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

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Fractional Derivatives with Mittag-Leffler Kernel

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 194))

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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References

  1. Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1, 73–85 (2015)

    Google Scholar 

  2. Losada, J., Nieto, J.J.: Properties of a new fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1, 87–92 (2015)

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kumar, S.: A new analytical modelling for telegraph equation via laplace transform. Appl. Math. Model 38(13), 3154–63 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Google Scholar 

  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)

    Google Scholar 

  18. Caputo, M., Fabrizio, M.: Applications of new time and spatial fractional derivatives with exponential kernels. Prog. Fract. Differ. Appl. 2, 1–11 (2016)

    Article  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Google Scholar 

  21. Atangana, A., Baleanu, D.: Caputo-Fabrizio derivative applied to groundwater flow within a confined aquifer. J. Eng. Mech. 1, 1–16 (2016)

    Google Scholar 

  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)

    Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  27. Akgül, A., Grow, D.: Existence of solutions to the telegraph equation in binary reproducing kernel Hilbert spaces (2017)

    Google Scholar 

  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  Google Scholar 

  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)

    Google Scholar 

  30. Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  31. Bergman, S.: The Kernel Function and Conformal Mapping. American Mathematical Society, New York (1950)

    Book  MATH  Google Scholar 

  32. Cui, M., Zhongxing, D.: On the best operator of interpolation. Math. Numer. Sin. 8, 209–216 (1986)

    MathSciNet  MATH  Google Scholar 

  33. Cui, M., Yingzhen, L.: Nonlinear Numerical Analysis in the Reproducing Kernel Space. Nova Science Publishers Inc., New York (2009)

    MATH  Google Scholar 

  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  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  37. Huanmin, Y., Lin, Y.: Solving singular boundary value problems of higher even order. J. Comput. Appl. Math. 223, 703–713 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  39. Geng, F., Minggen, C.: A reproducing kernel method for solving nonlocal fractional boundary value problems. Appl. Math. Lett. 25, 818–823 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  40. Wu, B.Y., Li, X.Y.: Iterative reproducing kernel method for nonlinear oscillator with discontinuity. Appl. Math. Lett. 23, 1301–1304 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  41. Geng, F., Minggen, C.: Solving a nonlinear system of second order boundary value problems. J. Math. Anal. Appl. 327, 1167–1181 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  42. Mustafa, I., Akgül, A.: Approximate solutions for MHD squeezing fluid flow by a novel method. Bound. Value Probl. 1, 1–18 (2014)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  44. Akgül, A., Mustafa, I., Esra, K.: Reproducing kernel functions for difference equations. Discret. Contin. Dyn. Syst. Ser. S 8, 1055–1064 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  45. Geng, F.: Solving integral equations of the third kind in the reproducing kernel space. Bull. Iran. Math. Soc. 38, 543–551 (2012)

    MathSciNet  MATH  Google Scholar 

  46. Lorenz, E.N.: Deterministic non-periodic flow. J. Atmos. Sci. 20(2), 130–141 (1963)

    Article  MATH  Google Scholar 

  47. Ivancevic Vladimir, G., Tijana, T.I.: Complex Nonlinearity: Chaos, Phase Transitions, Topology Change, and Path Integrals. Springer, Berlin (2008)

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  49. Vellekoop, M., Berglund, R.: On intervals, transitivity = chaos. Am. Math. Mon. 101(4), 353–355 (1994)

    MathSciNet  MATH  Google Scholar 

  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)

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  52. Šremr, J.: Absolutely continuous functions of two variables in the sense of Carathéodory. Electron. J. Differ. Equ. 1, 1–11 (2010)

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Siirt University Art and Science Faculty Department of Mathematics, 56100, Siirt, Turkey

    Ali Akgül

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  1. Ali Akgül
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Correspondence to Ali Akgül .

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Editors and Affiliations

  1. CONACYT-Tecnológico Nacional de México, Centro Nacional de Investigación y Desarrollo Tecnológico, Cuernavaca, Morelos, Mexico

    José Francisco Gómez

  2. CONACYT-Instituto de Ingeniería, Universidad Nacional Autónoma de México, Mexico City, Mexico

    Lizeth Torres

  3. Tecnológico Nacional de México, Centro Nacional de Investigación y Desarrollo Tecnológico, Cuernavaca, Morelos, Mexico

    Ricardo Fabricio Escobar

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Akgül, A. (2019). Reproducing Kernel Method for Fractional Derivative with Non-local and Non-singular Kernel. In: Gómez, J., Torres, L., Escobar, R. (eds) Fractional Derivatives with Mittag-Leffler Kernel. Studies in Systems, Decision and Control, vol 194. Springer, Cham. https://doi.org/10.1007/978-3-030-11662-0_1

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

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-11661-3

  • Online ISBN: 978-3-030-11662-0

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