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 converts 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

Almeida, Ricardo

Bastos, Nuno R.O.

arXiv

Repositório Institucional da Universidade de Aveiro

Journal of Mathematical AnalysisISSN: 2217-3412, URL: http://www.ilirias.comVolume 7 Issue 1 (2016), Pages 23–30.AN APPROXIMATION FORMULAFOR THE KATUGAMPOLA INTEGRALRICARDO ALMEIDA, NUNO R. O. BASTOSAbstract. The objective of this paper is to present an approximation for-mula for the Katugampola fractional integral, that allows us to solve fractionalproblems with dependence on this type of fractional operator. The formulaonly depends on first-order derivatives, and thus converts the fractional prob-lem into a standard one. With some examples, we show the accuracy of themethod, and then we present the utility of the method by solving a fractionalintegral equation.1. IntroductionMany 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 ofthe process, friction, flow in heterogenous porous media, viscoelasticity, etc [4,5,7,12,19]. Fractional derivative and fractional integral are generalizations of ordinarycalculus, by considering derivatives of arbitrary real or complex order, and a generalform for multiple integrals. Although mathematicians have wondered since the verybeginning of calculus about these questions, only recently they have proven theirusefulness and since then important results have appeared not only in mathematics,but also in physics, applied engineering, biology, etc. One question that is importantis what type of fractional operator should be considered, since we have in handseveral 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 isto consider general definitions of fractional derivatives and integrals, for which theknown ones are simply particular cases. We mention for example the approachusing kernels (see [6, 13–15,21]).The paper is organized in the following way. In Section 2 we present the def-initions of left and right Katugampola fractional integrals of order α > 0. Next,in Section 3, we prove the two new results of the paper, Theorems 3.1 and 3.2.2010 Mathematics Subject Classification. 26A33, 34A08, 34K28.Key words and phrases. Fractional calculus, Katugampola fractional integral, numericalmethods.c©2016 Ilirias Publications, Prishtine¨, Kosove¨.Submitted May 18, 2015. Revised Nov 24, 2015. Published Jan 04, 2016.The authors are supported by Portuguese funds through the CIDMA - Center for Re-search and Development in Mathematics and Applications, and the Portuguese Foundationfor Science and Technology (FCT-Fundac¸a˜o para a Cieˆncia e a Tecnologia), within projectUID/MAT/04106/2013.2324 R. ALMEIDA, N. R. O. BASTOSThe formula is simple to use, and uses only the function itself and an auxiliaryfamily of functions, where each of them is given by a solution of a Cauchy prob-lem. Finally, in Section 4, we present some examples where we compare the exactfractional integral of a test function with some numerical approximations as givenin the previous section. To end, we exemplify how we can apply the approximationto solve a fractional integral equation with an initial condition.2. Caputo–Katugampola fractional integralTo start, we review the main concept as presented in [8]. It generalizes theRiemann–Liouville and Hadamard fractional integrals by introducing in the defi-nition a new parameter ρ > 0, that allows us to obtain them for special values ofρ.Definition 2.1. Let a, b be two nonnegative real numbers with a < b, α, ρ two posi-tive real numbers, and x : [a, b]→ R be an integrable function. The left Katugampolafractional integral is defined asIα,ρa+ x(t) =ρ1−αΓ(α)∫ taτρ−1(tρ − τρ)α−1x(τ)dτ, (2.1)and the right Katugampola fractional integral is defined asIα,ρb− x(t) =ρ1−αΓ(α)∫ btτρ−1(τρ − tρ)α−1x(τ)dτ.These notions were motivated from the following relation. When α = n is aninteger, the left Katugampola fractional integral is a generalization of the n-foldintegrals ∫ taτρ−11 dτ1∫ τ1aτρ−12 dτ2 . . .∫ τn−1aτρ−1n x(τn) dτn=ρ1−n(n− 1)!∫ taτρ−1(tρ − τρ)n−1x(τ)dτ.We also notice that, taking ρ = 1, we obtain the left and right Riemann–Liouvillefractional integrals, and as ρ→ 0+, we get the left and right Hadamard fractionalintegrals (cf. [11,16,20]). We refer to [9], where a notion of Katugampola fractionalderivative is presented, generalizing the Riemann–Liouville and Hadamard frac-tional derivatives of order α ∈ (0, 1), and to [10] where an existence and uniquenessresult is proven for a fractional differential equation involving this new operator.For example, consider the functionx(t) = (tρ − aρ)v, v > −1.Then,Iα,ρa+ x(t) =ρ1−αΓ(α)∫ taτρ−1(tρ − τρ)α−1(τρ − aρ)vdτ=ρ1−αΓ(α)∫ taτρ−1(tρ − aρ)α−1[1− τρ − aρtρ − aρ]α−1(τρ − aρ)vdτ.With the change of variablesu =τρ − aρtρ − aρ ,AN APPROXIMATION FORMULA FOR THE KATUGAMPOLA INTEGRAL 25we arrive toIα,ρa+ x(t) =ρ−αΓ(α)(tρ − aρ)α+v∫ 10(1− u)α−1uvdu=ρ−αΓ(α)(tρ − aρ)α+vB(α, v + 1),where B(·, ·) is the Beta functionB(x, y) =∫ 10tx−1(1− t)y−1dt, x, y > 0.Using the useful propertyB(x, y) =Γ(x)Γ(y)Γ(x+ y),we get the formulaIα,ρa+ x(t) =ρ−αΓ(v + 1)Γ(α+ v + 1)(tρ − aρ)α+v.In a similar way, if we considery(t) = (bρ − tρ)v, v > −1,we have the followingIα,ρb− y(t) =ρ−αΓ(v + 1)Γ(α+ v + 1)(bρ − tρ)α+v.3. Caputo–Katugampola fractional integralIn this section, we present the main results of the paper. We prove an approxi-mation formula for the Katugampola fractional integrals, which will allow us laterto solve a fractional integral equation, by approximating it by an ordinary differ-ential equation. This idea was motivated by the recent works in [2, 3, 17, 18], andhas been developed in the recent book [1], where similar formula are proven for theRiemann–Liouville and Hadamard fractional operators.Theorem 3.1. Let N ∈ N and x : [a, b] → R be a function of class C1. Fork ∈ {1, . . . , N}, define the quantitiesA =ρ−αΓ(α+ 1)[1 +N∑k=1Γ(k − α)Γ(−α)k!], Bk =ρ1−αΓ(k − α)Γ(α+ 1)Γ(−α)(k − 1)! ,and the function Vk : [a, b]→ R byVk(t) =∫ taτρ−1(τρ − aρ)k−1x(τ)dτ.Then,Iα,ρa+ x(t) = A(tρ − aρ)αx(t)−N∑k=1Bk(tρ − aρ)α−kVk(t) + EN (t),withlimN→∞EN (t) = 0.26 R. ALMEIDA, N. R. O. BASTOSProof. Starting with formula (2.1), and integrating by parts choosingu˙(τ) = τρ−1(tρ − τρ)α−1 and v(τ) = x(τ),we obtainIα,ρa+ x(t) =ρ−αΓ(α+ 1)(tρ − aρ)αx(a) + ρ−αΓ(α+ 1)∫ ta(tρ − τρ)αx˙(τ)dτ. (3.1)By the binomial theorem, we have(tρ − τρ)α = ((tρ − aρ)− (τρ − aρ))α= (tρ − aρ)α(1− τρ − aρtρ − aρ)α= (tρ − aρ)α∞∑k=0Γ(k − α)Γ(−α)k!(τρ − aρtρ − aρ)k.(3.2)Replacing formula (3.2) into (3.1), and if we truncate the sum, we getIα,ρa+ x(t) =ρ−αΓ(α+ 1)(tρ − aρ)αx(a)+ρ−αΓ(α+ 1)(tρ − aρ)αN∑k=0Γ(k − α)Γ(−α)k!(tρ − aρ)k∫ ta(τρ − aρ)kx˙(τ)dτ + EN (t),withEN (t) =ρ−αΓ(α+ 1)(tρ − aρ)α∞∑k=N+1Γ(k − α)Γ(−α)k!∫ ta(τρ − aρtρ − aρ)kx˙(τ)dτ.If we split the sum into k = 0 and the remaining terms k = 1, . . . , N , we deducethe followingIα,ρa+ x(t) =ρ−αΓ(α+ 1)(tρ − aρ)αx(t)+ρ−αΓ(α+ 1)(tρ − aρ)αN∑k=1Γ(k − α)Γ(−α)k!(tρ − aρ)k∫ ta(τρ − aρ)kx˙(τ)dτ + EN (t).If we proceed with another integration by parts, choosing this timeu(τ) = (τρ − aρ)k and v˙(τ) = x˙(τ),we obtainIα,ρa+ x(t) =ρ−αΓ(α+ 1)[1 +N∑k=1Γ(k − α)Γ(−α)k!](tρ − aρ)αx(t)− ρ1−αΓ(α+ 1)N∑k=1Γ(k − α)Γ(−α)(k − 1)! (tρ − aρ)α−k∫ taτρ−1(τρ − aρ)k−1x(τ)dτ + EN (t),proving the desired formula. It remains to prove thatlimn→∞EN (t) = 0.LetM = maxτ∈[a,b]|x˙(τ)|.AN APPROXIMATION FORMULA FOR THE KATUGAMPOLA INTEGRAL 27Since (τρ − aρtρ − aρ)k≤ 1, ∀τ ∈ [a, t],and∞∑k=N+1∣∣∣∣Γ(k − α)Γ(−α)k!∣∣∣∣ ≤ ∞∑k=N+1exp(α2 + α)kα+1≤∫ ∞Nexp(α2 + α)kα+1dk =exp(α2 + α)αNα,we get|EN (t)| ≤ Mρ−αΓ(α+ 1)(tρ − aρ)α(t− a)exp(α2 + α)αNα,which converges to zero as N →∞, ending the proof. For the right Katugampola fractional integral, the formula is the following. Weomit the proof since it is similar to the proof of Theorem 3.1.Theorem 3.2. Let N ∈ N and x : [a, b] → R be a function of class C1. Fork ∈ {1, . . . , N}, define the quantitiesA =ρ−αΓ(α+ 1)[1 +N∑k=1Γ(k − α)Γ(−α)k!], Bk =ρ1−αΓ(k − α)Γ(α+ 1)Γ(−α)(k − 1)! ,and the function Wk : [a, b]→ R byWk(t) =∫ btτρ−1(bρ − τρ)k−1x(τ)dτ.Then,Iα,ρb− x(t) = A(bρ − tρ)αx(t)−N∑k=1Bk(bρ − tρ)α−kWk(t) + EN (t),withlimN→∞EN (t) = 0.4. Examples and applicationsTo test the efficiency of the purposed method, consider the test functionx(t) = t2ρ, t ∈ [0, 0.5].The expression of the fractional integral of x isIα,ρ0+ x(t) =2ρ−αΓ(α+ 3)tρ(α+2), t ∈ [0, 0.5].Below, in Figure 1, we present the graphs of the exact expression of the fractionalintegral of x, and some numerical approximations as given by Theorem 3.1 fordifferent values of α and ρ.Now we show how the purposed approximation can be useful to solve fractionalintegral equations with dependence on the Katugampola fractional integral. Con-sider the system {Iα,ρ0+ x(t) + x(t) = t2ρ + 2ρ−αΓ(α+3) tρ(α+2)x(0) = 028 R. ALMEIDA, N. R. O. BASTOS0 0.1 0.2 0.3 0.4 0.500.020.040.060.080.10.12tx(t)  AnalyticApproximate, N=5, E=1.9018e−005Approximate, N=10, E=1.3848e−005Approximate, N=15, E=1.3738e−0050.4697 0.4697 0.46970.10360.10360.10360.10360.1036(a) α = 0.1 and ρ = 2.30 0.1 0.2 0.3 0.4 0.500.0050.010.0150.020.0250.030.0350.040.0450.05tx(t)  AnalyticApproximate, N=5, E=0.00031914Approximate, N=10, E=0.00029989Approximate, N=15, E=0.000298060.4697 0.46970.04180.04180.04190.04190.04190.04190.0419(b) α = 0.9 and ρ = 0.20 0.1 0.2 0.3 0.4 0.500.0050.010.0150.020.0250.030.035t  AnalyticApproximate, N=5, E=0.006467Approximate, N=10, E=0.001253Approximate, N=15, E=0.000471350.4098 0.40980.01920.01940.01960.0198(c) α = 1.5 and ρ = 0.80 0.1 0.2 0.3 0.4 0.50123456789t  AnalyticApproximate, N=5, E=2.4693Approximate, N=10, E=0.2179Approximate, N=15, E=0.054250.4 0.4057.57.557.67.65(d) α = 2.4 and ρ = 0.1Figure 1. Analytic vs. numerical approximations.The solution for this problem is the function x(t) = t2ρ. The numerical procedureto solve the problem is the following. Using Theorem 3.1, we approximate Iα,ρ0+ x(t)by the sumIα,ρ0+ x(t) ≈ Atραx(t)−N∑k=1Bktρ(α−k)Vk(t),withA =ρ−αΓ(α+ 1)[1 +N∑k=1Γ(k − α)Γ(−α)k!], Bk =ρ1−αΓ(k − α)Γ(α+ 1)Γ(−α)(k − 1)! ,and Vk solution of the system{V˙k(t) = tρ−1tρ(k−1)x(t)Vk(0) = 0So, the initial fractional problem is replaced by the Cauchy problem(Atρα + 1)x(t)−∑Nk=1Bktρ(α−k)Vk(t) = t2ρ + 2ρ−αΓ(α+3) tρ(α+2)V˙k(t) = tρ−1tρ(k−1)x(t), k = 1, . . . , Nx(0) = 0Vk(0) = 0, k = 1, . . . , NAN APPROXIMATION FORMULA FOR THE KATUGAMPOLA INTEGRAL 29For different values of N , we obtain different accuracies of the method. Some resultsare exemplified next, in Figure 2, for different values of α and ρ.0 0.1 0.2 0.3 0.4 0.500.020.040.060.080.10.12tx(t)  AnalyticApproximate, N=5, E=1.9018e−005Approximate, N=10, E=1.3848e−005Approximate, N=15, E=1.3738e−0050.4697 0.4697 0.46970.10360.10360.10360.1036(a) α = 3.5 and ρ = 1.50 0.1 0.2 0.3 0.4 0.500.0050.010.0150.020.0250.030.0350.040.0450.05tx(t)  AnalyticApproximate, N=5, E=0.00031914Approximate, N=10, E=0.00029989Approximate, N=15, E=0.000298060.4697 0.46970.04180.04190.042(b) α = 1.8 and ρ = 2.10 0.1 0.2 0.3 0.4 0.500.010.020.030.040.050.060.07tx(t)  AnalyticApproximate, N=5, E=3.1399e−006Approximate, N=10, E=2.5497e−006Approximate, N=15, E=2.5326e−0060.46970.46970.46970.05660.05660.0566(c) α = 3.3 and ρ = 1.90 0.1 0.2 0.3 0.4 0.500.050.10.150.20.25tx(t)  AnalyticApproximate, N=5, E=0.0793Approximate, N=10, E=0.071603Approximate, N=15, E=0.0698990.468 0.47 0.4720.220.230.24(d) α = 1.4 and ρ = 1Figure 2. Analytic vs. numerical approximations.5. ConclusionDealing with fractional operators is in most cases extremely difficult, and so sev-eral numerical methods are purposed to overcome these problems. In our work, wesuggest 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 α and ρ, and observe thatas N increases, the error of the approximation decreases and the numerical approx-imations approaches the exact expression of the Caputo–Katugampola fractionalintegral, converging to it. Next, in Figure 2, we exemplify how it can be useful,by solving a fractional integral equation. For all numerical experiments presentedabove, we used MatLab to obtain the results.References1. R. Almeida, S. Pooseh and D. F. M. Torres, Computational Methods in the Fractional Calculusof Variations, Imperial College Press, London, 2015.30 R. ALMEIDA, N. R. O. BASTOS2. T. M. Atanackovic´ , M. Janevb, S. Pilipovicc and D. Zoricab, Convergence analysis of anumerical scheme for two classes of non-linear fractional differential equations, Appl. Math.Comput. 243 (2014), 611–623.3. T. M. Atanackovic´ and B. Stankovic, On a numerical scheme for solving differential equationsof fractional order, Mech. Res. Comm. 35 (2008), 429–438.4. 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 ongroundwater 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 ModelUsing Regularized Fractional Derivative Kernels with Application to the Human CalcanealFat Pad, Biomech Model Mechanobiol. 5 (2006) 203–215.7. J.-H. He, Approximate analytical solution for seepage flow with fractional derivatives in porousmedia, 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 fractionaldifferential equations, preprint, arXiv:1411.5229 [math.CA].11. A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differen-tial 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 tothe fractional order α 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 fractionalcalculus 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 withapplications 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 frac-tional 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-orderderivatives 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 ofderivatives, 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, translatedfrom the 1987 Russian original, Gordon and Breach, Yverdon, 1993.21. H. M. Srivastava, Zˇ. Tomovski, Fractional calculus with an integral operator containing ageneralized Mittag-Leffler function in the kernel, Appl. Math. Comput. 211 (2009), 198–210.Ricardo AlmeidaCenter for Research and Development in Mathematics and Applications (CIDMA), De-partment of Mathematics, University of Aveiro, 3810–193 Aveiro, PortugalE-mail address: ricardo.almeida@ua.ptNuno R. O. BastosCenter for Research and Development in Mathematics and Applications (CIDMA), De-partment of Mathematics, University of Aveiro, 3810–193 Aveiro, PortugalCentre for the Study of Education, Technologies and Health (CI&DETS), Departmentof Mathematics, School of Technology and Management of Viseu, Polytechnic Insti-tute of Viseu, 3504–510 Viseu, PortugalE-mail address: nbastos@estv.ipv.pt

An approximation formula for the Katugampola integral

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An approximation formula for the Katugampola integral

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