Numerical Simulation of Time-Space Fractional K(n,n) Equations using Sage-math
Keywords:
Time–space fractional equations, $K(n,n)$ equation, Fractional homotopy perturbation method, Traveling wave solutions, SageMath, Numerical simulation, Fractional calculus, Nonlinear wave propagationAbstract
This paper presents a modest approach for solving and simulating the time-space fractional $K(n,n)$ equations using SageMath programming. The time-space fractional $K(n,n)$ equations describe compacton-type traveling wave solutions that arise in various physical and engineering systems governed by nonlinear dispersion. The fractional homotopy perturbation method is employed to derive approximate analytical series solutions for the time-space fractional $K(2,2)$ and $K(3,3)$ equations. Python routines built in SageMath are used to evaluate the fractional derivatives and integrals required for the numerical analysis. The obtained results are presented graphically to illustrate the influence of fractional-order parameters on the propagation of compacton solutions. The numerical experiments confirm that the method is stable, accurate, and efficient for analyzing nonlinear fractional models.
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References
bibitem{hpm1}
S.~I.~Abdelsalam, W.~Abbas, A.~M.~Megahed, H.~M.~H.~Sadek, and M.~S.~Emam,
Numerical simulation via homotopy perturbation approach of a dissipative squeezed Carreau fluid flow due to a sensor surface,
emph{Frontiers in Heat and Mass Transfer} textbf{22} (2025), 69–359, doi{10.32604/fhmt.2025.069359}.
bibitem{hpm2}
J.~Biazar, Z.~Ayati, and H.~Ebrahimi,
Comparing Homotopy Perturbation Method and Adomian Decomposition Method,
emph{Proceedings of the Conference}, Psalidi, Kos (Greece), (2008), 80--86. doi{10.1063/1.2991054}
bibitem{kdv frac}
H.~Cao, X.~Cheng, and Q.~Zhang,
Numerical simulation methods and analysis for the dynamics of the time-fractional KdV equation,
emph{Physica D: Nonlinear Phenomena} textbf{457} (2024), 134050, doi{10.1016/j.physd.2024.134050}.
bibitem{FDE1}
K.~Diethelm,
emph{The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type}.
Springer, 2010.
DOI: href{https://doi.org/10.1007/978-3-642-14574-2}{10.1007/978-3-642-14574-2}.
bibitem{FDE2}
A.~Ercan and M.~L.~Kavvas,
Fractional governing equations of diffusion wave and kinematic wave open-channel flow in fractional time--space. II. Numerical simulations,
emph{Journal of Hydrologic Engineering} textbf{20}(9) (2015), 04015011, doi{10.1061/(ASCE)HE.1943-5584.0001081}.
bibitem{Ghode2022}
K.~Ghode, K.~Takale, S.~Gaikwad, and K.~Bondar,
Python: Powerful tool for solving space-time fractional traveling wave equation,
emph{Technical Report} (2022), 1--15.
bibitem{Ghode2023}
K.~Ghode, K.~Takale, and S.~Gaikwad,
Traveling wave solutions of fractional differential equations arising in warm plasma,
emph{Baghdad Science Journal} textbf{20}(3) (2023), 1--10, doi{10.21123/bsj.2023.20.3.xx}.
bibitem{hpm3}
J.-H.~He,
Homotopy perturbation technique,
emph{Computer Methods in Applied Mechanics and Engineering} textbf{178}(3--4) (1999), 257--262, doi{10.1016/S0045-7825(99)00018-3}.
bibitem{frac}
R.~Hilfer,
emph{Applications of Fractional Calculus in Physics},
World Scientific, (2000), 1--500, doi{10.1142/3779}.
bibitem{FDE3}
S.~Holm, S.~P.~Næsholm, F.~Prieur, and R.~Sinkus,
Deriving fractional acoustic wave equations from mechanical and thermal constitutive equations,
emph{Computers & Mathematics with Applications} textbf{66}(5) (2013), 621--629, doi{10.1016/j.camwa.2013.05.004}.
bibitem{Frac2}
A.~A.~Kilbas, H.~M.~Srivastava, and J.~J.~Trujillo,
emph{Theory and Applications of Fractional Differential Equations}.
Elsevier, 2006.
DOI: href{https://doi.org/10.1016/S0304-0208(06)80001-0}{10.1016/S0304-0208(06)80001-0}.
bibitem{hpm7}
J.-L.~Li,
Adomian’s decomposition method and homotopy perturbation method in solving nonlinear equations,
emph{Journal of Computational and Applied Mathematics} textbf{228}(1) (2009), 168--173, doi{10.1016/j.cam.2008.09.034}.
bibitem{Luchko2013}
Y.~Luchko,
Fractional wave equation and damped waves,
emph{Journal of Mathematical Physics} textbf{54}(3) (2013), 031505, doi{10.1063/1.4795741}.
bibitem{Mainardi2010}
F.~Mainardi,
emph{Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models}.
Imperial College Press, 2010.
DOI: href{https://doi.org/10.1142/p614}{10.1142/p614}.
bibitem{Pandey2017}
R.~K.~Pandey and H.~K.~Mishra,
Numerical simulation for solution of space--time fractional telegraph equations with local fractional derivatives via HAFSTM,
emph{New Astronomy} textbf{56} (2017), 95--104, doi{10.1016/j.newast.2017.06.009}.
bibitem{Podlubny1999}
I.~Podlubny,
emph{Fractional Differential Equations},
Academic Press, San Diego, (1999), 1--340.
DOI: href{https://doi.org/10.1016/S0076-5392(99)80003-3}{10.1016/S0076-5392(99)80003-3}.
bibitem{Pskhu2020}
A.~Pskhu and S.~Rekhviashvili,
Fractional Diffusion–Wave Equation with Application in Electrodynamics,
emph{Mathematics} textbf{8}(11) (2020), 1--18, doi{10.3390/math8111993}.
bibitem{Rihan2013}
F.~A.~Rihan,
Numerical Modeling of Fractional-Order Biological Systems,
emph{Abstract and Applied Analysis} textbf{2013} (2013), 1--11, doi{10.1155/2013/475016}.
bibitem{Scalas2000}
E.~Scalas, R.~Gorenflo, and F.~Mainardi,
Fractional calculus and continuous-time finance,
emph{Physica A} textbf{284}(1--4) (2000), 376--384, doi{10.1016/S0378-4371(00)00255-7}.
bibitem{Sekar2015}
S.~Sekar and A.~Sakthivel,
Numerical investigation of linear first-order fuzzy differential equations using He's homotopy perturbation method,
emph{Preprint} (2015).
bibitem{Shah2020}
K.~Shah, F.~Jarad, and T.~Abdeljawad,
Stable numerical results to a class of time--space fractional partial differential equations via spectral method,
emph{Journal of Advanced Research} textbf{24} (2020), 423--433, doi{10.1016/j.jare.2020.05.022}.
bibitem{Shalangwa2025}
A.~A.~Shalangwa, D.~John, M.~Cornelius, I.~Zubairu, and E.~Kessel,
Numerical solution of integral equations using homotopy perturbation method and series solution method,
emph{BIMA Journal} textbf{9}(1a) (2025), 1--10, doi{10.64290/bima.v9i1a.893}.
bibitem{Tapaswini2013}
S.~Tapaswini and S.~Chakraverty,
Numerical solution of (n)-th order fuzzy linear differential equations by homotopy perturbation method,
emph{International Journal of Computer Applications} textbf{64}(13) (2013), 1--6, doi{10.5120/10636-5376}.
bibitem{Tarasov2011}
V.~E.~Tarasov,
emph{Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media}.
Springer, 2011.
DOI: href{https://doi.org/10.1007/978-3-642-14003-7}{10.1007/978-3-642-14003-7}.
bibitem{Wazwaz2009}
A.-M.~Wazwaz,
emph{Partial Differential Equations and Solitary Waves Theory},
Springer, Berlin, (2009), 1--590, doi{10.1007/978-3-540-88162-8}.
bibitem{Yang2011}
Q.~Yang, T.~J.~Moroney, K.~Burrage, I.~Turner, and F.~Liu,
Novel numerical methods for time--space fractional reaction diffusion equations in two dimensions,
emph{ANZIAM Journal} textbf{52} (2011), 1--15, doi{10.21914/anziamj.v52i0.3791}.
bibitem{Yousif2016}
M.~A.~Yousif, B.~A.~Mahmood, K.~K.~Ali, and H.~F.~Ismael,
Numerical simulation using the homotopy perturbation method for a thin liquid film over an unsteady stretching sheet,
emph{International Journal of Pure and Applied Mathematics} textbf{107}(2) (2016), 189--200, doi{10.12732/IJPAM.V107I2.1}.
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