An approximation formula for the Katugampola integral

An approximation formula for the Katugampola integral


The objective of this paper is to present an approximation formula for the Katugampola fractional integral, that allows us to solve fractional problems with dependence on this type of fractional operator. The formula only depends on first-order derivatives, and thus we convert the fractional problem into a standard one. With some examples we show the accuracy of the method, and then we present the utility of the method by solving a fractional integral equation.

Keywords: fractional calculus, Katugampola fractional integral, numerical methods.

1 Introduction

Many real-world phenomena are described by non-integer order systems. In fact, they model several problems since they take into consideration e.g. the memory of the process, friction, flow in heterogenous porous media, viscoelasticity, etc [4, 5, 7, 12, 19]. Fractional derivative and fractional integral are generalizations of ordinary calculus, by considering derivatives of arbitrary real or complex order, and a general form for multiple integrals. Although mathematicians have wondered since the very beginning of calculus about these questions, only recently they have proven their usefulness and since then important results have appeared not only in mathematics, but also in physics, applied engineering, biology, etc. One question that is important is what type of fractional operator should be considered, since we have in hand several distinct definitions and the choice dependes on the considered problem. Because of this, we find in the literature several papers dealing with similar subjects, but for different type of fractional operators. So, to overcome this, one solution is to consider general definitions of fractional derivatives and integrals, for which the known ones are simply particular cases. We mention for example the approach using kernels (see [6, 13, 14, 15, 21]).

The paper is organized in the following way. In Section 2 we present the definitions of left and right Katugampola fractional integrals of order . Next, in Section 3, we prove the two new results of the paper, Theorems 3.1 and 3.2. The formula is simple to use, and uses only the function itself and an auxiliary family of functions, where each of them is given by a solution of a Cauchy problem. Finally, in Section 4, we present some examples where we compare the exact fractional integral of a test function with some numerical approximations as given in the previous section. To end, we exemplify how we can apply the approximation to solve a fractional integral equation with an initial condition.

2 Caputo–Katugampola fractional integral

To start, we review the main concept as presented in [8]. It generalizes the Riemann–Liouville and Hadamard fractional integrals by introducing in the definition a new parameter , that allows us to obtain them for special values of .

Definition 2.1.

Let be two nonnegative real numbers with , two positive real numbers, and be an integrable function. The left Katugampola fractional integral is defined as


and the right Katugampola fractional integral is defined as

These notions were motivated from the following relation. When is an integer, the left Katugampola fractional integral is a generalization of the -fold integrals

We also notice that, taking , we obtain the left and right Riemann–Liouville fractional integrals, and as , we get the left and right Hadamard fractional integrals (cf. [11, 16, 20]). We refer to [9], where a notion of Katugampola fractional derivative is presented, generalizing the Riemann–Liouville and Hadamard fractional derivatives of order , and to [10] where an existence and uniqueness result is proven for a fractional differential equation involving this new operator.

For example, consider the function


With the change of variables

we arrive to

where is the Beta function

Using the useful property

we get the formula

In a similar way, if we consider

we have the following

3 Caputo–Katugampola fractional integral

In this section, we present the main results of the paper. We prove an approximation formula for the Katugampola fractional integrals, which will allow us later to solve a fractional integral equation, by approximating it by an ordinary differential equation. This idea was motivated by the recent works in [1, 2, 17, 18], and has been developed in the recent book [3], where similar formula are proven for the Riemann–Liouville and Hadamard fractional operators.

Theorem 3.1.

Let and be a function of class . For , define the quantities

and the function by



Proof: Starting with formula 1, and integrating by parts choosing

we obtain


By the binomial theorem, we have


Replacing formula 3 into 2, and if we truncate the sum, we get


If we split the sum into and the remaining terms , we deduce the following

If we proceed with another integration by parts, choosing this time

we obtain

proving the desired formula. It remains to prove that




we get

which converges to zero as , ending the proof.

For the right Katugampola fractional integral, the formula is the following. We omit the proof since it is similar to the proof of Theorem 3.1.

Theorem 3.2.

Let and be a function of class . For , define the quantities

and the function by



4 Examples and applications

To test the efficiency of the purposed method, consider the test function

The expression of the fractional integral of is

Below, in Figure 1, we present the graphs of the exact expression of the fractional integral of , and some numerical approximations as given by Theorem 3.1 for different values of and .

(a) and
(b) and
(c) and
(d) and
Figure 1: Analytic vs. numerical approximations.

Now we show how the purposed approximation can be useful to solve fractional integral equations with dependence on the Katugampola fractional integral. Consider the system

The solution for this problem is the function . The numerical procedure to solve the problem is the following. Using Theorem 3.1, we approximate by the sum


and solution of the system

So, the initial fractional problem is replaced by the Cauchy problem

For different values of , we obtain different accuracies of the method. Some results are exemplified next, in Figure 2, for different values of and .

(a) and
(b) and
(c) and
(d) and
Figure 2: Analytic vs. numerical approximations.

5 Conclusion

Dealing with fractional operators is in most cases extremely difficult, and so several numerical methods are purposed to overcome these problems. In our work, we suggest a decomposition formula that depends only on the first-order derivative, and with this tool in hand we can transform the fractional problem into an integer-order one. In Figure 1 we considered different values of and , and observe that as increases, the error of the approximation decreases and the numerical approximations approaches the exact expression of the Caputo–Katugampola fractional integral, converging to it. Next, in Figure 2, we exemplify how it can be useful, by solving a fractional integral equation. For all numerical experiments presented above, we used MatLab to obtain the results.


This work was supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciência e a Tecnologia), within project UID/MAT/04106/2013.


  1. T. M. Atanacković , M. Janevb, S. Pilipovicc and D. Zoricab, Convergence analysis of a numerical scheme for two classes of non-linear fractional differential equations, Appl. Math. Comput. 243 (2014), 611–623.
  2. T. M. Atanacković and B. Stankovic, On a numerical scheme for solving differential equations of fractional order, Mech. Res. Comm. 35 (2008), 429–438.
  3. R. Almeida, S. Pooseh and D. F. M. Torres, Computational Methods in the Fractional Calculus of Variations, Imperial College Press, London, 2015.
  4. D. A. Benson, The Fractional Advection-Dispersion Equation: Development and Application. 1998. 144 p. University of Nevada, Reno. Ph.D. thesis.
  5. Y.-W. Cao, R.-Q. Huang, S.-L. Shen, Y.-S. Xu and L. Ma, Investigation of blocking effect on groundwater seepage of piles in aquifer, Yantu Lixue/Rock Soil Mech. 35 (2014), 1617–1622.
  6. A.D. Freed and K. Diethelm, Fractional Calculus in Biomechanics: A 3D Viscoelastic Model Using Regularized Fractional Derivative Kernels with Application to the Human Calcaneal Fat Pad, Biomech Model Mechanobiol. 5 (2006) 203–215.
  7. J.-H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comp Methods Appl Mech Eng 167 (1998) 57–68.
  8. U.N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput. 218 (2011), 860–865.
  9. U.N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. App. 6 (2014), 1–15.
  10. U. N. Katugampola, Existence and uniqueness result for a class of generalized fractional differential equations, submitted.
  11. A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.
  12. N. A. Malik, R. A. Ghanam and S. Al-Homidan, Sensitivity of the pressure distribution to the fractional order in the fractional diffusion equation, Canad. J. Phys. 93 (2015), 18–36.
  13. A. B. Malinowska, T. Odzijewicz and D. F. M. Torres, Advanced methods in the fractional calculus of variations, Springer Briefs in Applied Sciences and Technology, Springer, Cham, 2015.
  14. T. Odzijewicz, A. B. Malinowska and D. F. M. Torres, Generalized fractional calculus with applications to the calculus of variations, Comput. Math. Appl. 64 (2012) 3351–3366.
  15. T. Odzijewicz, A. B. Malinowska and D. F. M. Torres, Green’s theorem for generalized fractional derivatives, Fract. Calc. Appl. Anal. 16 (2013), 64–75.
  16. I. Podlubny, Fractional differential equations, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.
  17. S. Pooseh, R. Almeida and D.F.M. Torres, Expansion formulas in terms of integer-order derivatives for the Hadamard fractional integral and derivative, Numer. Funct. Anal. Optim. 33 (2012), 301–319.
  18. S. Pooseh, R. Almeida and D.F.M. Torres, Approximation of fractional integrals by means of derivatives, Comput. Math. Appl. 64 (2012), 3090–3100.
  19. F. Riewe, Mechanics with fractional derivatives, Phys. Rev. E 55 (1997), 3581–3592.
  20. S.G. Samko, A.A. Kilbas and O. I. Marichev, Fractional integrals and derivatives, translated from the 1987 Russian original, Gordon and Breach, Yverdon, 1993.
  21. H. M. Srivastava, Ž. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel, Appl. Math. Comput. 211 (2009), 198–210.
Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
Add comment
Loading ...
This is a comment super asjknd jkasnjk adsnkj
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test description