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

• Ali Akgül
Chapter
Part of the Studies in Systems, Decision and Control book series (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.

## Keywords

Fractional calculus Atangana–Baleanu fractional derivative Reproducing kernel method

## References

1. 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. 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. 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)
4. 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)Google Scholar
5. 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)Google Scholar
6. 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)
7. 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)
8. 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)
9. 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)Google Scholar
10. 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)
11. 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)
12. 12.
Kumar, S.: A new analytical modelling for telegraph equation via laplace transform. Appl. Math. Model 38(13), 3154–63 (2014)
13. 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)
14. 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)Google Scholar
15. 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)
16. 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. 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. 18.
Caputo, M., Fabrizio, M.: Applications of new time and spatial fractional derivatives with exponential kernels. Prog. Fract. Differ. Appl. 2, 1–11 (2016)Google Scholar
19. 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)
20. 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. 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. 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. 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)Google Scholar
24. 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)Google Scholar
25. 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)
26. 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)Google Scholar
27. 27.
Akgül, A., Grow, D.: Existence of solutions to the telegraph equation in binary reproducing kernel Hilbert spaces (2017)Google Scholar
28. 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)
29. 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. 30.
Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950)
31. 31.
Bergman, S.: The Kernel Function and Conformal Mapping. American Mathematical Society, New York (1950)
32. 32.
Cui, M., Zhongxing, D.: On the best operator of interpolation. Math. Numer. Sin. 8, 209–216 (1986)
33. 33.
Cui, M., Yingzhen, L.: Nonlinear Numerical Analysis in the Reproducing Kernel Space. Nova Science Publishers Inc., New York (2009)
34. 34.
Mustafa, I., Akgül, A., Kilicman, A.: On solving KdV equation using reproducing kernel Hilbert space method. Abstr. Appl. Anal. 1, 1–11 (2013)
35. 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)
36. 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)
37. 37.
Huanmin, Y., Lin, Y.: Solving singular boundary value problems of higher even order. J. Comput. Appl. Math. 223, 703–713 (2009)
38. 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)
39. 39.
Geng, F., Minggen, C.: A reproducing kernel method for solving nonlocal fractional boundary value problems. Appl. Math. Lett. 25, 818–823 (2012)
40. 40.
Wu, B.Y., Li, X.Y.: Iterative reproducing kernel method for nonlinear oscillator with discontinuity. Appl. Math. Lett. 23, 1301–1304 (2010)
41. 41.
Geng, F., Minggen, C.: Solving a nonlinear system of second order boundary value problems. J. Math. Anal. Appl. 327, 1167–1181 (2007)
42. 42.
Mustafa, I., Akgül, A.: Approximate solutions for MHD squeezing fluid flow by a novel method. Bound. Value Probl. 1, 1–18 (2014)
43. 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)
44. 44.
Akgül, A., Mustafa, I., Esra, K.: Reproducing kernel functions for difference equations. Discret. Contin. Dyn. Syst. Ser. S 8, 1055–1064 (2015)
45. 45.
Geng, F.: Solving integral equations of the third kind in the reproducing kernel space. Bull. Iran. Math. Soc. 38, 543–551 (2012)
46. 46.
Lorenz, E.N.: Deterministic non-periodic flow. J. Atmos. Sci. 20(2), 130–141 (1963)
47. 47.
Ivancevic Vladimir, G., Tijana, T.I.: Complex Nonlinearity: Chaos, Phase Transitions, Topology Change, and Path Integrals. Springer, Berlin (2008)
48. 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)
49. 49.
Vellekoop, M., Berglund, R.: On intervals, transitivity = chaos. Am. Math. Mon. 101(4), 353–355 (1994)
50. 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. 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)
52. 52.
Šremr, J.: Absolutely continuous functions of two variables in the sense of Carathéodory. Electron. J. Differ. Equ. 1, 1–11 (2010)