An Analytical Technique to Obtain Approximate Solutions of Nonlinear Fractional PDEs

Authors

  • Near East University, Operational Research Center in Healthcare, Nicosia, PC: 99138, TRNC Mersin 10, Turkey
  • Department of Mathematics, Faculty of Education for Pure Sciences, University of Thi-Qar, Nasiriyah, Iraq

Abstract

Abstract:

In this work we obtain analytical approximate solutions for the two dimensional nonlinear PDEs with Liouville-Caputo fractional derivative. Numerical simulations of alternative models are presented for evaluating  the  effectiveness  of  these  representations

References

[ 1] Jajarmi, S. Arshad, D. Baleanu, A new fractional modelling and control strategy for the outbreak of dengue fever, Physica A, 535 (2019), 122524.

[ 2] M. Bologna, P. Grigolini, B. J. West, Physics of fractal operators, New York: Springer, 2003. [ 3] C. Li, J. Lu, J. Wang, Observer-based robust stabilisation of a class of non-linear fractional-order

uncertain systems: an linear matrix inequalitie approach, IET Control Theory Appl., 6 (2012), 2757– 2764.

[ 4] W. R. Schneider, W. Wyess, Fractional diusion and wave equations, J. Math. Phys., 30 (1989)134– 144.

[ 5] H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagt, Digital computation of the fractional Fourier transform, IEEE Transactionson Signal Processing, 44 (1996), 2141–2150.

[ 6] J. Klafter, S. C. Lim, R. Metzler, Fractional dynamics: recent advances, Singapore: World Scientific, 2011.

[ 7] R. L. Magin, Fractional calculus in bioengineering, Crit. Rev. Biomed. Eng., 32 (2006), 1–104.

[ 8] A. Zappone, E. Jorswieck, Energy effciency in wireless networks via fractional programming theory found, Trends Commun. Inf. Theory, 11 (2014), 185-396.

[ 9] T. A. Yıldız, S. Arshad, D. Baleanu, New observations on optimal cancer treatments for a fractional tumor growth model with and without singular kernel, Chaos Soliton. Fract., 117 (2018), 226–239.

[ 10] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, London and New York: Elsevier, 2006.

[ 11] L. K. Alzaki, H. K. Jassim, Time-Fractional Differential Equations with an Approximate Solution, Journal of the Nigerian Society of Physical Sciences, 4 (3)(2022) 1-8.

[ 12] H. K. Jassim, M. A. Hussein, A Novel Formulation of the Fractional Derivative with the order α>0 and without the Singular Kernel, Mathematics, 10 (21) (2022), 1-18.

[ 13] H. G. Taher, N. J. Hassan, Approximate analytical solutions of differential equations with Caputo-Fabrizio fractional derivative via new iterative method, AIP Conference Proceedings, 2398 (060020) (2022) 1-16.

[ 14] S. A. Sachit, N. J. Hassan, Revised fractional homotopy analysis method for solving nonlinear fractional PDEs, AIP Conference Proceedings, 2398 (060044) (2022) 1-15.

[ 15] S. H. Mahdi, N. J. Hassan, A new analytical method for solving nonlinear biological population model, AIP Conference Proceedings, 2398 (060043) (2022) 1-12.

[ 16] M. Y. Zayir, H. K. Jassim, A unique approach for solving the fractional Navier–Stokes equation, Journal of Multiplicity Mathematics, 25(8-B) (2022) 2611-2616.

[ 17] H. Jafari, M. Y. Zayir, Analysis of fractional Navier-Stokes equations, Heat Transfer, 52(3)(2023) 2859-2877.

[ 18] S. A. Sachit, Solving fractional PDEs by Elzaki homotopy analysis method, AIP Conference Proceedings, 2414 (040074) (2023) 1-12.

[ 19] H. Adnan, N. J. Hassan, The Weibull Lindley Rayleigh distribution, AIP Conference Proceedings, 2414 (040064) (2023) 1-17.

[ 20] S. H. Mahdi, A new technique of using Adomian decomposition method for fractional order nonlinear differential equations, AIP Conference Proceedings, 2414 (040075) (2023) 1-12.

[ 21] H. K Jassim, M A. Hussein, A New Approach for Solving Nonlinear Fractional Ordinary Differential Equations, Mathematics, 11(7)(2023) 1565.

[ 22] D. Ziane, M. H. Cherif, K. Belghaba, A. Al-Dmour, Application of Local Fractional Variational Iteration Transform Method to Solve Nonlinear Wave-Like Equations within Local Fractional Derivative, Progress in Fractional Differentiation and Applications, 9(2) (2023) 311– 318.

[ 23] D. Kumar, J. Singh, V. P. Dubey. A Computational Study of Local Fractional Helmholtz and Coupled Helmholtz Equations in Fractal Media, Lecture Notes in Networks and Systems, 2023, 666 LNNS, pp. 286–298.

[ 24] N. H. Mohsin, et al., A New Analytical Method for Solving Nonlinear Burger’s and Coupled Burger’s Equations, Materials Today: Proceedings, 80 (3)(2023) 3193-3195 .

[ 25] M. A. Hussein, et al., Analysis of fractional differential equations with Antagana-Baleanu fractional operator, Progress in Fractional Differentiation and Applications, 9(4)(2023) 681-686.

[ 26] H. K. Jassim, The Approximate Solutions of Fredholm Integral Equations on Cantor Sets within Local Fractional Operators, Sahand Communications in Mathematical Analysis, Vol. 16, No. 1, 13- 20, 2016.

[ 27] H. Jafari, et al., Approximate Solution for Nonlinear Gas Dynamic and Coupled KdV Equations Involving Local Fractional Operator, Journal of Zankoy Sulaimani-Part A, vol. 18, no.1, pp.127-132, 2016

[ 28] H. Jafari, H. K. Jassim, A new approach for solving a system of local fractional partial differential equations, Applications and Applied Mathematics, Vol. 11, No. 1, pp.162-173, 2016.

[ 29] H. K. Jassim, Local Fractional Variational Iteration Transform Method for Solving Couple Helmholtz Equations within Local Fractional Operator, Journal of Zankoy Sulaimani-Part A, Vol. 18, No. 2, pp.249-258, 2016.

[ 30] H. K. Jassim, On Analytical Methods for Solving Poisson Equation, Scholars Journal of Research in Mathematics and Computer Science, Vol. 1, No. 1, pp. 26- 35, 2016.

[ 31] H. K. Kadhim, Application of Local Fractional Variational Iteration Method for Solving Fredholm Integral Equations Involving Local Fractional Operators, Journal of University of Thi-Qar, Vol. 11, No. 1, pp. 12-18, 2016.

[ 32] H. Jafari, et al., Application of Local Fractional Variational Iteration Method to Solve System of Coupled Partial Differential Equations Involving Local Fractional Operator, Applied Mathematics & Information Sciences Letters, Vol. 5, No. 2, pp. 1-6, 2017.

[ 33] H. K. Jassim, The Analytical Solutions for Volterra Integro-Differential Equations Involving Local fractional Operators by Yang-Laplace Transform, Sahand Communications in Mathematical Analysis, Vol. 6 No. 1 (2017), 69-76 .

[ 34] H. K. Jassim, Some Dynamical Properties of Rössler System, Journal of University of Thi-Qar, 3(1) (2017), 69-76.

[ 35] H. K. Jassim, A Coupling Method of Regularization and Adomian Decomposition for Solving a Class of the Fredholm Integral Equations within Local Fractional Operators, 2(3) (2017), 95-99.

[ 36] H. K. Jassim, On Approximate Methods for Fractal Vehicular Traffic Flow, TWMS Journal of Applied and Engineering Mathematics, 7(1)(2017), 58-65.

[ 37] H. K. Jassim, A Novel Approach for Solving Volterra Integral Equations Involving Local Fractional Operator, Applications and Applied Mathematics, 12(1) (2017), 496 – 505.

[ 38] H. K. Jassim, Extending Application of Adomian Decomposition Method for Solving a Class of Volterra Integro-Differential Equations within Local Fractional Integral Operators, Journal of college of Education for Pure Science, 7(1) (2017), 19-29.

[ 39] A. A. Neamah, et al., Analytical Solution of The One Dimensional Volterra Integro Differential Equations within Local Fractional Derivative, Journal of Kufa for Mathematics and Computer, 4 (1)(2017), 46-50

[ 40] H. K. Jassim, Approximate Methods for Local Fractional Integral Equations, The Journal of Hyperstructures, 6 (1) (2017), 40-51

[ 41] H. K. Jassim, An Efficient Technique for Solving Linear and Nonlinear Wave Equation within Local Fractional Operators, The Journal of Hyperstructures, 6( 2)(2017), 136-146.

[ 42] H. A. Naser and A. K. Jiheel, A New Efficient Method for solving Helmholtz and Coupled Helmholtz Equations Involving Local Fractional Operators, 6)4)(2018), 153-157.

[ 43] M. G. Mohammed, S. A. Khafif, The Approximate solutions of time-fractional Burger’s and coupled time-fractional Burger’s equations, International Journal of Advances in Applied Mathematics and Mechanics,6(4)(2019),64-70 .

[ 44] M. G. Mohammad, , S. M. Kadhim, Symmetry Classification of First Integrals for Scalar Linearizable, International Journal of Advances in Applied Mathematics and Mechanics, 7(1) (2019), 20-40.

[ 45] M. Zayir, et al., A Fractional Variational Iteration Approach for Solving Time-Fractional Navier- Stokes Equations. Mathematics and Computational Sciences, 3(2) (2022), 41-47 .

[ 46] A. T. Salman, A new approximate analytical method and its convergence for time-fractional differential equations, NeuroQuantology, 20(6) (2022) 3670-3689.

[ 47] M. A. Hussein, et al., New approximate analytical technique for the solution of two dimensional fractional differential equations, NeuroQuantology, 20(6) (2022) 3690-3705.

[ 48] A. T. Salman, N. J. Hassan, An application of the Elzaki homotopy perturbation method for solving fractional Burger's equations, International Journal of Nonlinear Analysis and Applications, 13(2) (2022) 21-30.

[ 49] M. A. Hussein, A New Numerical Solutions of Fractional Differential Equations with Atangana- Baleanu operator in Reimann sense, International Journal of Scientific Research and Engineering Development, 5(6)(2022) 843- 849.

[ 50] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal., 5 (2002), 367–386.

[ 51] A. Horani, R. Khalil, M. Sababheh, A. Yousef, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65–70.

[ 52] G. Jumarie, On the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion, Appl. Math. Lett., 18 (2005), 817–826.

Downloads

Published

2024-03-01