An approximation formula for the Katugampola integral
aa r X i v : . [ m a t h . G M ] N ov An approximation formula for the Katugampola integral
Ricardo Almeida [email protected] Nuno R. O. Bastos , [email protected] Center for Research and Development in Mathematics and Applications(CIDMA)Department of Mathematics, University of Aveiro, 3810–193 Aveiro, Portugal Department of Mathematics, School of Technology and Management of ViseuPolytechnic Institute of Viseu, 3504–510 Viseu, Portugal
Keywords: fractional calculus, Katugampola fractional integral, numerical methods.
Abstract
The objective of this paper is to present an approximation formula for the Katugampolafractional integral, that allows us to solve fractional problems with dependence on this type offractional operator. The formula only depends on first-order derivatives, and thus we convertthe fractional problem into a standard one. With some examples we show the accuracy ofthe method, and then we present the utility of the method by solving a fractional integralequation.
Many real-world phenomena are described by non-integer order systems. In fact, they modelseveral 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 andfractional integral are generalizations of ordinary calculus, by considering derivatives of arbitraryreal or complex order, and a general form for multiple integrals. Although mathematicians havewondered since the very beginning of calculus about these questions, only recently they haveproven 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 typeof fractional operator should be considered, since we have in hand several distinct definitions andthe choice dependes on the considered problem. Because of this, we find in the literature severalpapers dealing with similar subjects, but for different type of fractional operators. So, to overcomethis, one solution is to consider general definitions of fractional derivatives and integrals, for whichthe 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 leftand right Katugampola fractional integrals of order α >
0. Next, in Section 3, we prove the twonew results of the paper, Theorems 3.1 and 3.2. The formula is simple to use, and uses only thefunction itself and an auxiliary family of functions, where each of them is given by a solution ofa Cauchy problem. Finally, in Section 4, we present some examples where we compare the exactfractional integral of a test function with some numerical approximations as given in the previoussection. To end, we exemplify how we can apply the approximation to solve a fractional integralequation with an initial condition. 1
Caputo–Katugampola fractional integral
To start, we review the main concept as presented in [8]. It generalizes the Riemann–Liouvilleand Hadamard fractional integrals by introducing in the definition a new parameter ρ >
0, thatallows us to obtain them for special values of ρ . Definition 2.1.
Let a, b be two nonnegative real numbers with a < b , α, ρ two positive real num-bers, and x : [ a, b ] → R be an integrable function. The left Katugampola fractional integral isdefined as I α,ρa + x ( t ) = ρ − α Γ( α ) Z ta τ ρ − ( t ρ − τ ρ ) α − x ( τ ) dτ, (1) and the right Katugampola fractional integral is defined as I α,ρb − x ( t ) = ρ − α Γ( α ) Z bt τ ρ − ( τ ρ − t ρ ) α − x ( τ ) dτ. These notions were motivated from the following relation. When α = n is an integer, the leftKatugampola fractional integral is a generalization of the n -fold integrals Z ta τ ρ − dτ Z τ a τ ρ − dτ . . . Z τ n − a τ ρ − n x ( τ n ) dτ n = ρ − n ( n − Z ta τ ρ − ( t ρ − τ ρ ) n − x ( τ ) dτ. We also notice that, taking ρ = 1, we obtain the left and right Riemann–Liouville fractionalintegrals, and as ρ → + , 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 theRiemann–Liouville and Hadamard fractional derivatives of order α ∈ (0 , x ( t ) = ( t ρ − a ρ ) v , v > − . Then, I α,ρa + x ( t ) = ρ − α Γ( α ) Z ta τ ρ − ( t ρ − τ ρ ) α − ( τ ρ − a ρ ) v dτ = ρ − α Γ( α ) Z ta τ ρ − ( t ρ − a ρ ) α − (cid:20) − τ ρ − a ρ t ρ − a ρ (cid:21) α − ( τ ρ − a ρ ) v dτ. With the change of variables u = τ ρ − a ρ t ρ − a ρ , we arrive to I α,ρa + x ( t ) = ρ − α Γ( α ) ( t ρ − a ρ ) α + v Z (1 − u ) α − u v du = ρ − α Γ( α ) ( t ρ − a ρ ) α + v B ( α, v + 1) , where B ( · , · ) is the Beta function B ( x, y ) = Z t x − (1 − t ) y − dt, x, y > . Using the useful property B ( x, y ) = Γ( x )Γ( y )Γ( x + y ) ,
2e get the formula I α,ρa + x ( t ) = ρ − α Γ( v + 1)Γ( α + v + 1) ( t ρ − a ρ ) α + v . In a similar way, if we consider y ( t ) = ( b ρ − t ρ ) v , v > − , we have the following I α,ρb − y ( t ) = ρ − α Γ( v + 1)Γ( α + v + 1) ( b ρ − t ρ ) α + v . In this section, we present the main results of the paper. We prove an approximation formulafor the Katugampola fractional integrals, which will allow us later to solve a fractional integralequation, by approximating it by an ordinary differential equation. This idea was motivated bythe recent works in [1, 2, 17, 18], and has been developed in the recent book [3], where similarformula are proven for the Riemann–Liouville and Hadamard fractional operators.
Theorem 3.1.
Let N ∈ N and x : [ a, b ] → R be a function of class C . For k ∈ { , . . . , N } ,define the quantities A = ρ − α Γ( α + 1) " N X k =1 Γ( k − α )Γ( − α ) k ! , B k = ρ − α Γ( k − α )Γ( α + 1)Γ( − α )( k − , and the function V k : [ a, b ] → R by V k ( t ) = Z ta τ ρ − ( τ ρ − a ρ ) k − x ( τ ) dτ. Then, I α,ρa + x ( t ) = A ( t ρ − a ρ ) α x ( t ) − N X k =1 B k ( t ρ − a ρ ) α − k V k ( t ) + E N ( t ) , with lim N →∞ E N ( t ) = 0 . Proof:
Starting with formula 1, and integrating by parts choosing˙ u ( τ ) = τ ρ − ( t ρ − τ ρ ) α − and v ( τ ) = x ( τ ) , we obtain I α,ρa + x ( t ) = ρ − α Γ( α + 1) ( t ρ − a ρ ) α x ( a ) + ρ − α Γ( α + 1) Z ta ( t ρ − τ ρ ) α ˙ x ( τ ) dτ. (2)By the binomial theorem, we have( t ρ − τ ρ ) α = (( t ρ − a ρ ) − ( τ ρ − a ρ )) α = ( t ρ − a ρ ) α (cid:18) − τ ρ − a ρ t ρ − a ρ (cid:19) α = ( t ρ − a ρ ) α ∞ X k =0 Γ( k − α )Γ( − α ) k ! (cid:18) τ ρ − a ρ t ρ − a ρ (cid:19) k . (3)Replacing formula 3 into 2, and if we truncate the sum, we get I α,ρa + x ( t ) = ρ − α Γ( α + 1) ( t ρ − a ρ ) α x ( a )3 ρ − α Γ( α + 1) ( t ρ − a ρ ) α N X k =0 Γ( k − α )Γ( − α ) k !( t ρ − a ρ ) k Z ta ( τ ρ − a ρ ) k ˙ x ( τ ) dτ + E N ( t ) , with E N ( t ) = ρ − α Γ( α + 1) ( t ρ − a ρ ) α ∞ X k = N +1 Γ( k − α )Γ( − α ) k ! Z ta (cid:18) τ ρ − a ρ t ρ − a ρ (cid:19) k ˙ x ( τ ) dτ. If we split the sum into k = 0 and the remaining terms k = 1 , . . . , N , we deduce the following I α,ρa + x ( t ) = ρ − α Γ( α + 1) ( t ρ − a ρ ) α x ( t )+ ρ − α Γ( α + 1) ( t ρ − a ρ ) α N X k =1 Γ( k − α )Γ( − α ) k !( t ρ − a ρ ) k Z ta ( τ ρ − a ρ ) k ˙ x ( τ ) dτ + E N ( t ) . If we proceed with another integration by parts, choosing this time u ( τ ) = ( τ ρ − a ρ ) k and ˙ v ( τ ) = ˙ x ( τ ) , we obtain I α,ρa + x ( t ) = ρ − α Γ( α + 1) " N X k =1 Γ( k − α )Γ( − α ) k ! ( t ρ − a ρ ) α x ( t ) − ρ − α Γ( α + 1) N X k =1 Γ( k − α )Γ( − α )( k − t ρ − a ρ ) α − k Z ta τ ρ − ( τ ρ − a ρ ) k − x ( τ ) dτ + E N ( t ) , proving the desired formula. It remains to prove thatlim n →∞ E N ( t ) = 0 . Let M = max τ ∈ [ a,b ] | ˙ x ( τ ) | . Since (cid:18) τ ρ − a ρ t ρ − a ρ (cid:19) k ≤ , ∀ τ ∈ [ a, t ] , and ∞ X k = N +1 (cid:12)(cid:12)(cid:12)(cid:12) Γ( k − α )Γ( − α ) k ! (cid:12)(cid:12)(cid:12)(cid:12) ≤ ∞ X k = N +1 exp( α + α ) k α +1 ≤ Z ∞ N exp( α + α ) k α +1 dk = exp( α + α ) αN α , we get | E N ( t ) | ≤ M ρ − α Γ( α + 1) ( t ρ − a ρ ) α ( t − a ) exp( α + α ) αN α , which converges to zero as N → ∞ , ending the proof.For the right Katugampola fractional integral, the formula is the following. We omit the proofsince 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 C . For k ∈ { , . . . , N } ,define the quantities A = ρ − α Γ( α + 1) " N X k =1 Γ( k − α )Γ( − α ) k ! , B k = ρ − α Γ( k − α )Γ( α + 1)Γ( − α )( k − , nd the function W k : [ a, b ] → R by W k ( t ) = Z bt τ ρ − ( b ρ − τ ρ ) k − x ( τ ) dτ. Then, I α,ρb − x ( t ) = A ( b ρ − t ρ ) α x ( t ) − N X k =1 B k ( b ρ − t ρ ) α − k W k ( t ) + E N ( t ) , with lim N →∞ E N ( t ) = 0 . To test the efficiency of the purposed method, consider the test function x ( t ) = t ρ , t ∈ [0 , . . The expression of the fractional integral of x is I α,ρ x ( t ) = 2 ρ − α Γ( α + 3) t ρ ( α +2) , t ∈ [0 , . . Below, in Figure 1, we present the graphs of the exact expression of the fractional integral of x , and some numerical approximations as given by Theorem 3.1 for different values of α and ρ . x ( t ) AnalyticApproximate, N=5, E=1.9018e−005Approximate, N=10, E=1.3848e−005Approximate, N=15, E=1.3738e−0050.46970.46970.46970.10360.10360.10360.10360.1036 (a) α = 0 . ρ = 2 . x ( 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 . ρ = 0 . (c) α = 1 . ρ = 0 . (d) α = 2 . ρ = 0 . Figure 1: Analytic vs. numerical approximations.5ow we show how the purposed approximation can be useful to solve fractional integral equa-tions with dependence on the Katugampola fractional integral. Consider the system ( I α,ρ x ( t ) + x ( t ) = t ρ + ρ − α Γ( α +3) t ρ ( α +2) x (0) = 0The solution for this problem is the function x ( t ) = t ρ . The numerical procedure to solve theproblem is the following. Using Theorem 3.1, we approximate I α,ρ x ( t ) by the sum I α,ρ x ( t ) ≈ At ρα x ( t ) − N X k =1 B k t ρ ( α − k ) V k ( t ) , with A = ρ − α Γ( α + 1) " N X k =1 Γ( k − α )Γ( − α ) k ! , B k = ρ − α Γ( k − α )Γ( α + 1)Γ( − α )( k − , and V k solution of the system (cid:26) ˙ V k ( t ) = t ρ − t ρ ( k − x ( t ) V k (0) = 0So, the initial fractional problem is replaced by the Cauchy problem ( At ρα + 1) x ( t ) − P Nk =1 B k t ρ ( α − k ) V k ( t ) = t ρ + ρ − α Γ( α +3) t ρ ( α +2) ˙ V k ( t ) = t ρ − t ρ ( k − x ( t ) , k = 1 , . . . , Nx (0) = 0 V k (0) = 0 , k = 1 , . . . , N For different values of N , we obtain different accuracies of the method. Some results are exemplifiednext, in Figure 2, for different values of α and ρ . Dealing with fractional operators is in most cases extremely difficult, and so several numericalmethods are purposed to overcome these problems. In our work, we suggest a decompositionformula that depends only on the first-order derivative, and with this tool in hand we can transformthe fractional problem into an integer-order one. In Figure 1 we considered different values of α and ρ , and observe that as N increases, the error of the approximation decreases and the numericalapproximations 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 fractionalintegral equation. For all numerical experiments presented above, we used MatLab to obtain theresults. Acknowledgments
This work was supported by Portuguese funds through the CIDMA - Center for Research andDevelopment in Mathematics and Applications, and the Portuguese Foundation for Science andTechnology (FCT-Funda¸c˜ao para a Ciˆencia e a Tecnologia), within project UID/MAT/04106/2013.
References [1] T. M. Atanackovi´c , M. Janevb, S. Pilipovicc and D. Zoricab, Convergence analysis of anumerical scheme for two classes of non-linear fractional differential equations, Appl. Math.Comput. (2014), 611–623. 6 x ( 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 . ρ = 1 . x ( 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 . ρ = 2 . x ( 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 . ρ = 1 . x ( 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 . ρ = 1 Figure 2: Analytic vs. numerical approximations.[2] T. M. Atanackovi´c and B. Stankovic, On a numerical scheme for solving differential equationsof fractional order, Mech. Res. Comm. (2008), 429–438.[3] R. Almeida, S. Pooseh and D. F. M. Torres, Computational Methods in the Fractional Cal-culus 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 ongroundwater seepage of piles in aquifer, Yantu Lixue/Rock Soil Mech. (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. (2006) 203–215.[7] J.-H. He, Approximate analytical solution for seepage flow with fractional derivatives inporous media, Comp Methods Appl Mech Eng (1998) 57–68.[8] U.N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput. (2011), 860–865.[9] U.N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal.App. (2014), 1–15.[10] U. N. Katugampola, Existence and uniqueness result for a class of generalized fractionaldifferential equations, submitted. 711] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differ-ential Equations. North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amster-dam, 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. (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. (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. (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. (2012), 301–319.[18] S. Pooseh, R. Almeida and D.F.M. Torres, Approximation of fractional integrals by means ofderivatives, Comput. Math. Appl. (2012), 3090–3100.[19] F. Riewe, Mechanics with fractional derivatives, Phys. Rev. E (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, ˇZ. Tomovski, Fractional calculus with an integral operator containing ageneralized Mittag-Leffler function in the kernel, Appl. Math. Comput.211