On Applications Shehu Variational Iteration Method to Time Fractional Initial Boundary Value Problems

Abstract

The purpose of this study is to establish a semi analytical solution for time fractional linear or nonlinear mathematical problems by utilizing Shehu Variational Iteration Method (SVIM). SVIM is made up of two methods, called Shehu transform (ST) and variational iteration method (VIM). First of all, the time fractional differential equation is transformed into integer order differential equation by means of ST. Later, by taking VIM into account the solution of linear or nonlinear mathematical problem is acquired. The convergence analysis of the semi analytical solution is investigated and proves that SVIM is an accurate and effective method for fractional mathematical problems. The illustrated examples support analysis of this method.

Keywords

• Liouville-Caputo Derivative
• Nonlinear Mathematical Problem
• Shehu Transform
• Variational Iteration Method

References

1. [1] S. Abbasbandy, An approximation solution of a nonlinear equation with Riemann–Liouville’s fractional derivatives by He’s variational iteration method, J. Comput. Appl. Math., 207(1) (2007), 53–58.
2. [2] O. Acana, M.M. Al Qurashib and D. Baleanu, Reduced differential transform method for solving time and space local fractional partial differential equations, Nonlinear Sci. Appl., 10 (2017), 5230–5238.
3. [3] L. Akinyemi and O. Iyiola, Exact and approximate solutions of time-fractional models arising from physics via Shehu transform, Math. Methods Appl. Sci. , (2020).
4. [4] R. Belgacem, D. Baleanu and A. Bokhari, Shehu Transform and applications to Caputo-fractional differential equations, International Journal of Analysis and Applications, 17(6) (2019), 917–927.
5. [5] A. Bokhari, D. Baleanu and R. Belgacem, Application of Shehu transform to Atangana-Baleanu derivatives, J. Math. Computer Sci., 20 (2020), 101-–107.
6. [6] B. Ibis and M. Bayram, Numerical comparison of methods for solving fractional differential–algebraic equations (FDAEs), Comput. Math. Appl., 62(8) (2011), 3270—3278.
7. [7] B. Ibis and M. Bayram, Analytical approximate solution of time-fractional Fornberg–Whitham equation by the fractional variational iteration method, Alexandria Engineering Journal, 53(4) (2014), 911–915.
8. [8] M. Inc, The approximate and exact solutions of the space and time-fractional Burgers equations with initial conditions by variational iteration method, J. Math. Anal. Appl., 345 (2008), 476–484.

9. [9] H. Jafari, V. Daftardar-Gejji, Solving a system of nonlinear fractional differential equations using Adomian decomposition, Journal of Computational and Applied Mathematics, 196 (2006), 644–651.
10. [10] A.A. Kilbas, H.M. Srivastava and J.J. Trujıllo, Theory and Applications of Fractional Differential Equations Elsevier, 2006.
11. [11] S. Maitama and W. Zhao, New Integral Transform: Shehu Transform a Generalization of Sumudu and Laplace Transform for Solving Differential Equations, Int. J. Anal. Appl., 17(2) (2019), 167–190.
12. [12] Z. Odibat and S. Momani, The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics, Comput. Math. Appl., 58(11–12) (2009), 2199–2208.
13. [13] Z. M. Odibat, A study on the convergence of variational iteration method, Math. Comput. Model., 51 (2010), 1181–1192.
14. [14] I. Podlubny, Fractional Differential Equations, Elsevier, San Diego, 1998.
15. [15] M. G. Sakar and H. Ergoren, Alternative variation iteration method for solving the time– fractional Fornberg-Whitham equation, Appl. Math. Model, 39(14) (2015), 3972–3979.
16. [16] M. Senol, O.S. Iyiola, H.D. Kasmaei and L. Akinyemi, Efficient analytical techniques for solving time-fractional nonlinear coupled Jaulent–Miodek system with energy-dependent Schr¨odinger potential, Adv. Differ. Equ., 462 (2019).