## Tbilisi Mathematical Journal

### An efficient method for solving nonlinear time-fractional wave-like equations with variable coefficients

#### Abstract

In this paper, fractional residual power series method (FRPSM) is effectively applied for finding the approximate analytical solutions of general nonlinear time-fractional wave-like equations with variable coefficients. This method based on constructing series solutions in a form of rapidly convergent series with easily computable components and without need of linearization, discretization, perturbation or unrealistic assumptions. Numerical results are given and then they are compared with the exact solutions both numerically and graphically. By numerical examples, it is shown that the FRPSM is very simple, efficient and convenient for solving different forms of nonlinear fractional partial differential equations.

#### Note

The authors would like to thank Professor Hvedri Inassaridze (Editor-in-Chief) as well as the anonymous referees who has made valuable and careful comments, which improved the paper considerably.

#### Article information

Source
Tbilisi Math. J., Volume 12, Issue 4 (2019), 131-147.

Dates
Accepted: 30 October 2019
First available in Project Euclid: 3 January 2020

https://projecteuclid.org/euclid.tbilisi/1578020573

Digital Object Identifier
doi:10.32513/tbilisi/1578020573

Mathematical Reviews number (MathSciNet)
MR4047581

Zentralblatt MATH identifier
07104577

#### Citation

Khalouta, Ali; Kadem, Abdelouahab. An efficient method for solving nonlinear time-fractional wave-like equations with variable coefficients. Tbilisi Math. J. 12 (2019), no. 4, 131--147. doi:10.32513/tbilisi/1578020573. https://projecteuclid.org/euclid.tbilisi/1578020573

#### References

• O. Abu Arqub, Series solution of fuzzy differential equations under strongly generalized differentiability, Journal of Advanced Research in Applied Mathematics, 5 (2013), No. 3, 31–52.
• O. Abu Arqub, A. El-Ajou, Z. Al Zhour and Sh. Momani, Multiple Solutions of Nonlinear Boundary Value Problems of Fractional Order: A New Analytic Iterative Technique, Entropy, 16 (2014), 471–493.
• O.P. Agrawal and D. Baleanu, Hamiltonian formulation and a direct numerical scheme for Fractional Optimal Control Problems, Journal of Vibration and Control, 13 (2007), 1269–1281.
• M. Alquran, Analytical solutions of fractional foam drainage equation by residual power series method, Mathematical Sciences, 8 (2014), 153–160.
• M. Alquran, Analytical solutions of time-fractional two-component evolutionary system of order 2 by residual power series method, The Journal of Applied Analysis and Computation, 5 (2015), 589–599.
• A. Arikoglu and I. Ozkol, Solution of fractional differential equations by using differential transform method, Chaos, Solitons $\And$ Fractals, 34 (2007), 1473–1481.
• A. El-Ajou, O. Abu Arqub and S. Momani, Approximate analytical solution of the nonlinear fractional KdV-Burgers equation a new iterative algorithm, Journal of Computational Physics, 293 (2015), 81–94.
• D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Series on Complexity, Nonlinearity and Chaos in Fractional Calculus Models and Numerical Methods, World Scientific, (2012).
• M. Dalir and M. Bashour. Applications of fractional calculus, Applied Mathematical Sciences, 45 (2010), 1021–1032.
• K. Diethelm, The Mean Value Theorems and a Nagumo-Type Uniqueness Theorem for Caputo's Fractional Calculus, Fractional Calculus and Applied Analysis, 15 (2012), 304–313
• A. A. Hemeda, Homotopy Perturbation Method for Solving Partial Differential Equations of Fractional Order, International Journal of Mathematical Analysis, 6 (2012), No. 49, 2431–2448.
• R. Herrmann, Fractional Calculus: An Introduction for Physicists, World Scientific, New Jerzey, 2014.
• R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, New Jerzey, 2000.
• I. S. Jesus and J. A. Tenreiro Machado, Fractional control of heat diffusion systems, Nonlinear Dynamic, 54 (2008), 263–282.
• A. Khalouta and A. Kadem, Comparison of New Iterative Method and Natural Homotopy Perturbation Method for Solving Nonlinear Time-Fractional Wave-Like Equations with Variable Coefficients, Nonlinear Dynamics and Systems Theory, 19(1-SI) (2019), 160–169.
• A. Khalouta and A. Kadem, A New Technique for Finding Exact Solutions of Nonlinear Time-Fractional Wave-Like Equations with Variable Coefficients, Accepted by: Proceedings of Institute of Mathematics and Mechanics, (2019).
• A. Khalouta and A. Kadem, Fractional natural decomposition method for solving a certain class of nonlinear time-fractional wave-like equations with variable coefficients, Acta Universitatis Sapientiae Mathematica, 11 (2019), No. 1, 99–116.
• A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Application of Fractional Differential equations, Elsevier, North-Holland, 2006.
• V. Lakshmikantham, S. Leela and D. J. Vasundhara, Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, 2009.
• R. L. Magin, Fractional Calculus in Bioengineering, Begell House, Connecticut, 2006.
• K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley, NewYork, 1993.
• I. Podlubny, Fractional Differential Equations, Academic Press, New York, (1999).
• I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fractional Calculus and Applied Analysis, 5 (2002), 367–386.
• S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives. Theory and Applications, Gordon and Breach, Yverdon, 1993.
• A. M. Shukur, Adomian Decomposition Method for Certain Space-Time Fractional Partial Differential Equations, IOSR Journal of Mathematics, 11 (2015), No. 1, 55-65.
• V. E. Tarasov, Fractional vector calculus and fractional Maxwell's equations, Annals of Physics, 323 (2008), 2756–2778.
• Y. Zhang, Time-Fractional Generalized Equal Width Wave Equations: Formulation and Solution via Variational Methods, Nonlinear Dynamics and Systems Theory, 14 (2014), No. 4, 410–425.