MUET logo

Semi-Analytical Solutions of the Space–Time Fractional Wave Equation for Copper Droplets in Cavitating Flows

Authors

  • Shaukat Iqbal

    School of Systems and Technology, University of Management and Technology, Lahore, 54770, Pakistan
    Author

  • Muhammad Abid

    Department of Mechanical Engineering, Faculty of Engineering, Islamic University of Madinah, Madinah, 42351, Saudi Arabia, KSA
    Author

  • Umair Sabi Ullah

    Department of Computer Science, University of Edinburgh, Glasgow, UK
    Author

  • Hafiz Abdul Wajid

    Department of Electrical Engineering, Faculty of Engineering, Islamic University of Madinah, Madinah, 42351, Saudi Arabia, KSA
    Author

  • Tareq Manzoor

    Energy Research Centre, COMSATS University Islamabad, Lahore Campus, Pakistan
    Author

  • Ayaz Akbar

    Institute of Advanced Interdisciplinary Technology, Shenzhen MSU-BIT University, Shenzhen, China
    Author

  • Sanaulah Manzoor

    School of Computing, Glasgow Caledonian University, Glasgow, UK
    Author

DOI:

https://doi.org/10.22581/0808

Keywords:

Fractional Wave Equation, Adomian Decomposition Method (ADM), Caputo Derivative, Cavitating Flow, Copper Droplets, Semi-analytical Solution

Abstract

In this paper, series solutions of the space–time fractional wave equation modeling copper droplets in cavitating flows are constructed using an extended Adomian Decomposition Method (ADM). The proposed method employs time- and space-fractional derivatives of the Caputo type without linearization or discretization, making it straightforward to implement. Linear and nonlinear wave equation problems are solved to test the accuracy and stability of the method; at integer order (λ = τ = 2), absolute errors on the order of 10−11 (linear case) and 10−4 (nonlinear case) are achieved, respectively. However, the error grows slowly with time in both cases. At fractional orders 1.25, 1.5, and 1.75, the solutions remain smooth and stable.

Downloads

References

[1] S. Roy, J. M. Coyne, J. A. Novak, and M. A. Edwards, “Flow-induced failure mechanisms of copper pipe in potable water systems,” Corrosion Reviews, vol. 36, no. 5, pp. 449–481, 2018.

[2] Z. Benzarti, N. Arrousse, R. Serra, S. Cruz, A. Bastos, J. Tedim, and S. Carvalho, “Copper corrosion mechanisms, influencing factors, and mitigation strategies for water circuits of heat exchangers: Critical review and current advances,” Corrosion Reviews, vol. 43, no. 4, pp. 429–455, 2025.

[3] J. He, C. Li, and Y. Guo, “Modeling transient vaporous cavitating flow in pipelines by a two-phase homogeneous flow model,” Processes, vol. 13, no. 11, 2025. Art. no. 3510.

[4] J. W. Wright and B. S. Mitchell, “Reactive cavitation erosion as a technique for production of functionalized copper hydroxychloride nanomaterials,” Journal of Physics Communications, vol. 4, no. 5, 2020. Art. no. 051002.

[5] B. Guo, X. Pu, and F. Huang, Fractional Partial Differential Equations and Their Numerical Solutions. World Scientific, 2015.

[6] R. Herrmann, Fractional Calculus: An Introduction for Physicists. River Edge, NJ: World Scientific, 2011.

[7] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, 2006.

[8] H. A. Wajid and S. Ayub, “An optimally blended finite-spectral element scheme with minimal dispersion for Maxwell equations,” Journal of Computational Physics, vol. 231, no. 24, pp. 8176–8187, 2012.

[9] I. Podlubny, Fractional Differential Equations. Academic Press, 1999.

[10] G. Adomian, “A review of the decomposition method in applied mathematics,” Journal of Mathematical Analysis and Applications, vol. 135, no. 2, pp. 501–544, 1988.

[11] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method. Springer Science & Business Media, 2013.

[12] G. Adomian and R. Rach, “Inversion of nonlinear stochastic operators,” Journal of Mathematical Analysis and Applications, vol. 91, no. 1, pp. 39– 46, 1983.

[13] Y. Cherruault, G. Adomian, K. Abbaoui, and R. Rach, “Further remarks on convergence of decomposition method,” International Journal of Bio-Medical Computing, vol. 38, no. 1, pp. 89–93, 1995.

[14] G. Adomian and R. Rach, “Modified Adomian polynomials,” Mathematical and Computer Modelling, vol. 24, no. 11, pp. 39–46, 1996.

[15] J. S. Duan, R. Rach, D. Baleanu, and A. M. Wazwaz, “A review of the Adomian decomposition method and its applications to fractional differential equations,” Communications in Fractional Calculus, vol. 3, no. 2, pp. 73–99, 2012.

[16] G. Adomian, “Convergent series solution of nonlinear equations,” Journal of Computational and Applied Mathematics, vol. 11, no. 2, pp. 225–230, 1984.

[17] G. Adomian, “On the convergence region for decomposition solutions,” Journal of Computational and Applied Mathematics, vol. 11, no. 3, pp. 379– 380, 1984.

[18] G. Adomian, “Nonlinear stochastic dynamical systems in physical problems,” Journal of Mathematical Analysis and Applications, vol. 111, no. 1, pp. 105–113, 1985.

[19] G. Adomian, “Decomposition solution for Duffing and Van der Pol oscillators,” International Journal of Mathematics and Mathematical Sciences, vol. 9, no. 4, pp. 731–732, 1986.

[20] J. S. Duan, R. Rach, and A. M. Wazwaz, “A reliable algorithm for positive solutions of nonlinear boundary value problems by the multistage Adomian decomposition method,” Open Engineering, vol. 5, no. 1, pp. 1–14, 2015.

[21] J. S. Duan, R. Rach, and A. M. Wazwaz, “Simulation of the eigenvalue problem for tapered rotating beams by the modified decomposition method,” International Journal for Computational Methods in Engineering Science and Mechanics, vol. 23, no. 1, pp. 20–28, 2022.

[22] R. Rach, J. S. Duan, and A. M. Wazwaz, “Simulation of large deflections of a flexible cantilever beam fabricated from functionally graded materials by the Adomian decomposition method,” International Journal of Dynamical Systems and Differential Equations, vol. 10, no. 4, pp. 287–298, 2020.

[23] A. M. Wazwaz, R. Rach, and J. S. Duan, “Adomian decomposition method for solving the Volterra integral form of the Lane–Emden equations with initial values and boundary conditions,” Applied Mathematics and Computation, vol. 219, no. 10, pp. 5004–5019, 2013.

[24] R. Rach, “A bibliography of the theory and applications of the Adomian decomposition method, 1961– 2011,” Kybernetes, vol. 41, no. 7–8, pp. 1087– 1148, 2012.

[25] V. Daftardar-Gejji and H. Jafari, “Adomian decomposition: A tool for solving a system of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 301, no. 2, pp. 508– 518, 2005.

[26] N. T. Shawagfeh, “Analytical approximate solutions for nonlinear fractional differential equations,” Applied Mathematics and Computation, vol. 131, no. 2–3, pp. 517–529, 2002.

[27] D. B. Dhaigude and G. A. Birajdar, “Numerical solution of system of fractional partial differential equations by discrete Adomian decomposition method,” Journal of Fractional Calculus and Applications, vol. 3, no. 12, pp. 1–11, 2012.

[28] D. B. Dhaigude, G. A. Birajdar, and V. R. Nikam, “Adomian decomposition method for fractional Benjamin–Bona–Mahony–Burger’s equations,” International Journal of Applied Mathematics and Mechanics, vol. 8, no. 12, pp. 42–51, 2012.

[29] V. N. Bhadgaonkar and B. R. Sontakke, “Exact solution of space–time fractional partial differential equations by Adomian decomposition method,” Journal of Advances in Mathematics and Computer Science, vol. 36, no. 6, pp. 75–87, 2021.

[30] R. Rach, A. M. Wazwaz, and J. S. Duan, “A reliable modification of the Adomian decomposition method for higher-order nonlinear differential equations,” Kybernetes, vol. 42, no. 2, pp. 282–308, 2013.

[31] N. Khodabakhshi, S. Mansour Vaezpour, and D. Baleanu, “Numerical solutions of the initial value problem for fractional differential equations by modification of the Adomian decomposition method,” Fractional Calculus and Applied Analysis, vol. 17, no. 2, pp. 382–400, 2014.

[32] D. B. Dhaigude and V. N. Bhadgaonkar, “Analytical solution of nonlinear nonhomogeneous space and time fractional physical models by improved Adomian decomposition method,” Punjab University Journal of Mathematics, vol. 54, no. 2, pp. 89–104, 2022.

[33] Z. Odibat and S. Momani, “A generalized differential transform method for linear partial differential equations of fractional order,” Applied Mathematics Letters, vol. 21, no. 2, pp. 194–199, 2008.

[34] G. Kear, B. D. Barker, and F. C. Walsh, “Electrochemical corrosion of unalloyed copper in chloride media—A critical review,” Corrosion Science, vol. 46, no. 1, pp. 109–135, 2004.

Published

2026-10-01

Issue

Vol. 45 No. 4 (2026): October Issue

Section

Articles

License

Copyright (c) 2026 Mehran University Research Journal of Engineering and Technology

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

How to Cite

Semi-Analytical Solutions of the Space–Time Fractional Wave Equation for Copper Droplets in Cavitating Flows. (2026). Mehran University Research Journal of Engineering and Technology, 45(4), 156-161. https://doi.org/10.22581/0808

Similar Articles

1-10 of 64

You may also start an advanced similarity search for this article.