A comparative study of approximate solutions to solving second-order linear complex partial differential equations

Obaid, Raad A.; Kadhim, Murtadha A.; Abed, Alaa M.

doi:10.47974/JIM-2535

Open Access Research Article

A comparative study of approximate solutions to solving second-order linear complex partial differential equations

^*Raad A. ObaidCorresponding authorraada.altharwanee@student.uokufa.edu.iqGeneral Directorate of Education Babylon, Ministry of EducationBabylon, 51001, IraqView full profile → , Murtadha A. Kadhimmurtadha.abduljawad.iku@atu.edu.iqAl-Furat Al-Awsat Technical UniversityKufa, Najaf, 54001, IraqView full profile → , Alaa M. AbedAlaamuhsien3030@gmail.comGeneral Directorate of Education Diwaniyah, Ministry of EducationDiwaniyah, 58001, Iraq0009-0004-1222-0219View full profile →

* Corresponding author · click or hover a name for details

pp. 1475–1486Vol. 29Issue 5-BMay 2026DOI: 10.47974/JIM-2535XML

Received:: 01 Nov 2025
Published Online:: 27 Jun 2026
Article type:: Research Article
Language:: EN
Article no.:: JIM-2535
Pages:: 1475–1486

Abstract

This study addresses linear complex partial differential equations (LCPDEs), which constitute vital mathematical frameworks across diverse fields in physics and applied science. We utilize the modified Variational Iteration Method (VIM) and the Homotopy Perturbation Method (HPM), techniques ideally suited for generalized LCPDEs. The efficacy and precision of both methods were rigorously evaluated on various problem sets. The findings substantiate the method’s ability to generate accurate and reliable approximate solutions, achieving excellent concordance with established analytical results. This confirms the competence of the modified VIM and HPM in effectively resolving LCPDEs.

Keywords

Variational iteration method Homotopy perturbation method Complex partial differential equation Approximate solution

Subject Classifications

Primary 93A30Secondary 49K15

References

[1] A. Al-Haboobi, G. A. Al-Juaifri, and A. H. Al-Muslimawi, “Numerical study of Newtonian laminar Flow around circular and square cylinders,” Results in Control and Optimization, vol. 14, p. 100328 (2024).
[2] A. Al-Haboobi and A. H. Al-Muslimawi, “Novel algorithm for compressible Newtonian axisymmetric thermal flow,” International Journal of Modern Physics C, vol. 35, no. 03, p. 2450025 (2024).
[3] G. A. Al-Juaifri and A. J. Harfash, “Analysis of a nonlinear reaction-diffusion system of the Fitzhugh-Nagumo type with Robin boundary conditions,” Ricerche di Matematica, vol. 72, no. 1, pp. 335–357 (2023).
[4] R. S. Teppawar, R. N. Ingle, and R. A. Muneshwar, “Analysis of system of fractional partial differential equations using Laplace reduced differential transform with decomposition method,” Journal of Statistics and Management Systems, vol. 27, no. 8, pp. 1577-1594 (2024), doi: 10.47974/JSMS-1112.
[5] V. V. Mane, T. A. Wani, P. Ghutke, V. Patil, M. Kurulekar, and G. M. Reddy, “Nonlinear partial differential equations in quantum fluid dynamics,” Journal of Interdisciplinary Mathematics, vol. 28, no. 6, pp. 2181-2188 (2025), doi: 10.47974/JIM-2359.
[6] A. H. A. Khitam and G. A. Al-Juaifri, “Strong Solutions of Brusselator System,” Malaysian Journal of Mathematical Sciences, vol. 18, no. 3 (2024).
[7] A. Al-Haboobi, I. A. Fadhel, A. A. Sharhan, and A. H. Al-Muslimawi, “Computational simulation of compressible Newtonian flow past circular and square bluff bodies,” Ocean Systems Engineering, vol. 14, no. 4, pp. 315–330 (2024), doi: 10.12989/ose.2024.14.4.315.
[8] A. J. Harfash, G. A. Al-Juaifri, W. K. Ghafil, and A. J. Harfash, “Slip boundary conditions effect on bidispersive convection with local thermal non-equilibrium: Significant findings,” Chinese Journal of Physics, vol. 89, pp. 144–159 (2024).
[9] H. R. Hashim, F. Luca, H. B. Shelash, and A. A. Shukur, “Generalized lucas graphs,” Afrika Matematika, vol. 34, no. 1, p. 10 (2023).
[10] S. A. Al-Ameedee and A. J. Obaid, “Sandwich results of meromorphic multivalent functions defined by multiplier transform,” Journal of Interdisciplinary Mathematics, vol. 26, no. 4, pp. 651–658 (2023), doi: 10.47974/JIM-1484.
[11] H. R. Hashim, “Solutions of the Markoff Equation in Tribonacci Numbers,” Proceedings of the Croatian Academy of Sciences and Arts, Mathematical Sciences, vol. 27, no. 555, pp. 71–79 (2023), doi: 10.21857/yq32ohx069.
[12] H. R. Hashim, “On the Solutions of 2x+ 2y= z2 in the Fibonacci and Lucas Numbers,” Journal of Prime Research in Mathematics, vol. 19, no. 1, pp. 27–33 (2023).
[13] A. N. Jasim and A. A. Najim, “Edges deletion problem of hypercube graphs for some n,” Discrete Mathematics, Algorithms and Applications, vol. 17, no. 3 (2025), doi: 10.1142/S1793830924500459.
[14] A. N. Jasim and A. A. Najim, “Solving Edges Deletion Problem of Complete Graphs,” Baghdad Science Journal, vol. 21, no. 12, p. 45 (2024).
[15] L. C. Evans, “Partial differential equations,” in Graduate Studies in Mathematics, 2nd ed., vol. 19, Providence, RI, USA: American Mathematical Society, p. 749 (2010).
[16] K. A. Abd and G. A. Al-Juaifri, “Analysis of a cross-diffusion Brusselator system,” AIP Conference Proceedings, vol. 3264, no. 1 (Mar. 2025), doi: 10.1063/5.0258982.
[17] A. S. Mohammed, I. S. Ahmed, and S. H. Asaad, “Study of the rings in which each element express as the sum of an idempotent and pure,” International Journal of Nonlinear Analysis and Applications, vol. 12, no. 2, pp. 1719–1724 (2021), doi: 10.22075/IJNAA.2021.5311.
[18] H. F. Abbas, H. H. Ebrahim, and A. Al-Fayadh, “Soft P *–field and some of its properties,” Journal of Discrete Mathematical Sciences and Cryptography, vol. 26, no. 7, pp. 1875–1882 (2023).
[19] Y. Keskin and G. Oturanc, “Reduced differential transform method for partial differential equations,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 6, pp. 741–750 (2009).
[20] Y. Guo, X. Cao, B. Liu, and M. Gao, “Solving partial differential equations using deep learning and physical constraints,” Applied Sciences, vol. 10, no. 17, p. 5917 (2020).
[21] B. Deconinck, T. Trogdon, and V. Vasan, “The method of Fokas for solving linear partial differential equations,” SIAM Rev., vol. 56, no. 1, pp. 159–186 (2014).
[22] Z. T. Abdulqader, N. E. Arif, and R. B. Esmaeel, “JX–L–open and some of their generalizations,” Journal of Interdisciplinary Mathematics, vol. 28, no. 3, pp. 1125–1130 (2025).
[23] T. Taylor, F. Palacios, K. Duraisamy, and J. Alonso, “Towards a hybrid adjoint approach for arbitrarily complex partial differential equations,” in 42nd AIAA Fluid Dynamics Conference and Exhibit, p. 3342 (2012).
[24] V. K. Patel, S. Singh, and V. K. Singh, “Numerical wavelets scheme to complex partial differential equation arising from Morlet continuous wavelet transform,” Numerical methods for partial differential equations, vol. 37, no. 2, pp. 1163–1199 (2021).
[25] S. H. Asaad, I. S. Ahmed, and H. H. Ebrahim, “On Finitely Null-additive and Finitely Weakly Null-additive Relative to the σ–ring,” Baghdad Science Journal, vol. 19, no. 5, pp. 1148–1148 (2022).
[26] M. S. Mechee, O. I. Al-Shaher, and G. A. Al-Juaifri, “Haar wavelet technique for solving fractional differential equations with an application,” AIP Conference Proceedings, no. 1, p. 030025 (Apr. 2019).
[27] J. H. He, “Variational iteration method–a kind of non-linear analytical technique: some examples,” International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 699–708 (1999).
[28] S. A. Al-Ameedee and A. J. Obaid, “New results and application of differential quasi subordinations for higher-order derivatives of meromorphic multivalent functions,” Journal of Interdisciplinary Mathematics, vol. 26, no. 4, pp. 643–650 (2023), doi: 10.47974/JIM-1483.
[29] N. Anjum and J.-H. He, “Homotopy perturbation method for N/MEMS oscillators,” Mathematical Methods in the Applied Sciences, vol. 43, no. 10, pp. 1–15 (2020), doi: 10.1002/mma.6583.

Views: 68Downloads: 81Citations: 0