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2019 | OriginalPaper | Buchkapitel

Reproducing Kernel Method for Fractional Derivative with Non-local and Non-singular Kernel

verfasst von : Ali Akgül

Erschienen in: Fractional Derivatives with Mittag-Leffler Kernel

Verlag: Springer International Publishing

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Abstract

Atangana and Baleanu introduced a derivative with fractional order to answer some outstanding questions that were posed by many investigators within the field of fractional calculus. Their derivative has a non-singular and nonlocal kernel. Therefore, we apply the reproducing kernel method to fractional differential equations with non-local and non-singular kernel. In this work, a new method has been developed for the newly established fractional differentiation. Examples are given to illustrate the numerical effectiveness of the reproducing kernel method when properly applied in the reproducing kernel space. The comparison of approximate and exact solutions leaves no doubt believing that the reproducing kernel method is very efficient and converges toward exact solution very rapidly.

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Nächstes Kapitel Necessary and Sufficient Optimality Conditions for Fractional Problems Involving Atangana–Baleanu’s Derivatives
Literatur
1.
Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1, 73–85 (2015)
2.
Losada, J., Nieto, J.J.: Properties of a new fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1, 87–92 (2015)
3.
Atangana, A.: On the new fractional derivative and application to nonlinear Fisher’s reaction-diffusion equation. Appl. Math. Comput. 1(273), 948–956 (2016)MathSciNetMATH
4.
Morales-Delgado, V.F., Taneco-Hernández, M.A., Gómez-Aguilar, J.F.: On the solutions of fractional order of evolution equations. Eur. Phys. J. Plus 132(1), 1–17 (2017)CrossRef
5.
Hristov, J.: Transient heat diffusion with a non-singular fading memory: from the Cattaneo constitutive equation with Jeffrey’s kernel to the Caputo-Fabrizio time-fractional derivative. Therm. Sci. 20(2), 757–762 (2016)CrossRef
6.
Yépez-Martínez, H., Gómez-Aguilar, J.F.: A new modified definition of Caputo-Fabrizio fractional-order derivative and their applications to the multi step homotopy analysis method (MHAM). J. Comput. Appl. Math. 346, 247–260 (2019)MathSciNetMATHCrossRef
7.
Gómez-Aguilar, J.F., López-López, M.G., Alvarado-Martínez, V.M., Baleanu, D., Khan, H.: Chaos in a cancer model via fractional derivatives with exponential decay and Mittag-Leffler law. Entropy 19(12), 1–21 (2017)MathSciNetCrossRef
8.
Doungmo Goufo, E.F., Pene, M.K., Jeanine, N.: Duplication in a model of rock fracture with fractional derivative without singular kernel. Open Math. 13, 839–846 (2015)MathSciNetMATH
9.
Gómez-Aguilar, J.F., Escobar-Jiménez, R.F., López-López, M.G., Alvarado-Martínez, V.M.: Atangana-Baleanu fractional derivative applied to electromagnetic waves in dielectric media. J. Electromagn. Waves Appl. 30(15), 1937–1952 (2016)CrossRef
10.
Brzezinski, D.W.: Accuracy problems of numerical calculation of fractional order derivatives and integrals applying the Riemann-Liouville/Caputo formulas. Appl. Math. Nonlinear Sci. 1, 23–43 (2016)MathSciNetMATHCrossRef
11.
Jiang, J., Cao, D., Chen, H.: Boundary value problems for fractional differential equation with causal operators. Appl. Math. Nonlinear Sci. 1, 11–22 (2016)MathSciNetMATHCrossRef
12.
Kumar, S.: A new analytical modelling for telegraph equation via laplace transform. Appl. Math. Model 38(13), 3154–63 (2014)MathSciNetMATHCrossRef
13.
Coronel-Escamilla, A., Gómez-Aguilar, J.F., Alvarado-Méndez, E., Guerrero-Ramírez, G.V., Escobar-Jiménez, R.F.: Fractional dynamics of charged particles in magnetic fields. Int. J. Mod. Phys. C 27(08), 1–16 (2016)MathSciNetCrossRef
14.
Gómez-Aguilar, J.F., Yépez-Martínez, H., Escobar-Jiménez, R.F., Astorga-Zaragoza, C.M., Morales-Mendoza, L.J., González-Lee, M.: Universal character of the fractional space-time electromagnetic waves in dielectric media. J. Electromagn. Waves Appl. 29(6), 727–740 (2015)CrossRef
15.
Kumar, S., Rashidi, M.M.: New analytical method for gas dynamics equation arising in shock fronts. Comput. Phys. Commun. 185(7), 1947–1954 (2014)MathSciNetMATHCrossRef
16.
Kumar, S., Yao, J.J., Kumar, A.: A fractioanal model to describing the Brownian motion of particles and its analytical solution. Adv. Mech. Eng. 7(12), 1–11 (2015)
17.
Kumar, S., Yin, X.B., Kumar, D.: A modified homotopy analysis method for solution of fractional wave equations. Adv. Mech. Eng. 7(12), 1–8 (2015)
18.
Caputo, M., Fabrizio, M.: Applications of new time and spatial fractional derivatives with exponential kernels. Prog. Fract. Differ. Appl. 2, 1–11 (2016)CrossRef
19.
Alsaedi, A., Baleanu, D., Etemad, S., Rezapour, S.: On coupled systems of time-fractional differential problems by using a new fractional derivative. J. Funct. Spaces 1, 1–8 (2016)MathSciNetMATH
20.
Gómez-Aguilar, J.F.: Behavior characteristics of a cap-resistor, memcapacitor, and a memristor from the response obtained of RC and RL electrical circuits described by fractional differential equations. Turk. J. Electr. Eng. Comput. Sci. 24(3), 1–16 (2016)
21.
Atangana, A., Baleanu, D.: Caputo-Fabrizio derivative applied to groundwater flow within a confined aquifer. J. Eng. Mech. 1, 1–16 (2016)
22.
Atangana, A., Baleanu, D.: New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Therm. Sci. 18, 1–10 (2016)
23.
Coronel-Escamilla, A., Gómez-Aguilar, J.F., Baleanu, D., Córdova-Fraga, T., Escobar-Jiménez, R.F., Olivares-Peregrino, V.H., Qurashi, M.M.A.: Bateman-Feshbach tikochinsky and Caldirola-Kanai oscillators with new fractional differentiation. Entropy 19(2), 1–21 (2017)CrossRef
24.
Atangana, A., Gómez-Aguilar, J.F.: Decolonisation of fractional calculus rules: breaking commutativity and associativity to capture more natural phenomena. Eur. Phys. J. Plus 133, 1–22 (2018)CrossRef
25.
Atangana, A., Koca, I.: Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order. Chaos Solitons Fractals 89, 447–454 (2016)MathSciNetMATHCrossRef
26.
Toufik, M., Atangana, A.: New numerical approximation of fractional derivative with non-local and non-singular kernel: application to chaotic models. Eur. Phys. J. Plus 132, 1–14 (2017)CrossRef
27.
Akgül, A., Grow, D.: Existence of solutions to the telegraph equation in binary reproducing kernel Hilbert spaces (2017)
28.
Zaremba, S.: L’équation biharmonique et une classe remarquable de fonctions fondamentales harmoniques. Bulletin International l’Académia des Sciences de Cracovie 1, 147–196 (1907)MATH
29.
Zaremba, S.: Sur le calcul numérique des fonctions demandées dan le probléme de dirichlet et le probleme hydrodynamique. Bulletin International l’Académia des Sciences de Cracovie 1, 125–195 (1908)
30.
31.
Bergman, S.: The Kernel Function and Conformal Mapping. American Mathematical Society, New York (1950)MATHCrossRef
32.
Cui, M., Zhongxing, D.: On the best operator of interpolation. Math. Numer. Sin. 8, 209–216 (1986)MathSciNetMATH
33.
Cui, M., Yingzhen, L.: Nonlinear Numerical Analysis in the Reproducing Kernel Space. Nova Science Publishers Inc., New York (2009)MATH
34.
Mustafa, I., Akgül, A., Kilicman, A.: On solving KdV equation using reproducing kernel Hilbert space method. Abstr. Appl. Anal. 1, 1–11 (2013)MATH
35.
Wang, Y.-L., Chao, L.: Using reproducing kernel for solving a class of partial differential equation with variable coefficients. Appl. Math. Mech. 29, 129–137 (2008)MathSciNetMATHCrossRef
36.
Wu, B.Y., Li, X.Y.: A new algorithm for a class of linear nonlocal boundary value problems based on the reproducing kernel method. Appl. Math. Lett. 24, 156–159 (2011)MathSciNetMATHCrossRef
37.
Huanmin, Y., Lin, Y.: Solving singular boundary value problems of higher even order. J. Comput. Appl. Math. 223, 703–713 (2009)MathSciNetMATHCrossRef
38.
Akgül, A., Mustafa, I., Esra, K., Baleanu, D.: Numerical solutions of fractional differential equations of lane-emden type by an accurate technique. Adv. Differ. Equ.S 1, 1–20 (2015)MathSciNetMATH
39.
Geng, F., Minggen, C.: A reproducing kernel method for solving nonlocal fractional boundary value problems. Appl. Math. Lett. 25, 818–823 (2012)MathSciNetMATHCrossRef
40.
Wu, B.Y., Li, X.Y.: Iterative reproducing kernel method for nonlinear oscillator with discontinuity. Appl. Math. Lett. 23, 1301–1304 (2010)MathSciNetMATHCrossRef
41.
Geng, F., Minggen, C.: Solving a nonlinear system of second order boundary value problems. J. Math. Anal. Appl. 327, 1167–1181 (2007)MathSciNetMATHCrossRef
42.
Mustafa, I., Akgül, A.: Approximate solutions for MHD squeezing fluid flow by a novel method. Bound. Value Probl. 1, 1–18 (2014)MathSciNetMATH
43.
Mustafa, I., Akgül, A., Geng, F.: Reproducing kernel Hilbert space method for solving Bratu’s problem. Bull. Malays. Math. Sci. Soc. 38, 271–287 (2015)MathSciNetMATHCrossRef
44.
Akgül, A., Mustafa, I., Esra, K.: Reproducing kernel functions for difference equations. Discret. Contin. Dyn. Syst. Ser. S 8, 1055–1064 (2015)MathSciNetMATHCrossRef
45.
Geng, F.: Solving integral equations of the third kind in the reproducing kernel space. Bull. Iran. Math. Soc. 38, 543–551 (2012)MathSciNetMATH
46.
Lorenz, E.N.: Deterministic non-periodic flow. J. Atmos. Sci. 20(2), 130–141 (1963)MATHCrossRef
47.
Ivancevic Vladimir, G., Tijana, T.I.: Complex Nonlinearity: Chaos, Phase Transitions, Topology Change, and Path Integrals. Springer, Berlin (2008)MATHCrossRef
48.
Safonov Leonid, A., Tomer, E., Strygin Vadim, V., Ashkenazy, Y., Havlin, S.: Multifractal chaotic attractors in a system of delay-differential equations modeling road traffic. Chaos 12, 1–11 (2002)MathSciNetMATHCrossRef
49.
Vellekoop, M., Berglund, R.: On intervals, transitivity = chaos. Am. Math. Mon. 101(4), 353–355 (1994)MathSciNetMATH
50.
Mustafa, I., Akgül, A.: The reproducing kernel Hilbert space method for solving Troesch’s problem. J. Assoc. Arab. Univ. Basic Appl. Sci. 14, 19–27 (2013)
51.
Mustafa, I., Akgül, A., Kilicman, A.: Numerical solutions of the second order one-dimensional telegraph equation based on reproducing kernel Hilbert space method. Abstr. Appl. Anal. 1, 1–13 (2013)MathSciNetMATH
52.
Šremr, J.: Absolutely continuous functions of two variables in the sense of Carathéodory. Electron. J. Differ. Equ. 1, 1–11 (2010)MATHMathSciNet
Metadaten
Titel
Reproducing Kernel Method for Fractional Derivative with Non-local and Non-singular Kernel
verfasst von
Ali Akgül
Copyright-Jahr
2019
Buch
Fractional Derivatives with Mittag-Leffler Kernel

Print ISBN: 978-3-030-11661-3

Electronic ISBN: 978-3-030-11662-0

Copyright-Jahr: 2019

DOI
https://doi.org/10.1007/978-3-030-11662-0_1

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