Numerical Investigation of Nonlinear Parabolic Dynamical Wave Equations Using Modified Variational Iteration Algorithm-II
DOI:
https://doi.org/10.30871/jaic.v10i1.12107Keywords:
Modified variational iteration algorithm-II, Allen-Cahn equation, Newell-Whitehead equation, Nonlinear PDEs, Numerical simulationAbstract
In this study, the Modified Variational Iteration Algorithm-II (MVIA-II) is implemented as a robust numerical scheme for solving nonlinear Parabolic partial differential equations. The study focuses on the implementation of an auxiliary parameter h into the correction functional to control the convergence region of the approximate series solution. To validate the efficiency of this semi-numerical approach, two fundamental models arising in mathematical physics and biology are investigated: The Allen-Cahn equation and the Newell-Whitehead equation. The results are compared with exact analytical solutions and other existing numerical methods. The error analysis demonstrates that the proposed algorithm yields high accuracy with minimal computational overhead, making it a promising tool for simulating nonlinear dynamical wave phenomena.
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[1] A. R. Seadawy and K. El-Rashidy, "Dispersive solitary wave solutions of Kadomtsev-Petviashvili and modified Kadomtsev-Petviashvili dynamical equations in unmagnetized dust plasma," Results Phys., vol. 8, pp. 1216–1222, 2018.
[2] A. M. Wazwaz, Partial Differential Equations and Solitary Waves Theory. Springer Science & Business Media, 2010.
[3] S. M. Allen and J. W. Cahn, "A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening," Acta Metallurgica, vol. 27, no. 6, pp. 1085–1095, 1979.
[4] A. C. Newell and J. A. Whitehead, "Finite bandwidth, finite amplitude convection," J. Fluid Mech., vol. 38, no. 2, pp. 279–303, 1969.
[5] M. Modanli, S. T. Abdulazeez, and A. M. Husien, "A new approach for pseudo hyperbolic partial differential equations with nonlocal conditions using Laplace Adomian decomposition method," Appl. Math. J. Chinese Univ., vol. 39, no. 4, pp. 750–758, 2024.
[6] S. T. Abdulazeez, M. Modanli, and A. M. Husien, "Numerical scheme methods for solving nonlinear pseudo-hyperbolic partial differential equations," J. Appl. Math. Comput. Mech., vol. 21, no. 4, 2022.
[7] A. M. Wazwaz, "The Hirota’s bilinear method and the tanh–coth method for multiple-soliton solutions of the Sawada–Kotera–Kadomtsev–Petviashvili equation," Appl. Math. Comput., vol. 200, no. 1, pp. 160–166, 2008.
[8] H. M. Patil and R. Maniyeri, "Finite difference method based analysis of bio-heat transfer in human breast cyst," Therm. Sci. Eng. Prog., vol. 10, pp. 42–47, 2019.
[9] R. Costa, J. M. Nóbrega, S. Clain, G. J. Machado, and R. Loubère, "Very high‐order accurate finite volume scheme for the convection‐diffusion equation with general boundary conditions on arbitrary curved boundaries," Int. J. Numer. Methods Eng., vol. 117, no. 2, pp. 188–220, 2019.
[10] M. Modanli, S. T. Abdulazeez, and H. Goksu, "Solutions of fractional order time-varying linear dynamical systems using the residual power series method," Honam Math. J., vol. 45, no. 4, pp. 619–628, 2023.
[11] A. Kumar and A. Bhardwaj, "A local meshless method for time fractional nonlinear diffusion wave equation," Numer. Algorithms, vol. 85, no. 4, pp. 1311–1334, 2020.
[12] J. H. He, "Variational iteration method—a kind of non-linear analytical technique: some examples," Int. J. Non-Linear Mech., vol. 34, no. 4, pp. 699–708, 1999.
[13] H. Ahmad, A. R. Seadawy, T. A. Khan, and P. Thounthong, "Analytic approximate solutions for some nonlinear parabolic dynamical wave equations," J. Taibah Univ. Sci., vol. 14, no. 1, pp. 346–358, 2020.
[14] H. Ahmad and T. A. Khan, "Variational iteration Algorithm-I with an auxiliary parameter for wave-like vibration equations," J. Low Freq. Noise Vib. Active Control, vol. 38, no. 3–4, pp. 1113–1124, 2019.
[15] M. Nadeem, F. Li, and H. Ahmad, "Modified Laplace variational iteration method for solving fourth-order parabolic partial differential equation with variable coefficients," Comput. Math. Appl., vol. 78, no. 6, pp. 2052–2062, 2019.
[16] J. H. He and C. Sun, "A variational principle for a thin film equation," J. Math. Chem., vol. 57, no. 9, pp. 2075–2081, 2019.
[17] W. Zahra, "Trigonometric B-spline collocation method for solving PHI-four and Allen–Cahn equations," Mediterr. J. Math., vol. 14, no. 3, Art. no. 122, 2017.
[18] A. Prakash and M. Kumar, "He’s variational iteration method for the solution of nonlinear Newell–Whitehead–Segel equation," J. Appl. Anal. Comput., vol. 6, no. 3, pp. 738–748, 2016.
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