Dynamics of the nonlinear Systems in the Frame of the Caputo-Fabrizio Fractional Derivative
Abstract
In this manuscript, efficient numerical technique called the fractional homotopy perturbation transform method (FHPTM) for solving linear and nonlinear systems. Caputo fabrizio derivative (CFD) is used to define fractional operator in this class. To get approximate solution for four case of the homogeneous linear system and inhomogeneous nonlinear system of equation. The FHPTM is used with the Laplace transform technique in this method. To demonstrate the capabilities and efficiency of the FHPT approach, several applicable problems from various domains of science and Engineering, such as Physics and Biology, are presented. We exhibit various two and three dimensional figures to demonstrate that these solutions have wave-like features
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References
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control for nonlinear parabolic PDE-ODE coupled systems, Fuzzy Sets and Systems,
402, 105-123, 2021.
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169, 1-28, 2021.
[23] P. Veeresha, D. G. Prakash, Solution for Fractional Kuramoto-Sivashinsky Equation
Using Novel Computational Technique, Int. J. Appl. Comput. Math, 7(33),1-23,2021.
[24] R. Hilfer, Applications of Fractional Calculus in Physics, World Sci. Publ., River
Edge, NJ, 2000.
[25] R.Nawaz, N.Ali, L.Zada, K.S.Nisar, M.R. Alharthi, W. Jamshed, Extension of natural transform method with Daftardar-Jafari polynomials for fractional order differential equations, Alex. Eng. J, 60, 3205-3217,2021.
[26] S. Kumara, S. Dasa, S.H. Ong, Analysis of tumor cells in the absence and presence of chemotherapeutic treatment: The case of Caputo-Fabrizio time fractional
derivative,Math. Comput. Simulation, 190, 1-14, 2021.
[27] S.M. Hosseini, Z.Asgari, Solution of stochastic nonlinear time fractional PDEs using polynomial chaos expansion combined with an exponential integrator, Comp. &
Math. with Appl, 73(6), 997-1007, 2007.
[28] Y. Liu, E. Fan, B. Yin, H. Li, Fast algorithm based on the novel approximation
formula for the Caputo-Fabrizio fractional derivative, AIMS Math, 5, 1729-1744,
2020.
[2] A.Kumar, P. Fartyal, Dynamical behavior for the approximate solutions and different wave profiles nonlinear fractional generalised pochhammer-chree equation in
mathematical physics, Opt. and Quan. Elect, 55(13) , 1128, 2023.
[3] A. M. Wazwaz, The variational iteration method for solving linear and nonlinear
systems of PDEs, Comp. and Math. with Appl, 54, 895-902, 2007.
[4] A. M. Wazwaz, Partial Differential Equations: Methods and Applications, Balkema
Publishers, The Netherlands, 2002.
[5] A. Prakash, A. Kumar, H. M. Baskonus , A.Kumar, Numerical analysis of nonlinear
fractional Klein-Fock-Gordon equation arising in quantum field theory via CaputoFabrizio fractional operator, Math. Sci, 15(3),269-281 2021.
[6] A. Alshabanat, M. Jleli, S. Kumar, B. Samet, Generalization of Caputo-Fabrizio
Fractional Derivative and Applications to Electrical Circuits, Front. Phys, 8(64),1-
10, 2020.
[7] A. Kumar, E.Ilhan, A.Ciancio, G. Yel, H.M. Baskonus, Extractions of some new
travelling wave solutions to the conformable Date-Jimbo-Kashiwara-Miwa equation,
AIMS Math, 6(5), 4238-4264, 2021.
[8] B. S. Kala, M. S. Rawat, A. Kumar, Numerical Analysis of the Flow of a Casson
Fluid in Magnetic Field over an Inclined Nonlinearly Stretching Surface with Velocity
Slip in a Forchheimer Porous Medium, A. Rese. J. Math, 16(7), 34-58, 2020.
[9] Caputo M, Fabrizio M. Applications of new time and spatial fractional derivatives
with exponential kernels. Prog. Fract. Differ. Appl, 2,1-11, 2016.
[10] J.L.G. Guirao, H. M. Baskonus, A. Kumar, F. S. V.Causanilles , G. R. Bermudez, Instability modulation properties of the (2+ 1)-dimensional Kundu-Mukherjee-Naskar
model in travelling wave solutions, Modern Phys. Lett. B, 1-15, 2021.
[11] J.H. He, A new approach to nonlinear partial differential equations, Commun. Nonlinear Sci. Numer. Simul, 2 (4), 203-205, 1997.
[12] J.L.G. Guirao,H.M.Baskonus,A.Kumar,F. S. V.Causanilles , G. R. Bermudez, Complex mixed dark-bright wave patterns to the modified α and modified VakhnenkoParkes equations, Alex. Eng. J, 1-12, 2020.
[13] J.L.G. Guirao,H.M.Baskonus,A.Kumar, Regarding New Wave Patterns of the Newly
Extended Nonlinear (2+1)-Dimensional Boussinesq Equation with Fourth Order,
Mathematics, 8(341), 1-9, 2020.
[14] Kilbas, A.A, Srivastava, H.M, Trujillo, J.J, Theory and Applications of Fractional
Differential Equations, Elsevier, Amsterdam, 2006.
[15] Li.Y, A. Kumar, J.L.G. Guirao, H.C.Baskonus, W.Gao, Deeper properties of the nonlinear Phi-four and Gross-Pitaevskii equations arising mathematical physics, mathematical physics, Modern Phys. Lett. B, 1-19,2021.
[16] L.Yan, G.Yel, A.Kumar, H.M.Baskonus, W. Gao, Newly Developed Analytical
Scheme and Its Applications to the Some Nonlinear Partial Differential Equations
with the Conformable Derivative, Frac. and Fract, 5(4), 2021.
[17] H.M.Baskonus, A. Kumar,A.Kumar,W. Gao, Deeper investigations of the (4 + 1)-
dimensional Fokas and (2 + 1)-dimensional Breaking soliton equations, I. J. of Modern Phys. B, 2050152, 1-16, 2020.
[18] M.S. Hashmi, U. Aslam, J. Singh, K. S.Nisar, An efficient numerical scheme for
fractional model of telegraph equation, Alex. Eng. J, 61(8), 6383-6393, 2022.
[19] M. Al-Refai, K. Pal, New Aspects of Caputo-Fabrizio Fractional Derivative, Progr.
Fract. Differ. Appl, 5(2), 157-166, 2019.
[20] H. Y. Martinez, J.F. G. Aguilar, A new modified definition of Caputo-Fabrizio
fractional-order derivative and their applications to the Multi Step Homotopy Analysis Method,J. Comput. Appl. Math, 346(15), 247-260, 2019.
[21] H.N. Wu, X.M.Zhang, J.W. Wang, H.Y.Zhu, Observer-based output feedback fuzzy
control for nonlinear parabolic PDE-ODE coupled systems, Fuzzy Sets and Systems,
402, 105-123, 2021.
[22] P. Bonneau, E.Mazzilli, Complex associated to some systems of PDE, J. Geom. Phys,
169, 1-28, 2021.
[23] P. Veeresha, D. G. Prakash, Solution for Fractional Kuramoto-Sivashinsky Equation
Using Novel Computational Technique, Int. J. Appl. Comput. Math, 7(33),1-23,2021.
[24] R. Hilfer, Applications of Fractional Calculus in Physics, World Sci. Publ., River
Edge, NJ, 2000.
[25] R.Nawaz, N.Ali, L.Zada, K.S.Nisar, M.R. Alharthi, W. Jamshed, Extension of natural transform method with Daftardar-Jafari polynomials for fractional order differential equations, Alex. Eng. J, 60, 3205-3217,2021.
[26] S. Kumara, S. Dasa, S.H. Ong, Analysis of tumor cells in the absence and presence of chemotherapeutic treatment: The case of Caputo-Fabrizio time fractional
derivative,Math. Comput. Simulation, 190, 1-14, 2021.
[27] S.M. Hosseini, Z.Asgari, Solution of stochastic nonlinear time fractional PDEs using polynomial chaos expansion combined with an exponential integrator, Comp. &
Math. with Appl, 73(6), 997-1007, 2007.
[28] Y. Liu, E. Fan, B. Yin, H. Li, Fast algorithm based on the novel approximation
formula for the Caputo-Fabrizio fractional derivative, AIMS Math, 5, 1729-1744,
2020.
Published
2025-09-30
Section
Advanced Computational Methods for Fractional Calculus
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