Ukrainian Mathematical Journal

, Volume 70, Issue 5, pp 687–701

# Numerical Solutions of Fractional Systems of Two-Point BVPs by Using the Iterative Reproducing Kernel Algorithm

• Z. Altawallbeh
• I. Komashynska
• A. Ateiwi
Article

We propose an efficient computational method, namely, the iterative reproducing kernel method for the approximate solution of fractional-order systems of two-point time boundary-value problems in the Caputo sense. Two extended inner-product spaces are constructed in which the boundary conditions are satisfied for the analyzed systems. The reproducing kernel functions are constructed to get an accurate algorithm for the investigation of fractional systems. The developed procedure is based on generating the orthonormal basis with an aim to formulate the solution via the evolution of the algorithm. The analytic solution is represented in the form of a series in the reproducing kernel Hilbert space with readily computed components. In this connection, some numerical examples are presented to show the good performance and applicability of the developed algorithm. The numerical results indicate that the proposed algorithm is a powerful tool for the solution of fractional models encountered in various fields of sciences and engineering.

## References

1. 1.
X. J. Yang, Advanced Local Fractional Calculus and Its Applications, World Scientific Publ., New York (2012).Google Scholar
2. 2.
I. Podlubny, Fractional Differential Equations, Academic Press, New York (1999).
3. 3.
K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York (1993).
4. 4.
M. Caputo, “Linear models of dissipation whose Q is almost frequency independent, Part II,” Geophys. J. Int., 13, 529–539 (1967).Google Scholar
5. 5.
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, “Theory and applications of fractional differential equations,” in: North-Holland Mathematics Studies, Elsevier, New York (2006), 204.Google Scholar
6. 6.
F. Geng, “Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method,” Appl. Math. Comput., 215, 2095–2102 (2009).
7. 7.
F. Geng and S. P. Qian, “Reproducing kernel method for singularly perturbed turning point problems having twin boundary layers,” Appl. Math. Lett., 26, 998–1004 (2013).
8. 8.
G. Gumah, K. Moaddy, M. AL-Smadi, and I. Hashim, “Solutions of uncertain Volterra integral equations by fitted reproducing kernel Hilbert space method,” J. Funct. Spaces, Article ID 2920463, 11 p. (2016).Google Scholar
9. 9.
I. Komashynska and M. Al-Smadi, “Iterative reproducing kernel method for solving second-order integrodifferential equations of Fredholm type,” J. Appl. Math., 2014, 1–11 (2014).
10. 10.
O. Abu Arqub, M. Al-Smadi, and S. Momani, “Application of reproducing kernel method for solving nonlinear Fredholm–Volterra integrodifferential equations,” Abstr. Appl. Anal., 2012, 1–16 (2012).
11. 11.
O. Abu Arqub and M. Al-Smadi, “Numerical algorithm for solving two-point, second-order periodic boundary value problems for mixed integro-differential equations,” Appl. Math. Comput., 243, 911–922 (2014).
12. 12.
M. Al-Smadi, O. Abu Arqub, and S. Momani, “A computational method for two-point boundary value problems of fourth-order mixed integrodifferential equations,” Math. Probl. Eng., 2013, 1–10 (2013).
13. 13.
O. Abu Arqub, M. Al-Smadi, and N. Shawagfeh, “Solving Fredholm integrodifferential equations using reproducing kernel Hilbert space method,” Appl. Math. Comput., 219, 8938–8948 (2013).
14. 14.
Z. Altawallbeh, M. Al-Smadi, and R. Abu-Gdairi, “Approximate solution of second-order integrodifferential equation of Volterra type in RKHS method,” Int. J. Math. Anal., 7, No 44, 2145–2160 (2013).
15. 15.
O. Abu Arqub, M. Al-Smadi, S. Momani, and T. Hayat, “Numerical solutions of fuzzy differential equations using reproducing kernel Hilbert space method,” Soft Comput., 20, No. 8, 3283–3302 (2016).
16. 16.
O. Abu Arqub, M. AL-Smadi, S. Momani, and T. Hayat, “Application of reproducing kernel algorithm for solving second-order, two-point fuzzy boundary value problems,” Soft Comput., 1–16 (2016).Google Scholar
17. 17.
R. Saadeh, M. Al-Smadi, G. Gumah, H. Khalil, and R. A. Khan, “Numerical investigation for solving two-point fuzzy boundary value problems by reproducing kernel approach,” Appl. Math. Inf. Sci., 10, No. 6, 2117–2129 (2016).
18. 18.
M. Al-Smadi, O. Abu Arqub, N. Shawagfeh, and S. Momani, “Numerical investigations for systems of second-order periodic boundary value problems using reproducing kernel method,” Appl. Math. Comput., 291, 137–148 (2016).
19. 19.
M. Al-Smadi, O. Abu Arqub, and A. El-Ajou, “A numerical iterative method for solving systems of first-order periodic boundary value problems,” J. Appl. Math., 2014, 1–10 (2014).
20. 20.
M. Inc and A. Akgül, “Approximate solutions for MHD squeezing fluid flow by a novel method,” Bound. Value Probl., 18, 1–17 (2014).
21. 21.
A. Akgül, M. Inc, and E. Karatas, “Reproducing kernel functions for difference equations,” Discrete Contin. Dyn. Syst. Ser. S, 8, No. 6, 1055–1064 (2015).
22. 22.
F. Geng and M. Cui, “New method based on the HPM and RKHSM for solving forced Duffing equations with integral boundary conditions,” J. Comput. Appl. Math., 233, 165–172 (2009).
23. 23.
Y. Zhou, M. Cui, and Y. Lin, “Numerical algorithm for parabolic problems with nonclassical conditions,” J. Comput. Appl. Math., 230, 770–780 (2009).
24. 24.
K. Moaddy, M. Al-Smadi, and I. Hashim, “A novel representation of the exact solution for differential algebraic equations system using residual power-series method,” Discrete Dynam. Nat. Soc., Article ID 205207, 12 p. (2015).Google Scholar
25. 25.
M. Al-Smadi and Z. Altawallbeh, “Solution of system of Fredholm integrodifferential equations by RKHS method,” Int. J. Contemp. Math. Sci., 8, No. 11, 531–540 (2013).
26. 26.
M. Inc, A. Akgül, and F. Geng, “Reproducing kernel Hilbert space method for solving Bratu’s problem,” Bull. Malays. Math. Sci. Soc., 38, 271–287 (2015).Google Scholar

## Authors and Affiliations

• Z. Altawallbeh
• 1