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Approximate calculation of the Caputo-type fractional derivative from inaccurate data. Dynamical approach

  • Platon G. Surkov
Veröffentlicht/Copyright: 23. Juni 2021
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Fractional Calculus and Applied Analysis
Aus der Zeitschrift Fractional Calculus and Applied Analysis Band 24 Heft 3

Abstract

A specific formulation of the “classical” problem of mathematical analysis is considered. This is the problem of calculating the derivative of a function. The purpose of this work is to construct an algorithm for the approximate calculation of the Caputo-type fractional derivative based on the methods of control theory. The input data of the algorithm is represented by inaccurate measured function values at discrete, frequently enough, times. The proposed algorithm is based on two aspects: a local modification of the Tikhonov regularization method from the theory of ill-posed problems and the Krasovskii extremal shift method from the guaranteed control theory, both of which ensure the stability to informational noises and computational errors. Numerical experiments were carried out to illustrate the operation of the algorithm.

MSC 2010: Primary 34A08; 65D25; Secondary 26A33; 49N45; 93C40
Key Words and Phrases: Caputo-type fractional derivative; reconstruction; incomplete information; error estimate; Mittag-Leffler type functions
Appendix A

Appendix A Appendices

A.1. Monotonicity of functions fi(t). Let us investigate the following functions for monotonicity:

fk(t)=(t−τkh)γ−(τih−τkh)γ−(t−τk+1h)γ+(τih−τk+1h)γ,    t∈δih,                                      k=0,…,i−1,    i∈Nh.

At the left point of the interval δih the functions have the values fk(τih)=0 . The derivatives of these functions satisfy the inequalities

f′k(t)=γ(t−τk+1)1−γ−(t−τk)1−γ(t−τk)1−γ(t−τk+1)1−γ<0,    t∈δih,     k=0,…,i−1,    i∈Nh.

Therefore, the functions fk(t) are strictly monotone and reach the smallest values, which are the largest in absolute values, as t→τi+1h . Let us calculate these values

|fk(τi+1h)|=|(τi+1h−τkh)γ−(τih−τkh)γ−(τi+1h−τk+1h)γ+(τih−τk+1h)γ|=|(i+1)δ−kδ)γ−(iδ−kδ)γ−((i+1)δ−(k+1)δ)γ +(iδ−(k+1)δ)γ|=δγ|(i−k−1)γ−2(i−k)γ+(i−k+1)γ|=δγcik,

where cik are defined by formulas (4.12).

A.2. Estimate of sum ∑k=0i−1cik. . The considered sum consists of positive terms (see (4.12)), therefore, the following relations are valid

∑k=0i−1cik=∑k=0i−1|(i−k−1)γ−2(i−k)γ+(i−k+1)γ|=∑m=1i|(m−1)γ−2mγ+(m+1)γ|≤∑m=1∞|(m−1)γ−2mγ+(m+1)γ|=:Σ.

Since the common term of the series Σ can be represented as |(m+1)γ–mγ–(mγ–(m –1)γ)|, we consider the function

g(x)=(x+1)γ−xγ,    x>0,

Examining the monotonicity of g(x) we have

g′(x)=γ((x+1)γx+1−xγx)=γxγx+1((1+1x)γ−(1+1x)).

Since 1 + 1/x >1 for x > 0, then for γ ϵ (0,1) the inequality

(1+1x)γ<(1+1x)

holds. Consequently, g'(x) < 0, which means g(x) decreases monotonically, and g(m + 1) – g(m) < 0. Then the expression for the series Σ can be rewritten without the modulus sign in the form

Σ=∑m=1∞(2mγ−(m+1)γ−(m−1)γ).

Now, we write the elements of this series

a1=2−2γ,a2=2⋅2γ−3γ−1,a3=2⋅3γ−4γ−2γ,a4=2⋅4γ−5γ−3γ,a5=2⋅5γ−6γ−4γ,…

then we obtain a formula for partial sums:

Sm=1+mγ−(m+1)γ.

Transforming the expression for Sm and using the equivalence of infinitesimal functions, we calculate the limit

Σ=limm→∞Sm=1+limm→∞(1+−1m+1)γ−1(m+1)−γ=1+limm→∞−γ(m+1)1−γ=1,

this implies

∑k=0i−1cik≤Σ=1.

References

[1] V.V. Arestov, Approximation of unbounded operators by bounded operators and related extremal problems. Russ. Math. Surv. 51, No 6 (1996), 1093-1126.10.1070/RM1996v051n06ABEH003001Suche in Google Scholar

[2] V.V. Arestov, Best uniform approximation of the differentiation operator by operators bounded in the space. Tr. Inst. Mat. Mekh. UrO RAN 24, No 4 (2018), 34-56.10.1134/S0081543820020029Suche in Google Scholar

[3] Y. Bar-Shalom, X.R. Li, Estimation and Tracking: Principles, Techniques, and Software. Boston, Artech House (1993).Suche in Google Scholar

[4] H. Bateman (ds. A. Erdèlyi et al.), Higher Transcendental Functions. Vol. III. McGraw-Hill Book Co., Inc., New York (1955).Suche in Google Scholar

[5] E.E. Berdysheva, M.A. Filatova, On the best approximation of the infinitesimal generator of a contraction semigroup in a Hilbert space. Ural Math. J. 3, No 2 (2017), 40-45.10.15826/umj.2017.2.006Suche in Google Scholar

[6] M.S. Blizorukova, V.I. Maksimov, L. Pandolfi, Dynamic input reconstruction for a nonlinear time-delay system.Autom. Remote Control 63 (2002), 171-180.10.1023/A:1014233405265Suche in Google Scholar

[7] BourdinL. Cauchy-Lipschitz theory for fractional multi-order dynamics: State-transition matrices, Duhamel formulas and duality theorems. Differ. Int. Equ. 31, No 7/8 (2018), 559-594.Suche in Google Scholar

[8] Yu.G. Bulychev, I.V. Burlaj, Repeated differentiation of finite functions with the use of the sampling theorem in problems of control, estimation, and identification. Autom. Remote Control 57, No 4 (1996), 499-509.Suche in Google Scholar

[9] S. Butera, M. Di Paola, A physically based connection between fractional calculus and fractal geometry. Ann. Phys. 350 (2014), 146-158.10.1016/j.aop.2014.07.008Suche in Google Scholar

[10] M. Caputo, F. Mainardi, A new dissipation model based on memory mechanism. Pure Appl. Geophys. 91 (1971), 134-147.10.1007/BF00879562Suche in Google Scholar

[11] R. Chartrand, Numerical differentiation of noisy, nonsmooth data. ISRN Applied Mathematics 2011 (2011), # 164564.10.5402/2011/164564Suche in Google Scholar

[12] Y. Chen, Y. Wei, D. Liu, D. Boutat, X. Chen, Variable-order fractional numerical differentiation for noisy signals by wavelet denoising. J. Comput. Phys. 311 (2016), 338-347.10.1016/j.jcp.2016.02.013Suche in Google Scholar

[13] K. Diethelm, The Analysis of Fractional Differential Equations. Springer-Verlag, Berlin (2010).10.1007/978-3-642-14574-2Suche in Google Scholar

[14] A.M.A. El-Sayed, A.E.M. El-Mesiry, H.A.A. El-Saka, On the fractional-order logistic equation. Appl. Math. Lett. 20, No 7 (2007), 817-823.10.1016/j.aml.2006.08.013Suche in Google Scholar

[15] S. Elaydi, An Introduction to Difference Equations. Springer-Verlag, New York (2005).Suche in Google Scholar

[16] Y. Ferdi, J.P. Herbeuval, A. Charef, B. Boucheham, R wave detection using fractional digital differentiation. ITBM-RBM 24, No 5-6 (2003), 273-280.10.1016/j.rbmret.2003.08.002Suche in Google Scholar

[17] A.N. Gerasimov, A generalization of linear laws of deformation and its application to the problems of internal friction. Prikl.Mat. i Mekhanika (PMM) 12, No 3 (1948), 251-260 (in Russian).Suche in Google Scholar

[18] M.I. Gomoyunov, Fractional derivatives of convex Lyapunov functions and control problems in fractional order systems. Fract. Calc. Appl. Anal. 21, No 5 (2018), 1238-1261; DOI: 10.1515/fca-2018-0066; https://www.degruyter.com/journal/key/FCA/21/5/html.Suche in Google Scholar

[19] V.A. Gordin, A.V. Khalyavin, Projection methods for error suppression in the meteorological fields before calculation of derivatives. Russ. Meteorol. Hydrol. 32, No 10 (2007), 643-650.10.3103/S1068373907100056Suche in Google Scholar

[20] R. Gorenflo, A.A. Kilbas, F. Mainardi, S.V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications. Springer-Verlag, Berlin (2014), 2nd Ed. (2020).10.1007/978-3-662-61550-8Suche in Google Scholar

[21] C.W. Groetsch, Differentiation of approximately specified functions. Amer. Math. Mon. 98, No 9 (1991), 847-850.10.1080/00029890.1991.12000802Suche in Google Scholar

[22] M. Hanke, O. Scherzer, Inverse problems light: numerical differentiation. Am. Math. Mon. 108, No 6 (2001), 512-521.10.1080/00029890.2001.11919778Suche in Google Scholar

[23] V.K. Ivanov, V.V. Vasin, V.P. Tanana, Theory of Linear Ill-Posed Problems and its Applications. VSP, Utrecht (2002).10.1515/9783110944822Suche in Google Scholar

[24] S.I. Kabanikhin, Inverse and Ill-Posed Problems: Theory and Application. De Gruyter, Berlin (2011).10.1515/9783110224016Suche in Google Scholar

[25] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier Science, New York 2006.Suche in Google Scholar

[26] N.N. Krasovskii, A.I. Subbotin, Game-Theoretical Control Problems. Springer Verlag, New York-Berlin (1988).10.1007/978-1-4612-3716-7Suche in Google Scholar

[27] A.V. Kryazhimskii, Yu.S. Osipov, Best approximation of the differen-tiation operator in the class of nonanticipatory operators.Math. Notes 37 (1985), 109-114.10.1007/BF01156754Suche in Google Scholar

[28] S.N. Kudryavtsev, Recovering a function with its derivatives from function values at a given number of points. Izv. Math. 45, NO 3 (1995), 505-528.10.1070/IM1995v045n03ABEH001666Suche in Google Scholar

[29] Y. Li, Y. Chen, I. Podlubny, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Comput. Math. Appl. 59 (2010), 1810-1821.10.1016/j.camwa.2009.08.019Suche in Google Scholar

[30] Y. Li, C. Pan, X. Meng, Y. Ding, H. Chen, A method of approximate fractional order differentiation with noise immunity. Chemom. Intell. Lab. Syst. 144 (2015), 31-38.10.1016/j.chemolab.2015.03.009Suche in Google Scholar

[31] V.I. Maksimov, Dynamical Inverse Problems of Distributed Systems. VSP, Zeist (2002).10.1515/9783110944839Suche in Google Scholar

[32] V.I. Maksimov, Method of controlled models in the problem of re-constructing a boundary input. Proc. Steklov Inst. Math. 262 (2008), 170-178.10.1134/S0081543808030139Suche in Google Scholar

[33] V. I. Maksimov, Calculation of the derivative of an inaccurately de-fined function by means of feedback laws. Proc. Steklov Inst. Math. 291 (2015), 219-231.10.1134/S0081543815080179Suche in Google Scholar

[34] V.I. Maksimov, N.A. Fedina, The method of controlled models in the problem of reconstruction of a nonlinear delay system. Diff. Eqns. 43 (2007), 37-42.10.1134/S0012266107010065Suche in Google Scholar

[35] V.I. Maksimov, L. Pandolfi, The reconstruction of unbounded controls in non-linear dynamical systems. J. Appl. Math. Mech. 65, No 3 (2001), 371-376.10.1016/S0021-8928(01)00042-9Suche in Google Scholar

[36] E.L. Melanson, J.P. Ingebrigtsen, A. Bergouignan, K. Ohkawara, W.M. Kohrt, J.R. Lighton, A new approach for flow-through respirometry measurements in humans. Amer. J. Physiol. Regul. Integr. Comp. Phys-iol. 298, No 6 (2010), 1571-1579.10.1152/ajpregu.00055.2010Suche in Google Scholar PubMed PubMed Central

[37] L. Melnikova, V. Rozenberg, One dynamical input reconstruction problem: Tuning of solving algorithm via numerical experiments. AIMS Math. 4, No 3 (2019), 699-713.10.3934/math.2019.3.699Suche in Google Scholar

[38] R.R. Nigmatullin, A fractional integral and its physical interpretation. Theoret. Math. Phys. 90, No 3 (1992), 242-251.10.1007/BF01036529Suche in Google Scholar

[39] J.P. Norton, An Introduction to Identification. Academic Press, Lon-don (1986).Suche in Google Scholar

[40] O.G. Novozhenova, Life and science of Alexey Gerasimov, one of the pioneers of fractional calculus in Soviet Union. Fract. Calc. Appl. Anal. 20, No 3 (2017), 790-809; DOI: 10.1515/fca-2018-006610.1515/fca-2017-0040 https://www.degruyter.com/journal/key/FCA/20/3/html.Suche in Google Scholar

[41] Yu.S. Osipov, A.V. Kryazhimskii, Inverse Problems for Ordinary Differential Equations: Dynamical Solutions. Gordon and Breach, Basel (1995).Suche in Google Scholar

[42] I. Podlubny, Fractional Differential Equations. Academic Press, SanDiego (1998).Suche in Google Scholar

[43] I. Podlubny, Geometrical and physical interpretation of fractional integration and fractional differentiation. Fract. Calc. Appl. Anal. 5, No 4 (2002), 367-386.Suche in Google Scholar

[44] H. Pollard, The completely monotonic character of the Mittag-Leffler function Ea(-x). Bull. Amer. Math. Soc. 54, No 12 (1948), 1115-1116.10.1090/S0002-9904-1948-09132-7Suche in Google Scholar

[45] V.G. Romanov, Inυestigation Methods for Inυerse Problems. DeGruyter, Berlin (2014).Suche in Google Scholar

[46] B. Ross, S.G. Samko, E.R. Love, Functions that have no first order derivative might have fractional derivatives of all orders less then one. Real Anal. Exchange 20, No 2 (1994), 140-157.10.2307/44152475Suche in Google Scholar

[47] R.S. Rutman, On the paper by R.R. Nigmatullin “Fractional integral and its physical interpretation”. Theoret. Math. Phys. 100, No 3 (1994),1154-1156.10.1007/BF01018580Suche in Google Scholar

[48] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yveron (1993).Suche in Google Scholar

[49] C. Schäfer, M.G. Rosenblum, H.-H. Abel, J. Kurths, Synchronization in the human cardiorespiratory system. Phys. Rev. E 60, No 1 (1999), 857-870.10.1103/PhysRevE.60.857Suche in Google Scholar PubMed

[50] G. Schmeisser, Numerical differentiation inspired by a formula of R.P. Boas. J. Approx. Theory 160 (2009), 202-222.10.1016/j.jat.2008.04.005Suche in Google Scholar

[51] L. Schumaker, Spline Functions: Basic Theory. Cambridge University Press, Cambridge (2007).10.1137/1.9781611973907Suche in Google Scholar

[52] G.G. Skorik, On the best error estimate for the method of averaging kernels in the problem of the differentiation of a noisy function. Russian Math. (Iz. VUZ) 48, No 3 (2004), 70-74.Suche in Google Scholar

[53] A.A. Stanislavsky, Probability interpretation of the integral of fractional order. Theoret. Math. Phys. 138 (2004), 418-431.10.1023/B:TAMP.0000018457.70786.36Suche in Google Scholar

[54] S.B. Stechkin, Best approximation of linear operators. Math. Notes 1, No 2 (1967), 91-99.10.1007/BF01268056Suche in Google Scholar

[55] P.G. Surkov, Application of the residual method in the right hand side reconstruction problem for a system of fractional order.Comput. Math. and Math. Phys. 59, No 11 (2019), 1781-1790.10.1134/S0965542519110113Suche in Google Scholar

[56] P.G. Surkov, Dynamic right-hand side reconstruction problem for a system of fractional differential equations. Diff.Eqns. 55, No 6 (2019), 849-858.10.1134/S0012266119060120Suche in Google Scholar

[57] V.E. Tarasov, Geometric interpretation of fractional-order derivative. Fract. Calc. Appl. Anal. 19, No 5 (2016), 1200-1221; DOI: 10.1515/fca-2018-006610.1515/fca-2016-0062 https://www.degruyter.com/journal/key/FCA/19/5/html.Suche in Google Scholar

[58] F.B. Tatom, The relationship between fractional calculus and fractals. Fractals 3, No 1 (1995), 217-229.10.1142/S0218348X95000175Suche in Google Scholar

[59] J.A. Tenreiro Machado, A probabilistic interpretation of thefractional-order differentiation. Fract. Calc. Appl. Anal. 6, No 1 (2003), 73-79.Suche in Google Scholar

[60] A.N. Tikhonov, V.Ya. Arsenin,Solution of Ill-posed Problems. John Wiley, New York (1977).Suche in Google Scholar

[61] H. Unbehauen, G.P. Rao, Identification of Continuous Systems. Elsevier, Amsterdam (1987).Suche in Google Scholar

[62] V.V. Vasin, The stable evaluation of a derivative in space C(-∞,∞). U.S.S.R. Comput. Math. Math. Phys. 13, No 6 (1973), 16-24.10.1016/0041-5553(73)90002-5Suche in Google Scholar

[63] J. Wang, Wavelet approach to numerical differentiation of noisy function. Commun. Pure Appl. Anal. 6, No 3 (2007), 873-897.10.3934/cpaa.2007.6.873Suche in Google Scholar

[64] Y.B. Wang, X.Z. Jia, J. Cheng, A numerical differentiation method and its application to reconstruction of discontinuity. Inverse Problems 18, No 6 (2002), 1461-1476.10.1088/0266-5611/18/6/301Suche in Google Scholar
Received: 2020-09-30
Revised: 2021-05-14
Published Online: 2021-06-23
Published in Print: 2021-06-25

© 2021 Diogenes Co., Sofia

Artikel in diesem Heft

  1. Frontmatter
  2. Editorial Survey
  3. In memory of the honorary founding editors behind the FCAA success story
  4. Research Paper
  5. Short time coupled fractional fourier transform and the uncertainty principle
  6. (N + α)-Order low-pass and high-pass filter transfer functions for non-cascade implementations approximating butterworth response
  7. Sharp asymptotics in a fractional Sturm-Liouville problem
  8. Multi-term fractional integro-differential equations in power growth function spaces
  9. Galerkin method for time fractional semilinear equations
  10. Müntz sturm-liouville problems: Theory and numerical experiments
  11. Simultaneous inversion for the fractional exponents in the space-time fractional diffusion equation ∂tβ u = −(− Δ)α/2 u − (− Δ)γ/2 u
  12. Nonlinear convolution integro-differential equation with variable coefficient
  13. An efficient localized collocation solver for anomalous diffusion on surfaces
  14. Approximate calculation of the Caputo-type fractional derivative from inaccurate data. Dynamical approach
  15. Sliding methods for the higher order fractional laplacians
  16. Global stability of fractional different orders nonlinear feedback systems with positive linear parts and interval state matrices
https://doi.org/10.1515/fca-2021-0038
Schlagwörter für diesen Artikel
Caputo-type fractional derivative; reconstruction; incomplete information; error estimate; Mittag-Leffler type functions

Artikel in diesem Heft

  1. Frontmatter
  2. Editorial Survey
  3. In memory of the honorary founding editors behind the FCAA success story
  4. Research Paper
  5. Short time coupled fractional fourier transform and the uncertainty principle
  6. (N + α)-Order low-pass and high-pass filter transfer functions for non-cascade implementations approximating butterworth response
  7. Sharp asymptotics in a fractional Sturm-Liouville problem
  8. Multi-term fractional integro-differential equations in power growth function spaces
  9. Galerkin method for time fractional semilinear equations
  10. Müntz sturm-liouville problems: Theory and numerical experiments
  11. Simultaneous inversion for the fractional exponents in the space-time fractional diffusion equation ∂tβ u = −(− Δ)α/2 u − (− Δ)γ/2 u
  12. Nonlinear convolution integro-differential equation with variable coefficient
  13. An efficient localized collocation solver for anomalous diffusion on surfaces
  14. Approximate calculation of the Caputo-type fractional derivative from inaccurate data. Dynamical approach
  15. Sliding methods for the higher order fractional laplacians
  16. Global stability of fractional different orders nonlinear feedback systems with positive linear parts and interval state matrices
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