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Numerical Solution of Bratu-Type Initial Value Problems by Aboodh Adomian Decomposition Method

Year 2023, Volume: 20 Issue: 2, 64 - 75, 01.11.2023

Abstract

This study presents an efficient method for solving Bratu equations with an initial value problem. The procedure is based on the use of Aboodh transform and Adomian decomposition method. Adomian polynomials for the index n replace the nonlinear term. The dependent variable components are also replaced in the recurrence relation by their corresponding Aboodh transform components of the same index. Therefore, the nonlinear problem is solved directly without any linearization or discretization.
Examples are presented to show the effectiveness and validity of this method. The derived results are compared with the existing exact solution.

References

  • [1] A. Wazwaz, “The Successive differentiation method for solving Bratu equation and Bratu-type equations,” Rom. Journ. Phys., vol. 61, no. 5-6, pp. 774-783, 2016.
  • [2] L. H. Koudahoun, J. Akande, D. K. Adjai, Y. F. Kpomahou, and M.D. Monsia, “On the general solution to Bratu and generalized Bratu equations,” Journal of Mathematics and Statistics, vol. 14, pp. 193-200, 2018.
  • [3] M. Abukhaled, S. Khuri, and A. Sayfy, “Spline-based numerical treatments of Bratu-type equations,” Palestine Journal of Mathematics, vol. 1, pp. 63-70, 2012.
  • [4] M. I. Syam, and A. Hamdan, “An efficient method for solving Bratu equations,” Applied Mathematics and Computation, vol. 176, pp. 704- 713, 2006.
  • [5] S. H. Kashkari, and S. Z. Abbas, “Solution of initial value problem of Bratu-type equation using modifications of homotopy perturbation method,” International Journal of Computer Applications, vol. 162, no. 5, 2017.
  • [6] A. Batiha, “Numerical solution of Bratu-type equations by the variational iteration method,” Journal of Mathematics and Statistics, vol. 39, no. 1, pp. 23-29, 2010.
  • [7] N. Das, R. Singh, A. Wazwaz, and J. Kumar, “An algorithm based on the variational iteration technique for the Bratu-type and the Lane-Emden problems,” Journal of Mathematical Chemistry, vol. 54, pp. 527-551, 2015.
  • [8] X. Feng, Y. He, and J. Meng, “Application of homotopy perturbation method to the Bratu-type equation,” Journal of Julius Schauder Center, vol. 31, pp. 243-252, 2008.
  • [9] K. H. Sedeeg, and M. M. A. Mahgoub, “Aboodh transform homotopy perturbation method for solving system of nonlinear partial differential equations,” Mathematical Theory and Modeling, vol. 6, no. 8, 2016.
  • [10] J. Ahmad, and J. Taric, “Application of Aboodh differential transform method on some higher order problems,” Journal of Science and Arts, no. 1, no. 42, pp. 5-18, 2018.
  • [11] K. S. Aboodh, R. A. Farah, I. A. Almardy, and A. K. Osman, “Solving delay differential equations by Aboodh transformation method,” International Academy of Science, Engineering and Technology, vol. 7, pp. 55-64, 2018.
  • [12] S. Qureshi, M. S. Chandio, A. A. Shaikh, and R. A. Memon, “On the Use of Aboodh Transform for Solving Noninteger Order Dynamical Systems,” Sindh University Research Journal (Social Series), vol. 51, no. 1, pp. 53-58, 2019.
  • [13] D. Baleanu, S. Qureshi, A. Soomro, and A. A. Shaikh, “Variable Stepsize Construction of a Two-step Optimized Hybrid Block Method with a Relative Stability,” Open Physics, vol. 20, no. 1, pp. 1112-1126, 2022.
  • [14] S. Qureshi, A. Soomro, E. Hincal, J. R. Lee, C. Park, and M. S. Osman, “An Efficient Variable Stepsize Rational Method for Stiff, Singular and Singularly Perturbed Problems,” Alexandria University Journal, vol. 61, pp. 10953-10963, 2022.
  • [15] S. Qureshi, A. Soomro, S. E. Fadugba, and E. Hincal, “A New Family of L-stable Block Methods with Relative Measure of Stability,” International Journal of Applied nonlinear Science, vol. 3, no. 3, pp. 197-222, 2022.
Year 2023, Volume: 20 Issue: 2, 64 - 75, 01.11.2023

Abstract

References

  • [1] A. Wazwaz, “The Successive differentiation method for solving Bratu equation and Bratu-type equations,” Rom. Journ. Phys., vol. 61, no. 5-6, pp. 774-783, 2016.
  • [2] L. H. Koudahoun, J. Akande, D. K. Adjai, Y. F. Kpomahou, and M.D. Monsia, “On the general solution to Bratu and generalized Bratu equations,” Journal of Mathematics and Statistics, vol. 14, pp. 193-200, 2018.
  • [3] M. Abukhaled, S. Khuri, and A. Sayfy, “Spline-based numerical treatments of Bratu-type equations,” Palestine Journal of Mathematics, vol. 1, pp. 63-70, 2012.
  • [4] M. I. Syam, and A. Hamdan, “An efficient method for solving Bratu equations,” Applied Mathematics and Computation, vol. 176, pp. 704- 713, 2006.
  • [5] S. H. Kashkari, and S. Z. Abbas, “Solution of initial value problem of Bratu-type equation using modifications of homotopy perturbation method,” International Journal of Computer Applications, vol. 162, no. 5, 2017.
  • [6] A. Batiha, “Numerical solution of Bratu-type equations by the variational iteration method,” Journal of Mathematics and Statistics, vol. 39, no. 1, pp. 23-29, 2010.
  • [7] N. Das, R. Singh, A. Wazwaz, and J. Kumar, “An algorithm based on the variational iteration technique for the Bratu-type and the Lane-Emden problems,” Journal of Mathematical Chemistry, vol. 54, pp. 527-551, 2015.
  • [8] X. Feng, Y. He, and J. Meng, “Application of homotopy perturbation method to the Bratu-type equation,” Journal of Julius Schauder Center, vol. 31, pp. 243-252, 2008.
  • [9] K. H. Sedeeg, and M. M. A. Mahgoub, “Aboodh transform homotopy perturbation method for solving system of nonlinear partial differential equations,” Mathematical Theory and Modeling, vol. 6, no. 8, 2016.
  • [10] J. Ahmad, and J. Taric, “Application of Aboodh differential transform method on some higher order problems,” Journal of Science and Arts, no. 1, no. 42, pp. 5-18, 2018.
  • [11] K. S. Aboodh, R. A. Farah, I. A. Almardy, and A. K. Osman, “Solving delay differential equations by Aboodh transformation method,” International Academy of Science, Engineering and Technology, vol. 7, pp. 55-64, 2018.
  • [12] S. Qureshi, M. S. Chandio, A. A. Shaikh, and R. A. Memon, “On the Use of Aboodh Transform for Solving Noninteger Order Dynamical Systems,” Sindh University Research Journal (Social Series), vol. 51, no. 1, pp. 53-58, 2019.
  • [13] D. Baleanu, S. Qureshi, A. Soomro, and A. A. Shaikh, “Variable Stepsize Construction of a Two-step Optimized Hybrid Block Method with a Relative Stability,” Open Physics, vol. 20, no. 1, pp. 1112-1126, 2022.
  • [14] S. Qureshi, A. Soomro, E. Hincal, J. R. Lee, C. Park, and M. S. Osman, “An Efficient Variable Stepsize Rational Method for Stiff, Singular and Singularly Perturbed Problems,” Alexandria University Journal, vol. 61, pp. 10953-10963, 2022.
  • [15] S. Qureshi, A. Soomro, S. E. Fadugba, and E. Hincal, “A New Family of L-stable Block Methods with Relative Measure of Stability,” International Journal of Applied nonlinear Science, vol. 3, no. 3, pp. 197-222, 2022.
There are 15 citations in total.

Details

Primary Language English
Subjects Partial Differential Equations, Engineering
Journal Section Articles
Authors

Olutunde Odetunde 0000-0003-1706-4978

Abiodun Ajani This is me 0000-0001-8639-1142

Abasss Taiwo 0000-0001-6186-082X

Sefiu Onitilo This is me 0000-0002-4418-197X

Publication Date November 1, 2023
Published in Issue Year 2023 Volume: 20 Issue: 2

Cite

APA Odetunde, O., Ajani, A., Taiwo, A., Onitilo, S. (2023). Numerical Solution of Bratu-Type Initial Value Problems by Aboodh Adomian Decomposition Method. Cankaya University Journal of Science and Engineering, 20(2), 64-75.
AMA Odetunde O, Ajani A, Taiwo A, Onitilo S. Numerical Solution of Bratu-Type Initial Value Problems by Aboodh Adomian Decomposition Method. CUJSE. November 2023;20(2):64-75.
Chicago Odetunde, Olutunde, Abiodun Ajani, Abasss Taiwo, and Sefiu Onitilo. “Numerical Solution of Bratu-Type Initial Value Problems by Aboodh Adomian Decomposition Method”. Cankaya University Journal of Science and Engineering 20, no. 2 (November 2023): 64-75.
EndNote Odetunde O, Ajani A, Taiwo A, Onitilo S (November 1, 2023) Numerical Solution of Bratu-Type Initial Value Problems by Aboodh Adomian Decomposition Method. Cankaya University Journal of Science and Engineering 20 2 64–75.
IEEE O. Odetunde, A. Ajani, A. Taiwo, and S. Onitilo, “Numerical Solution of Bratu-Type Initial Value Problems by Aboodh Adomian Decomposition Method”, CUJSE, vol. 20, no. 2, pp. 64–75, 2023.
ISNAD Odetunde, Olutunde et al. “Numerical Solution of Bratu-Type Initial Value Problems by Aboodh Adomian Decomposition Method”. Cankaya University Journal of Science and Engineering 20/2 (November 2023), 64-75.
JAMA Odetunde O, Ajani A, Taiwo A, Onitilo S. Numerical Solution of Bratu-Type Initial Value Problems by Aboodh Adomian Decomposition Method. CUJSE. 2023;20:64–75.
MLA Odetunde, Olutunde et al. “Numerical Solution of Bratu-Type Initial Value Problems by Aboodh Adomian Decomposition Method”. Cankaya University Journal of Science and Engineering, vol. 20, no. 2, 2023, pp. 64-75.
Vancouver Odetunde O, Ajani A, Taiwo A, Onitilo S. Numerical Solution of Bratu-Type Initial Value Problems by Aboodh Adomian Decomposition Method. CUJSE. 2023;20(2):64-75.