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Semi-Analytical Solutions of the Fractional-Order Burgers’ and Diffusion Equations Via the Homotopy Perturbation Method

Precious Chikaedum Agina, Doris Ijeoma Ezeora, Kingsley Kelechi Ibeh

Abstract

This paper presents semi-analytical solutions of the fractional-order Burgers’ equation and the fractional diffusion equation using the Homotopy Perturbation Method . The fractional derivatives are considered in the Caputo sense. The solutions are obtained in rapidly convergent series forms. The effectiveness and accuracy of the method are demonstrated through comparison with classical cases. Numerical simulations for different fractional orders illustrate the impact of memory effects on solution behavior.

Keywords

Fractional partial derivativeBurgers’ equationDiffusion equationHomotopy perturbation method.

References

[1] Taite, G., & DiNapoli, J. (2025). Perspectives on Mathematical Modeling Education: Conceptions and Research. Encyclopedia, 5(3), 138. https://doi.org/10.3390/encyclopedia5030138 [2] Silva, C.J., Chyba, M. & Cuellar G.H. (2024). Editorial: Mathematical modeling and optimization for real life phenomena. Front. Appl. Math. Stat. 10:1389061. doi: 10.3389/fams.2024.1389061 [3] Abdou, M. (2021). Tourism Demand Modelling and Forecasting: A Review of Literature. African Journal of Hospitality, Tourism and Leisure, 10(4), 1370-1393. [4] Agina, P.C., Agina, E.K., & Chukwuma, E.I. (2026). A Mathematical Modeling Approach to Tourism Destination Analysis in Nigeria Using the Tourism Attractiveness Index. Journal of Hotel Management and Tourism Research, 7(2), 33-43. [5] Singh, V.P. & Singh, S. (2025). Fractional Calculus and Its Applications: A Comprehensive Review. International Journal on Science and Technology, 16(1), 1-7. [6] Korus, P. & Valdes, J.E.N. (2024). Recent Advances in Fractional Calculus. Axioms, 13(5), 310. https://doi.org/10.3390/axioms13050310 [7] Agina, P.C., Ibeh, K.K., Chukwuma, E.I., Ezeora, D.I., Otugene, V.B. (2026). A Fractional- Order Nonlinear Model for The Transmission Dynamics of Ebola Virus Disease with Quarantine, Vaccination, and Condom Use. International Journal of Computer Science and Mathematical Theory, 12(3), 220-237. [8] Volos, C. (2025). Introductory Chapter: Fractional Calculus – From Theory to Applications. In Fractional Calculus - From Theory to Applications. IntechOpen. https://doi.org/10.5772/intechopen.1010158 [9] Boulaaras, S., Jan, R., & Pham, V.T. (2023). Recent advancement of fractional calculus and its applications in physical systems. The European Physical Journal Special Topics, 232, 2347-2350. [10] Otugene, V.B., Agina, P.C., Ezeora, D.I., & Ibeh, K.K. (2026). Modeling Inflation Dynamics with Atangana-Baleanu Fractional Derivatives: A Memory-Driven Approach to Consumer Price Index Forecasting. Iconic Research and Engineering Journals, 9(9), 2000–2022. [11] Roy, P. (2025). The Rise of Fractional Calculus: Novel Applications in Engineering and Biological Systems. Recent Trends in Mathematics, 1(2), 7-11. [12] Agina, P.C., Chukwuma, E.I., Chuka-Obidiegwu, B.E., Otugene, V.B., Ibeh, K.K., & Mbah, G.C.E. (2026). Local and Global Stability Analysis of the Disease-Free Equilibrium in a Fractional-Order Ebola Virus Transmission Model. International Journal of Applied Science and Mathematical Theory, 12(3), 30-48. [13] Otugene, V.B., Mbah, G.C.E, Owhenagbo, P., Idoko, P.I., Adedoyin, J.D., Enyejo, L.A., & Agina, P.C. (2024). Mathematical Analysis of Hepatitis B Virus Transmission Dynamics in the Absence of Therapy with Atangana-Baleanu Fractional-order SPQWXY Model. Journal of Advances in Mathematics and Computer Science, 39(11), 1-28 [14] Otugene, V.B., Agina, P.C., Edogbanya, H., & Digwo, D.C. (2022). Existence and uniqueness analysis of Atangana-Baleanu fractional order network model of Noro virus. International Journal of Mathematical Analysis and Modelling, 5(2), 63–75. [15] Vázquez, J.L. (2017). The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion. In: Bonforte, M., Grillo, G. Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions. Lecture Notes in Mathematics, 2186, 205-278. Springer, Cham. https://doi.org/10.1007/978-3-319-61494-6_5 [16] Laglands, T. (2010). An Introduction to Fractional Diffusion. Complex Physical, Biophysical and Econophysical Systems - Proceedings of the 22nd Canberra International Physics Summer School. https://doi.org/10.1142/9789814277327_0002 [17] Alshehry, A.S. & Shah, R. (2025). Exploring Fractional Damped Burgers’ Equation: A Comparative Analysis of Analytical Methods. Fractal Fract, 9(2), 107. https://doi.org/10.3390/fractalfract9020107 [18] Bonkile, M.P., Awasthi, A., & Lakshmi, C., Mukundan, V., & Aswin, V.S. (2018). A systematic literature review of Burgers’ equation with recent advances. Pramana - J Phys, 90(69), https://doi.org/10.1007/s12043-018-1559-4 [19] Balachandran, K. (2023). Fractional Partial Differential Equations. In: An Introduction to Fractional Differential Equations. Industrial and Applied Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-99-6080-4_5 [20] Murillo-Arcila, M., Peris, A. & Vargas-Moreno, A. (2025). Dynamics of the Caputo fractional derivative. Fract Calc Appl Anal 28, 1717–1731. https://doi.org/10.1007/s13540-025- 00430-4 [21] Nchama, G.A.M., Lau-Alfonso, L.D., Mercias, A.M.L., & Ricard, R.M. (2020). Properties of the Caputo-Fabrizio Fractional Derivative. Applied Mathematics & Information Sciences, 14(5), 761-769. [22] Atangana, A. (2018). Chapter 5 - Fractional Operators and Their Applications. In: Fractional Operators with Constant and Variable Order with Application to Geo-Hydrology. Academic Press, 79-112. https://doi.org/10.1016/B978-0-12-809670-3.00005-9 [23] Podlubny, I. (1999). Chapter 2 - Fractional Derivatives and Integrals. In: Fractional Differential Equations, 198, 41-119. Elsevier. https://doi.org/10.1016/S0076- 5392(99)80021-6 [24] Agina, P.C., Otugene, V. B., Adedoyin, J. D., Chukwuma, E.I., Chuka-Obidiegwu, B. E., Ezeora, D. I., & Mbah, G. C. E. (2025). Derivation of the basic reproduction number using the next generation matrix for a non-linear Ebola Virus transmission model. International Journal of Mathematical Analysis and Modelling, 8(2), 555-573. [25] Mitchell, C., and Kribs, C. (2017). A Comparison of Methods for Calculating the Basic Repro- ductive Number for Periodic Epidemic Systems. Bulletin of Mathematical Biology, 79, 1846– 1869. https://doi.org/10.1007/s11538-017-0309-y [26] Agina, P.C., Ezeora, D.I., Ibeh, K.K., Chukwuma, E.I. (2026). An Optimal Control Study of Ebola Transmission Incorporating Enlightenment and Drug Effectiveness. International Journal of Computer Science and Mathematical Theory, 12(3), 238-252. [27] Salkuyeh, D.K. (2007). Generalized Jacobi and Gauss-Seidel Methods for Solving Linear System of Equations. Numer. Math. J. Chinese Univ. (English Ser.), 16(2), 164-170. [28] Kong, Q., Siauw, T., & Bayen, A.M. (2021). Chapter 14 - Linear Algebra and Systems of Linear Equations. In: Python Programming and Numerical Methods, 235-263. Academic Press. https://doi.org/10.1016/B978-0-12-819549-9.00024-5 [29] Agina, P.C. & Echezona, G.N. (2022). Comparative analysis of the Gauss-Seidel and Jacobi methods for solving linear systems. COOU Journal of Physical Sciences, 5(1), 60-79. [30] He, J. H. (1999). Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering, 178(3–4), 257–262. https://doi.org/10.1016/S0045-7825(99)00018-3 [31] Momani, S., & Odibat, Z. (2007). Homotopy perturbation method for nonlinear partial differential equations of fractional order. Physics Letters A, 365(5–6), 345–350. https://doi.org/10.1016/j.physleta.2007.01.046 [32] Marinca, V. & Herisanu, N. (2012). The Optimal Homotopy Perturbation Method. Nonlinear Dynamical Systems in Engineering, Springer, Berlin, Heidelberg; 211-157. https://doi.org/10.1007/978-3-642-22735-6_7 [33] Agina, P.C., Otugene, V.B., Ibeh K.K., Ezeora, D.I. (2026). Application of the Homotopy Perturbation Method to Selected Nonlinear and Fractional Differential Equations with Comparative Analysis. Iconic Research and Engineering Journals, 9(10), 1760-1773. [34] He, J. H. (2000). A coupling method of homotopy technique and perturbation technique for nonlinear problems. International Journal of Non-Linear Mechanics, 35(1), 37–43. https://doi.org/10.1016/S0020-7462(98)00085-7 [35] Marinca, V. & Herisanu, N. (2012). The Optimal Homotopy Perturbation Method. Nonlinear Dynamical Systems in Engineering, Springer, Berlin, Heidelberg; 211-157. https://doi.org/10.1007/978-3-642-22735-6_7 [36] Agina, P.C., Chuka-Obidiegwu, B.E., Chukwuma, E.I., Ibeh, K.K., Otugene, V.B., & Mbah, G.C.E. (2026). An Approximate Solution of a Fractional-order Epidemic Model using the Homotopy Perturbation Method. International Journal of Applied Science and Mathematical Theory, 12(3), 58-77. [37] He, J. H. (2006). Some asymptotic methods for strongly nonlinear equations. International Journal of Modern Physics B, 20(10), 1141–1199. https://doi.org/10.1142/S0217979206033796 [38] He, J.H. (2003). Homotopy Perturbation Method: a new nonlinear analytical technique. Applied Mathematics and Computation, 135, 73-79.

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