Eulers Method for Fractional Differential Equations

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1 Eulers Method for Frctionl Differentil Equtions PING TONG School of Science Wuhn University of Science Technology Wuhn 4365 CHINA YUQIANG FENG School of Science Wuhn University of Science Technology Wuhn 4365 CHINA HENGJIN LV School of Science Wuhn University of Science Technology Wuhn 4365 CHINA Abstrct: This pper presents numericl method for solving frctionl differentil equtions in the Riemnn- Liouville sense. The pproch is bsed on the Eulers method. The min chrcteristic behind the pproch is tht Euler method hs intuitive geometric mening. The lgorithm is presented the convergence of the lgorithm is proved. As pplictions of min results, three specific numericl exmples re given. Key Words: Frctionl differentil equtions, Initil vlue problem, Solution, Existence, Eulers method Introduction With the rpid development of high-tech, the frctionl clculus gets involved in more more res, especilly in control theoryviscoelstic theoryelectronic chemiclsfrctl theory so on. See reference []-[5]. The Existence uniqueness for frctionl differentil equtions hs been investigted by mny uthors (see, e.g., [6]-[8]). Finding ccurte efficient methods for solving FDEs hs been n ctive reserch undertking. In the pst few decdes, mny methods hve been developed for solving FDEs from the numericl point of view, such s the Legendre wvelet method, the spectrl method qurtered shifted Legendre method bsed on GussCLbtt. See reference [9]-[]. Eulers method hs been proven to be efficient solving ordinry differentil equtions (ODEs) other kinds of equtions. See reference [2, 3]. A question rise nturlly: cn we hve Euler method to derive numericl solution of FDEs? This pper is concerned with the numericl solution of following initil vlue problem of FDE D+ α = f(x, y) Where < α < frctionl derivtive is in Riemnn-Liouville sense. In this pper, we give the Euler method for the frctionl differentil equtions. This pper is orgnized s follows. In section 2 we introduce some definitions some relevnt properties of Riemnn-Liouville derivtive Cputo derivtive. In section 3 we present the proof of convergence of the lgorithm error nlysis of the lgorithm. In section 4 improved lgorithms re given. In section 5 we give three specific numericl exmples equipped with compring figure of numericl solution nlyticl solution. Finlly we conclude the pper with some remrks. 2 Preliminries There re gret number of definitions of frctionl integrtion frctionl derivtive (see, [4]-[7]). We will only present Riemnn-Liouville Cputo. Let f : [, b] R be function, α is positive rel number stisfying n α < n, Γ the Euler gmm function. Definition The left right Riemnn-Liouville frctionl integrtion of order α is defined by I α +f(x) = Γ(α) I α b f(x) = Γ(α) b x (x t) α f(t)dt (t x) α f(t)dt Definition 2 The left right Riemnn-Liouville frctionl derivtive of order α is defined by D+f(x) α = Γ(n α) dx n d n d n Db α f(x) = ( )n Γ(n α) dx n b x (x t) n α f(t)dt (t x) n α f(t)dt Definition 3 The left right Cputo frctionl derivtive of order α is defined by C D α +f(x) = Γ(n α) (x t) n α f (n) (t)dt E-ISSN: Issue 2, Volume 2, December 23

2 = C D α b f(x) Γ(n α) b x ( ) n (t x) n α f (n) (t)dt There exists reltion between the Riemnn- Liouville frctionl derivtive Cputo frctionl derivtive. C D+f(x) α = n D+f(x) α f (k) () Γ(k α+) (x )k α () k= Lemm 4 [8] Let α, β, φ L [, b], then I+I α β + φ = Iα+β + φ holds lmost everywhere on [, b]. If φ C[, b] or α + β, the identity holds everywhere on [, b]. Lemm 5 [8] Let α, β, φ L [, b], f = I α+β + φ, then D α +D β + f = Dα+β + f. 3 Eulers method to frctionl differentil equtions error nlysis This pper is concerned with the numericl solution of following initil vlue problem of FDE D α +y = f(x, y) (2) y() = y (3) The frctionl derivtive is in Riemnn-Liouville sense with the order < α <. By using the properties of frctionl integrtion frctionl derivtive, we cn do nlogously trnsformtion s this pper. If we pply Riemnn-Liouville frctionl derivtive of α order on (2), we get the following eqution y = D+ α f(x, y) (4) According to Euler s method, we get the following lgorithm: x n+ h y n+ y n = x + nh = H n = hd α + f(x, y n) x=xn (5) With the Mtlb softwre, the lgorithm cn be chieved in computer. And the lgorithm is proved to be efficient convergent. Before the proof, we will give some relevnt definitions Lemm. Definition 6 [9] Let f (x), f 2 (x), f n (x), be sequence of functions on intervl I. It is clled uniformly bounded if there exists constnt K > such tht f n (x) K to ll x I n N +. Definition 7 [9] Let f (x), f 2 (x),, f n (x), be sequence of functions on intervl I. It is clled equicontinuous if rbitrry ε there exists δ such tht for rbitrry x, x 2 I such tht when x x 2 < δ, f n (x ) f n (x 2 ) < ε holds for ll n. Lemm 8 [9] Let f (x), f 2 (x),, f n (x), be sequence of functions on finite closed intervl I. If it is uniformly bounded equicontinuous, there is subsequence which is uniformly continuous. Lemm 9 [2] Function y hs continuous left frctionl derivtive, then it is necessrily tht y() = Theorem Let the function f(x, y) stisfies conditions tht f(x, y(x )) = f x (x, y) is continuous on R : x x c, y y b then the FDEs (2) (3) hve t lest one solution t the intervl x x H with H = min{c, b M } M > mx f(x, y). (x,y) R D α + Proof: Divide the intervl x x H into n equl prts. We cn get n + points: x k = x + kh n, k =,, 2 n. From the initil point P (x, y ), we denote intersection point of the direction of P (x, y ) verticl line x = x s P (x, y ), line segment [P, P ] s the first Euler line. Successively we get the Euler line γ n. For ny x stisfying x x H, there exists n integer s n such tht x s < x x s+. For ech n N, let {φ n (x)} denote the sequence: s φ n (x) = y + D+ α f(x, y k) x=xk (x k+ x k ) k= + D α + f(x, y s) x=xs (x x s ) (6) Since y y b, we hve φ n (x) y b, which implies {φ n (x)} is uniformly bounded. Due to f(x, y(x )) = eqution (), the Riemnn- Liouville frctionl derivtives is equl to the Cputo frctionl derivtive, i.e., D+ α f(x, y) = Γ(α) (x t) α d dt f(t, y)dt (7) E-ISSN: Issue 2, Volume 2, December 23

3 By continuity of f x (x, y) eqution (7), the term D+ α f(x, y) is continuous, D α + f(x, y) is bounded. So we hve φ n (s) φ n (t) M(x t) Nmely, {φ n (x)} is equicontinuous. According Ascoli Lemm, we cn choose subsequence of Eulers function which is uniformly convergent t the intervl x x H. Denote the chosen subsequence: φ n, φ n2,, φ nk,. Let F (x, y) = D+ α f(x, y). We shll prove φ n (x) = y + F (x, φ n (x))dx + δ n (x) x δ n (x). Noticing tht we hve F (x i, y i )(x i+ x i ) = F (x i, y i )(x i+ x i ) = where d n (i) = i+ x i i+ x i i+ for i =,,, s. For x s < x x s+, we hve F (x s, y s )(x x s ) = d n(x) = x i F (x i, y i )dx F (x, φ n (x)))dx+d n (i) [F (x i, y i ) F (x, φ n (x))]dx x s F (x, φ n (x)))dx + d n(x) x s [F (x s, y s ) F (x, ϕ n (x)]dx. Thus the identity (6) is equivlent to where φ n (x) = y + x F (x, φ n (x))dx + δ n (x) s δ n (x) = d n (i) + d n(x) i= According to the structure of Euler s method, we hve x x i H n ϕ n (x) y i MH n for x i < x x i+. Since D+ α f(x, y) is continuous, for rbitrry ε, there exists N such tht for rbitrry x i < x x i+, Therefore, d n (i) F (x i, y i ) F (x, φ n (x)) < ε H. i+ x i F (x i, y i ) F (x, ϕ n (x) dx < ε n. And for n > N, x s < x x s+, we hve δn(x) < ε n. So when n > N, δ n(x) < sε n + ε n ε, nmely, δ n (x). Thus φ nk (x) = y + x F (x, φ nk (x))dx + δ nk (x) Since the subsequence is uniformly convergent δ n (x), let ϕ(x) be the limit of {φ nk (x)}, then ϕ(x) = y + x F (x, ϕ(x))dx for x x H. Hence the FDEs (2) (3) hve t lest one solution on [x, x + H]. Remrk In ddition, if f x (x, y) stisfies Lipschitz condition f x (x, y ) f x (x, y 2 ) L y y 2, Then the solution of FDEs (2)-(3) is unique. Lemm 2 If f x (x, y) stisfies Lipschitz condition f x (x, y ) f x (x, y 2 ) L y y 2 conditions in Theorem, then F (x, y) = f(x, y) lso stisfies Lipschitz condition for y. D α + Proof: Noting tht F (x, y ) F (x, y 2 ) = Γ(α) L Γ(α) y y 2 (x t) α (f x (x, y ) f x (x, y 2 ))dt = L (x )α y y 2, αγ(α) (x t) α dt for x x c, there exists d which stisfies (x ) α d. Hence where M = F (x, y ) F (x, y 2 ) M y y 2 Ld αγ(α). E-ISSN: Issue 2, Volume 2, December 23

4 Theorem 3 If f x (x, y) stisfies Lipschitz condition f x (x, y ) f x (x, y 2 ) L y y 2 Then y(x n ) y n = O(h). Proof: The Euler itertion formul is bsed on y n = y(x n ), Then we cn get ȳ n+ = y(x n ) + hd α + f(x, y n) x=xn (8) So we cn esily get y(x n+ ) ȳ n+ = h2 2 y (ξ) Nmely, y(x n+ ) y n+ ch 2. According to eqution (8) Euler s itertion formul, we get Hence, ȳ n+ y n+ y(x n ) y n + h (D α f(x n, y(x n )) D α f(x n, y n ) ( + hm) y(x n ) y n. y(x n+ ) y n+ y(x n+ ) ȳ n+ + ȳ n+ y n+ ( + hm) y(x n ) y n + ch 2. Therefore, we hve estimte e n+ ( + hm) y(x n ) y n + ch 2. From bove we cn get the recursion formul e n ( + hm) n e + ch M [( + hm)n ]. Since x n x = nh H, ( + hl) n (e hl ) n e HL = g At the sme time we hve e =. Consequently e n ch M (g ), nmely, y(x n) y n = O(h). Remrk 4 Theorem 3 indictes the Euler s method effective with first-order error. 4 Improved Eulers method A question rises nturlly: cn we improve the ccurcy of the lgorithm? Firstly, we recll bckwrd Euler method. It is s follows: y n+ y n = hd α + f(x, y n+) x=xn+. It is obvious tht the bckwrd Euler s lgorithm is implicit. Euler s method bckwrd Euler s method hve their own chrcteristics. Euler s method is much more convenient. But tking the numericl stbility fctors into ccount, bckwrd Euler s method is often chosen. Bckwrd Euler s equtions re usully solved by itertion. And the essence of the itertive process is grdully explicit. The specific is: y () n+ y n = h D α + f(x, y n) x=xn+ y (k+) n+ y n = h D+ α f(x, y(k) n+ ) x=x n+ k =,, 2, By clcultion, we cn get locl trunction error of the two methods y(x n+ ) y n+ = h2 2 y (ξ) y(x n+ ) y n+ = h2 2 y (ξ) It is esy to see tht we cn get higher ccurcy method by the verge of the two methods. By the verge of the two methods, we get implicit trpezoidl method y n+ y n = h 2 D α + f(x, y n) x=xn + D α + f(x, y n) x=xn+ which cn be solved by itertion formul? y () n+ y n = hd α + f(x, y n) x=xn+ y (k+) n+ y n = h 2 D α + f(x, y(k) n+ ) x=x n+ + D α + f(x, y n) x=xn Although the trpezoidl method improves the ccurcy, the lgorithm is complex. In itertive formul, itertion opertion is repeted severl times which leds gret mount of computtion difficultly to predict the results. In order to decrese the mount of computtion, we hope the lgorithm trnsferred to the E-ISSN: Issue 2, Volume 2, December 23

5 next step clcultion fter only once or twice itertion opertion. Therefore, we propose improved Euler s method x n+ = x + nh y p y n = hd α + f(x, y n) x=xn y c y n = hd α + f(x, y p) x=xn+ y n+ = 2 (y p + y c ) Next we prove the Euler method is effective with firstorder error. Theorem 5 If f x (x, y) stisfies Lipschitz condition f x (x, y ) f x (x, y 2 ) L y y 2 We cn get y(x n ) y n = O(h 2 ) for improved Euler s method. Proof: Let Euler itertion formul be bsed on y n = y(x n ). We cn get ȳ n+ = y(x n ) + h 2 D α + f(x, y n) + h 2 D α + f(x, y n + hd+ α f(x, y n) x=xn ) x=xn+ (9) Set F (x, y) = D+ α f(x, y). We cn get F (x n+, y n + hy (x n )) = F (x n+, y(x n+ )) + F y (x n+, ξ)(y n + hy (x n ) y(x n+ )) F (x n+, y(x n+ )) = () y (x n ) + hy (x n ) + h2 2 y (x n ) + h3 3! y (x n ) + () y(x n+ ) = y(x n ) + hy (x n ) + h2 2 y (x n ) + h3 3! y (x n ) + By (9)-(), we get ȳ n+ = y(x n ) + h 2 (y (x n ) + y (x n ) + hy (x n ) + h2 2 y (x n ) + h2 2 F y(x n+, ξ)y (x n )) (2) (3) Combining Eq. (2) Eq. (3), we get y(x n+ ) ȳ n+ = O(h 3 ), nmely, y(x n+ ) ȳ n+ ch 2. (4) Let Then φ = (F (x, y) + F (x + h, y + hf (x, y)). 2 φ(x, y, h) φ(x, ȳ, h) F (x, y) F (x, ȳ) 2 + F (x + h, y + hf (x, y)) F (x + h, ȳ + hf (x, ȳ)) M( + h M) y ȳ 2 M( + h M) y ȳ 2 L φ y ȳ ȳ n+ y n+ y(x n ) y n + φ(x n, y(x n ), h φ(x n, y n,, h) ( + hl φ ) y(x n ) y n. Hence, we hve Nmely, y(x n+ ) y n+ y(x n+ ) ȳ n+ + ȳ n+ y n+ ( + hl φ ) y(x n ) y n + ch 3 e n+ ( + hl φ ) e n + ch 3. Thus we get the recursion formul e n ( + hl φ ) n e + ch2 L φ [( + hl φ ) n ] By x n x = nh H, then ( + hl φ ) n (e hlφ ) n e HLφ = g φ. At the sme time we hve e =. Consequently, we get tht e n ch2 L φ (g φ ), nmely, y(x n ) y n = O(h 2 ). 5 Exmples In this section, with the help of Mtlb, we give two exmples to illustrte the convergence of both Euler method improved Euler method by comprison figure of numericl solution under different segmenttion nlyticl solution. E-ISSN: Issue 2, Volume 2, December 23

6 Figure : Figure 3: Figure 2: Exmple 6 Consider the following frctionl differentil eqution [2]: The nlyticl solution is D.5 +y = x 2, x y() = y(x) = 6x 2.5 Γ(.5) 5 For ech method, we get three sets of numericl solution when the number of division is n = 2, n = n = 2 respectively. Using the Mtlb softwre, we get Fig. for Euler s method Fig.2 for improved Euler s method. In the both figures, the yellow, blue, red blck curve re corresponding to numericl solution of n = 2,, 2 the nlyticl solution respectively. From the bove two figures, we see tht the numericl solution is closer to nlyticl solution s the number of division increse for the sme method; the numericl solution of improved Euler method is closer to nlyticl solution thn the numericl solution of Euler method. In this exmple Euler s method is enough perfect. In ctul use, with improved Euler s method more stble ccurte but Euler s method opertion on the computer is fster; we cn choose more suitble upon request method. Exmple 7 Consider the following frctionl differentil eqution [2]: The nlyticl solution is D. +y = y, x y() =. y(x) = ( Γ(.25) Γ(.25) ).25 x.25. Using the MATLAB softwre, we get Fig.3 for Euler method Fig.4 for improved Euler method. The yellow, blue, red blck curve re corresponding to numericl solution of n = 2,, 2 the nlyticl solution respectively. According to the bove two figures, the numericl solution pper lrger devition using Euler s method but not mde up by improved Euler s method when n = 2 t the beginning certin number of points. This is to sy the improved Euler s method is more ccurte. Exmple 8 Consider the following frctionl differentil eqution: D+y.5 = 5x (5Γ(.5)y ).8, x 6 y() = E-ISSN: Issue 2, Volume 2, December 23

7 Figure 4: Figure 6: The nlyticl solution is Figure 5: y(x) = 6x 2.5 Γ(.5) 5. For ech method, we get three sets of numericl solution for the number of division is n = 2, n = n = 2 respectively. Using the Mtlb softwre, we get Fig.5 for Euler method Fig.6 for improved Euler method. In the two figure, the yellow, blue, red blck curve re corresponding to numericl solution of n = 2,, 2 the nlyticl solution respectively. 6 Conclusions In this pper we derive simple numericl method, Euler s method for solving frctionl differentil equtions in the Riemnn-Liouville sense, which hs intuitive geometric mening. And the numericl solution is closer to nlyticl solution s the number of division increse. In ctul use, when more stble ccurte is needed, we considered improve the method, which brings improved Euler s method. Compred with other lgorithms, the lgorithms in this pper re esier to underst more simple to be operted on the computer. In this pper we only consider the frctionl derivtives in Riemnn- Liouville sense with the order < α <, it cn be generlized to ny other order frctionl derivtives in other sense by using the reltionship mong vrious frctionl derivtives. Acknowledgements: We sincerely cknowledge the nonymous reviewers for their comments. This reserch is supported by the Doctorl Fund of Eduction Ministry of Chin( ); The Nturl Science Foundtion of Hubei Province (23CFA3) Hubei Province Key Lbortory of Systems Science in Metllurgicl Process(z232). References: [] J. T. Mchdo, V. Kirykov, F. Minrdi, Recent history of frctionl clculus, Commun. Nonliner. Sci. Numer. Simult. 6, 2, pp [2] G. Cooper, D. Cown, The ppliction of frctionl clculus to potentil field dt, Explor. Geophys. 34, 23, pp [3] Z. E. A. Fellh, C. Depollier, M. Fellh, Appliction of frctionl clculus to the sound wves propgtion in rigid porous mterils: Vlidtion vi ultrsonic mesurements, Act. Acust. united. Ac. 88, 22, pp [4] J. Sbtier, O. P. Agrwl, J.A. Mchdo, Advnces in frctionl clculus: Theoreticl devel- E-ISSN: Issue 2, Volume 2, December 23

8 opments pplictions in physics engineering, Springer Publishing Compny 27 [5] F. Minrdi, Frctionl clculus wves in liner viscoelsticity: n introduction to mthemticl models, Imperil College Press 2. [6] R. P. Agrwl, V. Lkshmiknthm, J. J. Nieto, On the concept of solution for frctionl differentil equtions with uncertinty, Nonliner. Anly. Theor. 72, 2, pp [7] R. P. Agrwl, D. O Regn, S. Stnëk, Positive solutions for Dirichlet problems of singulr nonliner frctionl differentil equtions, J. Mth. Anl. Appl. 37, 2, pp [8] A. Ykr, M. E. Koksl, Existences Results for Solutions of Nonliner Frctionl Differentil Equtions, Abstr. Appl. Anl. 22 [9] M. U. Rehmn, R. A. Khn, The Legendre wvelet method for solving frctionl differentil equtions, Commun. Nonliner. Sci. Numer. Simult. 6, 2, pp [] E. H. Doh, A. H. Bhrwy, S. S. Ezz-Eldien, Efficient Chebyshev spectrl methods for solving multi-term frctionl orders differentil equtions, Appl. Mth. Model. 35, 2, pp [] A. H. Bhrwy, A. S. Alofi, S. S. Ezz-Eldien, A qudrture tu method for frctionl differentil equtions with vrible coefficients, Appl. Mth. Lett. 24, 2, pp [2] D. J. Highm, X. Mo, A. M. Sturt, Strong convergence of Euler-type methods for nonliner stochstic differentil equtions, SIAM.J. Numer. Anly. 4, 22, pp [3] Z. Fn, M. Liu, W. Co, Existence uniqueness of the solutions convergence of semiimplicit Euler methods for stochstic pntogrph equtions, J. Mth. Anly. Appl. 325, 27, pp [4] V. Lkshmiknthm, A. S. Vtsl, Bsic theory of frctionl differentil equtions, Nonliner. Anly. Theor. 69, 28, pp [5] S. Ds, Functionl frctionl clculus, Springer Verlg, Berlin Heidelberg 2. [6] S. G. Smko, A. A. Kilbs, O. O. I. Mrichev, Frctionl integrls derivtives, Yverdon: Gordon Brech Science Publishers 993. [7] K. Diethelm, N.J. Ford, Anlysis of frctionl differentil equtions, J. Mth. Anl. Appl. 265, 22, pp [8] T. J. Xio, J. Ling, Approximtions of Lplce trnsforms integrted semigroups, J. Func. Anl. 72, 2, pp [9] L. Nirenberg, Topics in nonliner functionl nlysis, Americn Mthemticl Society 2. [2] R. Almeid, R. A. C. Ferreir, D. F. M. Torres, Isoperimetric problems of the clculus of vritions with frctionl derivtives, Act. Mth. Sci. 32, 22, pp [2] K. Diethelm, The nlysis of frctionl differentil equtions: n ppliction-oriented exposition using differentil opertors of Cputo type, Springer Verlg 2. E-ISSN: Issue 2, Volume 2, December 23