentiation and Applications. (2019). 5. 1-5.

K. Diethelm, N. Ford, A. Freed, “Detailed error analysis for a fractional Adams method,” Numerical Algorithms 36:31–52 May (2004).

Z. Odibat, “Approximations of fractional integrals and Caputo fractional derivative,” J Applied Mathematics and Computation, 178:527-533 (2006).

J. H. Mathews, K. D. Fink, “Numerical Methods Using Mathlab”, Prentice-Hall, (2004).

J. D. Daniel “Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach,” Wiley Online Books.

P. W. George Hornberger “Introduction to Finite Difference Methods for Partial Differential Equations,” (2005) .

M. M.Tai “A Mathematical model for the determination of total area under glucose tolerance and other metaboliccurves,” Diabetes Care 17(2): 152-154 (1994).






Graduate Research Articles

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.