Analytical Dynamics and Wave Evolution in a Time-Fractional Neural Transmission Model via Invariant Analysis

Anjali Kwatra; Vivek Sangwan; Rajesh Gupta

doi:10.32323/ujma.1823541

Analytical Dynamics and Wave Evolution in a Time-Fractional Neural Transmission Model via Invariant Analysis

Abstract

The intricate propagation of bioelectrical impulses in neural tissue can be effectively modeled using fractional reaction–diffusion frameworks that capture memory-dependent effects in neuronal signal transmission. In this study, a time-fractional form of the generalized Huxley equation is examined to obtain localized wave and series solutions. By employing the Riemann–Liouville fractional operator within an analytical reduction framework, the governing equation is transformed into a fractional ordinary differential equation characterized by Erdélyi–Kober-type derivatives. The reduced equation admits an explicit power series solution, whose convergence is rigorously analyzed to ensure analytical validity. Furthermore, exact traveling-wave solutions of the model are constructed through a fractional hyperbolic function approach, resulting in diverse localized wave profiles, including single and multiple singular peak structures. Graphical representations of the localized wave and series solutions reveal rich dynamical patterns, emphasizing the influence of the fractional order on the spatial profiles and responsiveness of the model.

Keywords

Convergence analysis, Erdélyi–Kober derivative, Fractional hyperbolic function method, Power series solution, Riemann–Liouville operator, Time-fractional reaction–diffusion equation

References

E. E. Holmes, M. A. Lewis, J. E. Banks, et al., Partial differential equations in ecology: spatial interactions and population dynamics, Ecology, 75(1) (1994), 17–29. https://doi.org/10.2307/1939378
T. Roubicek, Nonlinear Partial Differential Equations with Applications, 2nd ed., Birkhauser, Basel, 2005.
D. Baleanu, K. Diethelm, E. Scalas, et al., Fractional Calculus: Models and Numerical Methods, Vol. 3, World Scientific, Singapore, 2012.
V. S. Kiryakova, Generalized Fractional Calculus and Applications, Vol. 301, Longman, Harlow, UK; John Wiley & Sons, New York, USA, Pitman Research Notes in Mathematics, 1994.
I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, Methods of Their Solution and Some of Their Applications, Academic Press, 1998.
C. Li, F. Zeng, Numerical Methods for Fractional Calculus, CRC Press, Boca Raton, 2015.
K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
A. A. Kilbas, O. I. Marichev, S. G. Samko, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Switzerland, 1993.
R. A. El-Nabulsi, Fractional functional with two occurrences of integrals and asymptotic optimal change of drift in the Black-Scholes model, Acta Math. Vietnam., 40(4) (2015), 689–703. https://doi.org/10.1007/s40306-014-0079-7
K. Y. Chong, J. G. O’Hara, Lie symmetry analysis of a fractional Black-Scholes equation, AIP Conf. Proc., 2153(1) (2019), Article ID 020007. https://doi.org/10.1063/1.5125072

R. L. Magin, T. J. Royston, Fractional-order elastic models of cartilage: A multi-scale approach, Commun. Nonlinear Sci. Numer. Simul., 15(3) (2010), 657–664. https://doi.org/10.1016/j.cnsns.2009.05.008
R. A. El-Nabulsi, A cosmology governed by a fractional differential equation and the generalized Kilbas-Saigo-Mittag-Leffler function, Int. J. Theor. Phys., 55(2) (2016), 625–635. https://doi.org/10.1007/s10773-015-2700-5
R. A. El-Nabulsi, The fractional white dwarf hydrodynamical nonlinear differential equation and emergence of quark stars, Appl. Math. Comput., 218(6) (2011), 2837–2849. https://doi.org/10.1016/j.amc.2011.08.028
R. A. El-Nabulsi, Modifications at large distances from fractional and fractal arguments, Fractals, 18(2) (2010), 185–190. https://doi.org/10.1142/S0218348X10004828
R. Herrmann, Common aspects of q-deformed Lie algebras and fractional calculus, Physica A, 389(21) (2010), 4613–4622. https://doi.org/10.1016/j.physa.2010.07.004
T. M. Michelitsch, B. A. Collet, A. P. Riascos, et al., A fractional generalization of the classical lattice dynamics approach, Chaos Solitons Fractals, 92 (2016), 43–50. https://doi.org/10.1016/j.chaos.2016.09.009
M. F. Silva, J. A. Tenreiro Machado, A. M. Lopes, Fractional order control of a hexapod robot, Nonlinear Dyn., 38(1–4) (2004), 417–433. https://doi.org/10.1007/s11071-004-3770-8
R. A. El-Nabulsi, Fractional quantum Euler–Cauchy equation in the Schrödinger picture, complexified harmonic oscillators and emergence ofcomplexified Lagrangian and Hamiltonian dynamics, Mod. Phys. Lett. B, 23(28) (2009), 3369–3386. https://doi.org/10.1142/S0217984909021387
Y. Zhang, C. Papelis, Particle-tracking simulation of fractional diffusion-reaction processes, Phys. Rev. E, 84(6) (2011), Article ID 066704. https://doi.org/10.1103/PhysRevE.84.066704
P. Agarwal, M. Jleli, M. Tomar, Certain Hermite-Hadamard type inequalities via generalized k-fractional integrals, J. Inequal. Appl., 2017 (2017), Article ID 55. https://doi.org/10.1186/s13660-017-1318-y
G. Jumarie, Fractional partial differential equations and modified Riemann-Liouville derivative: New methods for solution, J. Appl. Math. Comput., 24(1–2) (2007), 31–48. https://doi.org/10.1007/BF02832299
M. M. Meerschaert, H. P. Scheffler, C. Tadjeran, Finite difference methods for two-dimensional fractional dispersion equation, J. Comput. Phys., 211(1) (2006), 249–261. https://doi.org/10.1016/j.jcp.2005.05.017
S. Momani, Z. Odibat, Numerical comparison of methods for solving linear differential equations of fractional order, Chaos Solitons Fractals, 31(5) (2007), 1248–1255. https://doi.org/10.1016/j.chaos.2005.10.068
S. Momani, Z. Odibat, Homotopy perturbation method for nonlinear partial differential equations of fractional order, Phys. Lett. A, 365(5–6) (2007), 345–350. https://doi.org/10.1016/j.physleta.2007.01.046
V. Daftardar-Gejji, H. Jafari, Adomian decomposition: A tool for solving a system of fractional differential equations, J. Math. Anal. Appl., 301(2) (2005), 508–518. https://doi.org/10.1016/j.jmaa.2004.07.039
S. Zhang, H. Q. Zhang, Fractional sub-equation method and its applications to nonlinear fractional PDEs, Phys. Lett. A, 375(7) (2011), 1069–1073. https://doi.org/10.1016/j.physleta.2011.01.029
R. K. Gazizov, A. A. Kasatkin, Construction of exact solutions for fractional order differential equations by the invariant subspace method, Comput. Math. Appl., 66(5) (2013), 576–584. https://doi.org/10.1016/j.camwa.2013.05.006
E. Buckwar, Y. Luchko, Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations, J. Math. Anal. Appl., 227(1) (1998), 81–97. https://doi.org/10.1006/jmaa.1998.6078
Z. Odibat, S. Momani, A generalized differential transform method for linear partial differential equations of fractional order, Appl. Math. Lett., 21(2) (2008), 194–199. https://doi.org/10.1016/j.aml.2007.02.022
R. L. Bagley, P. J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol., 27(3) (1983), 201–210. https://doi.org/10.1122/1.549724
L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, 2014.
G. Bluman, S. Anco, Symmetry and Integration Methods for Differential Equations, Vol. 154, Springer, New York, 2002.
M. Lakshmanan, P. Kaliappan, Lie transformations, nonlinear evolution equations, and Painleve forms, J. Math. Phys., 24(4) (1983), 795–806. https://doi.org/10.1063/1.525752
P. J. Olver, Applications of Lie Groups to Differential Equations, Vol. 107, 2nd ed., Springer, New York, 1993.
R. K. Gazizov, A. A. Kasatkin, S. Yu. Lukashchuk, Symmetry properties of fractional diffusion equations, Phys. Scr., 2009(T136) (2009), Article ID 014016. https://doi.org/10.1088/0031-8949/2009/T136/014016
G. W. Wang, T. Z. Xu, Invariant analysis and exact solutions of nonlinear time fractional Sharma–Tasso–Olver equation by Lie group analysis, Nonlinear Dyn., 76(1) (2014), 571–580. https://doi.org/10.1007/s11071-013-1150-y
Q. Huang, R. Zhdanov, Symmetries and exact solutions of the time fractional Harry-Dym equation with Riemann–Liouville derivative, Physica A, 409 (2014), 110–118. https://doi.org/10.1016/j.physa.2014.04.043
R. Sahadevan, T. Bakkyaraj, Invariant analysis of time fractional generalized Burgers and Korteweg–de Vries equations, J. Math. Anal. Appl., 393(2) (2012), 341–347. https://doi.org/10.1016/j.jmaa.2012.04.006
T. Bakkyaraj, R. Sahadevan, Group formalism of Lie transformations to time-fractional partial differential equations, Pramana J. Phys., 85(5) (2015), 849–860. https://doi.org/10.1007/s12043-015-1103-8
E. H. El Kinani, A. Ouhadan, Lie symmetry analysis of some time fractional partial differential equations, Int. J. Mod. Phys.: Conf. Ser., 38 (2015), Article ID 1560075. https://doi.org/10.1142/S2010194515600757
T. Bakkyaraj, R. Sahadevan, Invariant analysis of nonlinear fractional ordinary differential equations with Riemann–Liouville fractional derivative, Nonlinear Dyn., 80(1–2) (2015), 447–455. https://doi.org/10.1007/s11071-014-1881-4
G. F. Jefferson, J. Carminati, FracSym: Automated symbolic computation of Lie symmetries of fractional differential equations, Comput. Phys. Commun., 185(1) (2014), 430–441. https://doi.org/10.1016/j.cpc.2013.09.019
K. Singla, R. K. Gupta, Symmetry classification and exact solutions of (3+1)-dimensional fractional nonlinear incompressible non-hydrostatic coupled Boussinesq equations, J. Math. Phys., 62(1) (2021), Article ID 011504. https://doi.org/10.1063/5.0012954
K. Singla, R. K. Gupta, On invariant analysis of some time fractional nonlinear systems of partial differential equations. I, J. Math. Phys., 57(10) (2016), Article ID 101504. https://doi.org/10.1063/1.4964937
K. Singla, R. K. Gupta, On invariant analysis of space-time fractional nonlinear systems of partial differential equations. II, J. Math. Phys., 58(5) (2017), Article ID 051503. https://doi.org/10.1063/1.4982804
A. L. Hodgkin, A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol., 117(4) (1952), 500–544. https://doi.org/10.1113/jphysiol.1952.sp004764
A. L. Hodgkin, A. F. Huxley, B. Katz, Measurement of current-voltage relations in the membrane of the giant axon of Loligo, J. Physiol., 116(4) (1952), 424–448. https://doi.org/10.1113/jphysiol.1952.sp004716
V. Sangwan, B. V. R. Kumar, S. V. S. S. N. V. G. K. Murthy, et al., Three-step Taylor Galerkin method for singularly perturbed generalized Hodgkin–Huxley equation, Int. J. Model. Simul. Sci. Comput., 1(2) (2010), 257–276. https://doi.org/10.1142/S1793962310000183
A. Kumari, V. K. Kukreja, Robust septic Hermite collocation technique for singularly perturbed generalized Hodgkin–Huxley equation, Int. J. Comput. Math., 99(5) (2022), 909–923. https://doi.org/10.1080/00207160.2021.1939317
M. Mascagni, The backward Euler method for numerical solution of the Hodgkin–Huxley equations of nerve conduction, SIAM J. Numer. Anal., 27(4) (1990), 941–962. https://doi.org/10.1137/0727054
I. Hashim, M. S. M. Noorani, B. Batiha, A note on the Adomian decomposition method for the generalized Huxley equation, Appl. Math. Comput., 181(2) (2006), 1439–1445. https://doi.org/10.1016/j.amc.2006.03.011
B. Batiha, M. S. M. Noorani, I. Hashim, Numerical simulation of the generalized Huxley equation by He’s variational iteration method, Appl. Math. Comput., 186(2) (2007), 1322–1325. https://doi.org/10.1016/j.amc.2006.07.166
A. A. Aderogba, M. Chapwanya, Positive and bounded nonstandard finite difference scheme for the Hodgkin–Huxley equations, Japan J. Ind. Appl. Math., 35(2) (2018), 773–785. https://doi.org/10.1007/s13160-018-0302-3
K. Xiao, Y. Chen, J. Shen, Asymptotic behavior of non-autonomous fractional stochastic lattice FitzHugh–Nagumo system driven by linear mixed white noise, J. Math. Phys., 65(10) (2024), Article ID 102708. https://doi.org/10.1063/5.0195332
Shahid, M. Abbas, E. Kwessi, Nonstandard nearly exact analysis of the FitzHugh–Nagumo model, Symmetry, 16(5) (2024), Article ID 585. https://doi.org/10.3390/sym16050585
S. Sahoo, S. S. Ray, On the conservation laws and invariant analysis for time-fractional coupled FitzHugh–Nagumo equations using the Lie symmetry analysis, Eur. Phys. J. Plus, 134(2) (2019), Article ID 83. https://doi.org/10.1140/epjp/i2019-12440-6
T. Jamal, A. Jhangeer, M. Z. Hussain, An anatomization of pulse solitons of nerve impulse model via phase portraits, chaos and sensitivity analysis, Chin. J. Phys., 87 (2024), 496–509. https://doi.org/10.1016/j.cjph.2023.12.005
M. Elbrolosy, A. Elmandouh, Painlev´e analysis, Lie symmetry and bifurcation for the dynamical model of radial dislocations in microtubules, Nonlinear Dyn., 113(11) (2025), 13699–13714. https://doi.org/10.1007/s11071-025-11084-5
W. A. Faridi, M. Iqbal, H. A. Mahmoud, An invariant optical soliton wave study on integrable model: A Riccati–Bernoulli sub-optimal differential equation approach, Int. J. Theor. Phys., 64(3) (2025), Article ID 71. https://doi.org/10.1007/s10773-025-05929-3
H. H. Sherief, A. M. A. El-Sayed, S. H. Behiry, et al., Using fractional derivatives to generalize the Hodgkin–Huxley model, In Fractional dynamics and control, Springer, 2011, pp. 275–282. https://doi.org/10.1007/978-1-4614-0457-6 23
R. Magin, Fractional calculus in bioengineering, part 1, Crit. Rev. Biomed. Eng., 32(1) (2004). https://doi.org/10.1615/CritRevBiomedEng.v32.i1.10
A. S. Nuseir, A. Al-Hasson, Power series solution for nonlinear system of partial differential equations, Appl. Math. Sci., 6(104) (2012), 5147–5159.
F. W. J. Olver, D. W. Lozier, R. F. Boisvert, et al., NIST Handbook of Mathematical Functions, Cambridge University Press, New York, 2010.
G. Jumarie, Table of some basic fractional calculus formulae derived from a modified Riemann–Liouville derivative for non-differentiable functions, Appl. Math. Lett., 22(3) (2009), 378–385. https://doi.org/10.1016/j.aml.2008.06.003
W. Malfliet, W. Hereman, The tanh method: I. exact solutions of nonlinear evolution and wave equations, Phys. Scr., 54(6) (1996), 563. https://doi.org/10.1088/0031-8949/54/6/003
A. M. Wazwaz, New solitary wave solutions to the modified forms of Degasperis–Procesi and Camassa–Holm equations, Appl. Math. Comput., 186(1) (2007), 130–141. https://doi.org/10.1016/j.amc.2006.07.092

Details

Primary Language

English

Subjects

Numerical and Computational Mathematics (Other), Partial Differential Equations

Journal Section

Research Article

Authors

Anjali Kwatra
0000-0003-4145-1169
India

Vivek Sangwan ^*
0000-0002-3857-8479
India

Rajesh Gupta
0000-0002-7055-0839
India

Early Pub Date

May 22, 2026

Publication Date

-

Submission Date

November 17, 2025

Acceptance Date

April 8, 2026

Published in Issue

Year 2026 Number: Advanced Online Publication

DOI

https://doi.org/10.32323/ujma.1823541

IZ

https://izlik.org/JA66JD69JK

APA

Kwatra, A., Sangwan, V., & Gupta, R. (2026). Analytical Dynamics and Wave Evolution in a Time-Fractional Neural Transmission Model via Invariant Analysis. Universal Journal of Mathematics and Applications, Advanced Online Publication. https://doi.org/10.32323/ujma.1823541

AMA

1.Kwatra A, Sangwan V, Gupta R. Analytical Dynamics and Wave Evolution in a Time-Fractional Neural Transmission Model via Invariant Analysis. Univ. J. Math. Appl. 2026;(Advanced Online Publication). doi:10.32323/ujma.1823541

Chicago

Kwatra, Anjali, Vivek Sangwan, and Rajesh Gupta. 2026. “Analytical Dynamics and Wave Evolution in a Time-Fractional Neural Transmission Model via Invariant Analysis”. Universal Journal of Mathematics and Applications, no. Advanced Online Publication. https://doi.org/10.32323/ujma.1823541.

EndNote

Kwatra A, Sangwan V, Gupta R (May 1, 2026) Analytical Dynamics and Wave Evolution in a Time-Fractional Neural Transmission Model via Invariant Analysis. Universal Journal of Mathematics and Applications Advanced Online Publication

IEEE

[1]A. Kwatra, V. Sangwan, and R. Gupta, “Analytical Dynamics and Wave Evolution in a Time-Fractional Neural Transmission Model via Invariant Analysis”, Univ. J. Math. Appl., no. Advanced Online Publication, May 2026, doi: 10.32323/ujma.1823541.

ISNAD

Kwatra, Anjali - Sangwan, Vivek - Gupta, Rajesh. “Analytical Dynamics and Wave Evolution in a Time-Fractional Neural Transmission Model via Invariant Analysis”. Universal Journal of Mathematics and Applications. Advanced Online Publication (May 1, 2026). https://doi.org/10.32323/ujma.1823541.

JAMA

1.Kwatra A, Sangwan V, Gupta R. Analytical Dynamics and Wave Evolution in a Time-Fractional Neural Transmission Model via Invariant Analysis. Univ. J. Math. Appl. 2026. doi:10.32323/ujma.1823541.

MLA

Kwatra, Anjali, et al. “Analytical Dynamics and Wave Evolution in a Time-Fractional Neural Transmission Model via Invariant Analysis”. Universal Journal of Mathematics and Applications, no. Advanced Online Publication, May 2026, doi:10.32323/ujma.1823541.

Vancouver

1.Anjali Kwatra, Vivek Sangwan, Rajesh Gupta. Analytical Dynamics and Wave Evolution in a Time-Fractional Neural Transmission Model via Invariant Analysis. Univ. J. Math. Appl. 2026 May 1;(Advanced Online Publication). doi:10.32323/ujma.1823541