Adomian Decomposition and Variational Iteration Methods in the Context of Partial Differential Equations

Khadeejah  James Audu; Lawrence Linus; Yahaya Yusuph Amuda; Sıkırulaı Akande

doi:10.38088/jise.1563076

Research Article

Adomian Decomposition and Variational Iteration Methods in the Context of Partial Differential Equations

Year 2025, Volume: 9 Issue: 1, 39 - 53

Khadeejah James Audu , Lawrence Linus , Yahaya Yusuph Amuda , Sıkırulaı Akande

Abstract

Partial Differential Equations (PDE) help model problems in science and engineering, given their abilities to capture complex phenomena compared to Ordinary Differential Equations. This paper aims to investigate two semi-analytical techniques called the Adomian Decomposition Method (ADM) and the Variational Iteration Method (VIM), how these methods can be used in practice, and to make a comparative study of the two methods for solving linear and nonlinear PDEs. The efficiency of ADM and VIM is assessed by comparing their errors relative to the exact solutions of the examined numerical experiments. The results obtained from the numerical experiments revealed that ADM proved to be a more efficient and accurate method for solving PDEs than VIM.

Keywords

Partial Differential Equation, Semi-analytical methods, Decomposition Method, Variational Iteration Method, Comparative Analysis

References

1. Abbas, T., Haq, E. U., Hassan, Q. M. U., Majeed, A. and Ahmad, B. (2022). Application of Adomian Decomposition, Variational Iteration, and Series Solution Methods to Analysis of Integral Differential Equations. Journal of Science and Arts, 22(3), 655–662.
2. Adeniji, A. A., Mogbojuri, O. A., Kekana, M. C. and Fadugba, S. E. (2023). Numerical solution of rotavirus model using Runge-Kutta-Fehlberg method, differential transform method and Laplace Adomian decomposition method. Alexandria Engineering Journal, 82, 323–329.
3. Albert, B., Titus, R. and Michael, O. (2022). Variational iteration method for solving coupled nonlinear system of Klein-Gordon equations. International Journal of Statistics and Applied Mathematics, 7(1), 107–111.
4. Ahmad, N. and Ansari, A. (2023). Numerical Solution for nonlinear Volterra-Fredholm Integro-Differential Equations using Adomian and Modified Adomian Decomposition Method. Journal of Science and Arts, 23, 625-638.
5. Aibinu, M. O. and Moyo, S. (2023). Solutions of fractional differential equations by using a blend of variational iteration method with Sumudu transform and application to price adjustment equations. Partial Differential Equations in Applied Mathematics, 8, 100590.
6. Al-Sharai, A. A.A., Wedad M. and Journal, A. U. (2024). Exact Solution of Linear and Nonlinear System of Partial Differential Equations by Double Aboodh-Shehu and Adomian Decomposition Method. Albaydha University Journal, 5(5), 273-286.
7. Alao, S., Akinboro, F. S.and Oderinu, R. A. (2021). Numerical Solution of Integro-differential Equation using Adomian Decomposition Method and Variational Iteration Method. Journal of Mathematics, 10(4), 18-22
8. Altaie, H. O. (2020). Comparison the solutions for some kinds of differential equations using iterative methods. Journal of Interdisciplinary Mathematics, 24(5), 1113–1118.
9. Wazwaz, A. M. (2009). Partial Differential Equations and Solitary Wave, Springer.
10. Audu, K. J. and Ameh, S. (2023). Implementation of New Iterative Method for Solving Partial Differential Equations. International Physical Science Conference, 237-242
11. Audu, K. J. (2024). Numerical Solutions of Higher Order Differential Equations via New Iterative Method. Proceedings of International Conference on Mathematical Modelling Optimization and Analysis of Disease Dynamics, 285-294.
12. Audu, K. J. and Babatunde, O. (2024). A Comparative Analysis of Two Analytic Approaches in Solving Systems of First-Order Differential Equations, Scientific Journal of Mehmet Akif Ersoy University, 7(1), 8-24.
13. Audu, K. J., Tiamiyu, A. T., Akpabio, J. N., Ahmad, H. and Adebayo, M. (2024). Numerical Assessment of Some Semi-Analyical Techniques for Solving a Fractional Order Leptospirosis Model, Malaysian Journal of Science, 43(3), 68-85.
14. Bhadgaonkar, V. N. and Sontakke, B. R. (2021). Exact solution of space-time fractional partial differential equations by adomian decomposition method. Journal of Advances in Mathematics and Computer Science, 75–87.
15. Fatima, N. and Haque, S. (2023). Variational iteration method for solving nonlinear partial differential equations in fluid dynamics. MDPI AG, 201-217.
16. Hamoud, A. and Ghadle, K. (2021). A comparative study of VIM and ADM for solving volterra-fredholm integro-differential equations. MDPI AG, 116-129.
17. Liberty, E., Kubugha, W. B. and Onengiyeofori, A. D. (2023). Comparison of Adomian decomposition method with differential transformation method for unsteady MHD flow and heat transfer over a stretching/shrinking permeable sheet with ohmic heating. African Journal of Mathematics and Statistics Studies, 6(3), 70–85.
18. Li, W. and Pang, Y. (2020). Application of Adomian decomposition method to nonlinear systems. Journal of Advances in Difference Equations, 67(2020), 1-17.
19. M. Elzaki, T. and E. Ahmed, S. (2021). Solution of nonlinear partial differential equations by mixture Adomian decomposition method and sumudu transform. Advances in the Solution of Nonlinear Differential Equations. 16, 97-108.
20. Ozdogan N. (2024). Application of Mohand Transform. Journal of Informative Science and Engineering, 8(1), 18-24.
21. Oswaldo, G., G. (2022). Solution of nonlinear partial differential Equations by Adomian decomposition method. Studies in Engineering and Exact Sciences, 3(1), 61–78.
22. S, K. and Mulimani, M. (2022). Comparative study of Adomian decomposition method and Clique polynomial method. Partial Differential Equations in Applied Mathematics, 6, 100454.
23. Sahu, S. K., Derke, S. and Duressa, A. (2022). Solutions of 1D hyperbolic quasi-linear partial differential equations by variational iteration method. Journal of Science and Arts, 22(3), 432-445.
24. Singh, P. (2020). Accelerated adomian decomposition method for the system of nonlinear equations. Journal of Physics: Conference Series, 1531(1), 012084.
25. Shihab, M. A., Taha, W. M., Hameed, R. A., Jameel, A. and Sulaiman, I. M. (2023). Implementation of variational iteration method for various types of linear and nonlinear partial differential equations. International Journal of Electrical and Computer Engineering, 13(2), 21-31.
26. Yindoula, J. B. (2022). A comparative study of Adomian Decomposition Method and Variational Iteration Method.Universal Journal of Mathematics and Mathematical Sciences, 17, 1–30.
27. Falade, K. I, Tiamiyu A.T (2020). Numerical Solution of Partial Differential Equations with Fractional Variable Coefficients Using New Iterative Method (NIM), I.J. Mathematical Sciences and Computing, 3, 12-21.
28. Falade, K. I. (2022). Algorithm analytic-numeric solution for nonlinear gas dynamic partial differential equation, Engineering and Applied Science Letter, 5(2), 32-40.
29. Baghdadi, S. K. A., and Ahammad, N. A. (2024). A Comparative Study of Adomian Decomposition Method with Variational Iteration Method for Solving Linear and Nonlinear Differential Equations. Journal of Applied Mathematics and Physics, 12(8), 2789-2819.
30. Sinha, V. K., and Maroju, P. (2023). New Development of Variational Iteration Method Using Quasilinearization Method for Solving Nonlinear Problems. Mathematics, 11(4), 935.
31. Rilwan, M. A., and Akanbi, M. A. (2022). A Comparison of the Variational Iteration Method and Adomian Decomposition Methods in Solving the Problem of Squeezing Flow Between Two Circular Disks. The Journals of the Nigerian Association of Mathematical Physics, 64, 91-98.
32. Dehraj, S., Maitlo, A. A., Siyal, W. A., Memon, M., Arain, L. N., Arain, L., and Umrani, K. (2023). A Comparison of the Adomian Decomposition Method and Variational Iteration Method for a Two-Dimensional Nonlinear Wave Equation. Journal of Hunan University Natural Sciences, 50(2), 28-37.

Year 2025, Volume: 9 Issue: 1, 39 - 53

Khadeejah James Audu , Lawrence Linus , Yahaya Yusuph Amuda , Sıkırulaı Akande

Abstract

References

1. Abbas, T., Haq, E. U., Hassan, Q. M. U., Majeed, A. and Ahmad, B. (2022). Application of Adomian Decomposition, Variational Iteration, and Series Solution Methods to Analysis of Integral Differential Equations. Journal of Science and Arts, 22(3), 655–662.
2. Adeniji, A. A., Mogbojuri, O. A., Kekana, M. C. and Fadugba, S. E. (2023). Numerical solution of rotavirus model using Runge-Kutta-Fehlberg method, differential transform method and Laplace Adomian decomposition method. Alexandria Engineering Journal, 82, 323–329.
3. Albert, B., Titus, R. and Michael, O. (2022). Variational iteration method for solving coupled nonlinear system of Klein-Gordon equations. International Journal of Statistics and Applied Mathematics, 7(1), 107–111.
4. Ahmad, N. and Ansari, A. (2023). Numerical Solution for nonlinear Volterra-Fredholm Integro-Differential Equations using Adomian and Modified Adomian Decomposition Method. Journal of Science and Arts, 23, 625-638.
5. Aibinu, M. O. and Moyo, S. (2023). Solutions of fractional differential equations by using a blend of variational iteration method with Sumudu transform and application to price adjustment equations. Partial Differential Equations in Applied Mathematics, 8, 100590.
6. Al-Sharai, A. A.A., Wedad M. and Journal, A. U. (2024). Exact Solution of Linear and Nonlinear System of Partial Differential Equations by Double Aboodh-Shehu and Adomian Decomposition Method. Albaydha University Journal, 5(5), 273-286.
7. Alao, S., Akinboro, F. S.and Oderinu, R. A. (2021). Numerical Solution of Integro-differential Equation using Adomian Decomposition Method and Variational Iteration Method. Journal of Mathematics, 10(4), 18-22
8. Altaie, H. O. (2020). Comparison the solutions for some kinds of differential equations using iterative methods. Journal of Interdisciplinary Mathematics, 24(5), 1113–1118.
9. Wazwaz, A. M. (2009). Partial Differential Equations and Solitary Wave, Springer.
10. Audu, K. J. and Ameh, S. (2023). Implementation of New Iterative Method for Solving Partial Differential Equations. International Physical Science Conference, 237-242
11. Audu, K. J. (2024). Numerical Solutions of Higher Order Differential Equations via New Iterative Method. Proceedings of International Conference on Mathematical Modelling Optimization and Analysis of Disease Dynamics, 285-294.
12. Audu, K. J. and Babatunde, O. (2024). A Comparative Analysis of Two Analytic Approaches in Solving Systems of First-Order Differential Equations, Scientific Journal of Mehmet Akif Ersoy University, 7(1), 8-24.
13. Audu, K. J., Tiamiyu, A. T., Akpabio, J. N., Ahmad, H. and Adebayo, M. (2024). Numerical Assessment of Some Semi-Analyical Techniques for Solving a Fractional Order Leptospirosis Model, Malaysian Journal of Science, 43(3), 68-85.
14. Bhadgaonkar, V. N. and Sontakke, B. R. (2021). Exact solution of space-time fractional partial differential equations by adomian decomposition method. Journal of Advances in Mathematics and Computer Science, 75–87.
15. Fatima, N. and Haque, S. (2023). Variational iteration method for solving nonlinear partial differential equations in fluid dynamics. MDPI AG, 201-217.
16. Hamoud, A. and Ghadle, K. (2021). A comparative study of VIM and ADM for solving volterra-fredholm integro-differential equations. MDPI AG, 116-129.
17. Liberty, E., Kubugha, W. B. and Onengiyeofori, A. D. (2023). Comparison of Adomian decomposition method with differential transformation method for unsteady MHD flow and heat transfer over a stretching/shrinking permeable sheet with ohmic heating. African Journal of Mathematics and Statistics Studies, 6(3), 70–85.
18. Li, W. and Pang, Y. (2020). Application of Adomian decomposition method to nonlinear systems. Journal of Advances in Difference Equations, 67(2020), 1-17.
19. M. Elzaki, T. and E. Ahmed, S. (2021). Solution of nonlinear partial differential equations by mixture Adomian decomposition method and sumudu transform. Advances in the Solution of Nonlinear Differential Equations. 16, 97-108.
20. Ozdogan N. (2024). Application of Mohand Transform. Journal of Informative Science and Engineering, 8(1), 18-24.
21. Oswaldo, G., G. (2022). Solution of nonlinear partial differential Equations by Adomian decomposition method. Studies in Engineering and Exact Sciences, 3(1), 61–78.
22. S, K. and Mulimani, M. (2022). Comparative study of Adomian decomposition method and Clique polynomial method. Partial Differential Equations in Applied Mathematics, 6, 100454.
23. Sahu, S. K., Derke, S. and Duressa, A. (2022). Solutions of 1D hyperbolic quasi-linear partial differential equations by variational iteration method. Journal of Science and Arts, 22(3), 432-445.
24. Singh, P. (2020). Accelerated adomian decomposition method for the system of nonlinear equations. Journal of Physics: Conference Series, 1531(1), 012084.
25. Shihab, M. A., Taha, W. M., Hameed, R. A., Jameel, A. and Sulaiman, I. M. (2023). Implementation of variational iteration method for various types of linear and nonlinear partial differential equations. International Journal of Electrical and Computer Engineering, 13(2), 21-31.
26. Yindoula, J. B. (2022). A comparative study of Adomian Decomposition Method and Variational Iteration Method.Universal Journal of Mathematics and Mathematical Sciences, 17, 1–30.
27. Falade, K. I, Tiamiyu A.T (2020). Numerical Solution of Partial Differential Equations with Fractional Variable Coefficients Using New Iterative Method (NIM), I.J. Mathematical Sciences and Computing, 3, 12-21.
28. Falade, K. I. (2022). Algorithm analytic-numeric solution for nonlinear gas dynamic partial differential equation, Engineering and Applied Science Letter, 5(2), 32-40.
29. Baghdadi, S. K. A., and Ahammad, N. A. (2024). A Comparative Study of Adomian Decomposition Method with Variational Iteration Method for Solving Linear and Nonlinear Differential Equations. Journal of Applied Mathematics and Physics, 12(8), 2789-2819.
30. Sinha, V. K., and Maroju, P. (2023). New Development of Variational Iteration Method Using Quasilinearization Method for Solving Nonlinear Problems. Mathematics, 11(4), 935.
31. Rilwan, M. A., and Akanbi, M. A. (2022). A Comparison of the Variational Iteration Method and Adomian Decomposition Methods in Solving the Problem of Squeezing Flow Between Two Circular Disks. The Journals of the Nigerian Association of Mathematical Physics, 64, 91-98.
32. Dehraj, S., Maitlo, A. A., Siyal, W. A., Memon, M., Arain, L. N., Arain, L., and Umrani, K. (2023). A Comparison of the Adomian Decomposition Method and Variational Iteration Method for a Two-Dimensional Nonlinear Wave Equation. Journal of Hunan University Natural Sciences, 50(2), 28-37.

There are 32 citations in total.

Details

Primary Language	English
Subjects	Numerical Analysis
Journal Section	Research Articles
Authors	Khadeejah James Audu 0000-0002-6986-3491 Lawrence Linus 0009-0005-3316-9384 Yahaya Yusuph Amuda 0000-0002-3410-9173 Sıkırulaı Akande 0000-0002-8216-744X
Early Pub Date	May 2, 2025
Publication Date
Submission Date	October 17, 2024
Acceptance Date	February 14, 2025
Published in Issue	Year 2025Volume: 9 Issue: 1

Cite

APA	Audu, K. . J., Linus, L., Amuda, Y. Y., Akande, S. (2025). Adomian Decomposition and Variational Iteration Methods in the Context of Partial Differential Equations. Journal of Innovative Science and Engineering, 9(1), 39-53. https://doi.org/10.38088/jise.1563076
AMA	Audu KJ, Linus L, Amuda YY, Akande S. Adomian Decomposition and Variational Iteration Methods in the Context of Partial Differential Equations. JISE. May 2025;9(1):39-53. doi:10.38088/jise.1563076
Chicago	Audu, Khadeejah James, Lawrence Linus, Yahaya Yusuph Amuda, and Sıkırulaı Akande. “Adomian Decomposition and Variational Iteration Methods in the Context of Partial Differential Equations”. Journal of Innovative Science and Engineering 9, no. 1 (May 2025): 39-53. https://doi.org/10.38088/jise.1563076.
EndNote	Audu KJ, Linus L, Amuda YY, Akande S (May 1, 2025) Adomian Decomposition and Variational Iteration Methods in the Context of Partial Differential Equations. Journal of Innovative Science and Engineering 9 1 39–53.
IEEE	K. . J. Audu, L. Linus, Y. Y. Amuda, and S. Akande, “Adomian Decomposition and Variational Iteration Methods in the Context of Partial Differential Equations”, JISE, vol. 9, no. 1, pp. 39–53, 2025, doi: 10.38088/jise.1563076.
ISNAD	Audu, Khadeejah James et al. “Adomian Decomposition and Variational Iteration Methods in the Context of Partial Differential Equations”. Journal of Innovative Science and Engineering 9/1 (May 2025), 39-53. https://doi.org/10.38088/jise.1563076.
JAMA	Audu KJ, Linus L, Amuda YY, Akande S. Adomian Decomposition and Variational Iteration Methods in the Context of Partial Differential Equations. JISE. 2025;9:39–53.
MLA	Audu, Khadeejah James et al. “Adomian Decomposition and Variational Iteration Methods in the Context of Partial Differential Equations”. Journal of Innovative Science and Engineering, vol. 9, no. 1, 2025, pp. 39-53, doi:10.38088/jise.1563076.
Vancouver	Audu KJ, Linus L, Amuda YY, Akande S. Adomian Decomposition and Variational Iteration Methods in the Context of Partial Differential Equations. JISE. 2025;9(1):39-53.

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