Convergence and Ulam-Hyers-Rassias Stability Analyses with Error Estimation of Numerical Solutions of Bratu Type Equations
Keywords:
Bratu type equations, Picard's method, existence and uniqueness of solutions, convergence analysis, Ulam-Hyers-Rassias stability, error estimation.
Abstract
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A detailed investigation of properties and numerical solutions of Bratu type equations is performed in this article. The approximate solutions of Bratu differential equations are computed by Picard's iterative scheme. The existence, uniqueness and convergence of solutions are verified for every example. It has also been proved that Bratu type equations are Ulam-Hyers-Rassias stable. Furthermore, the derived results are compared with those already exist in the literature so as to validate that Picard's scheme estimates the actual solutions very precisely as compared to other numerical techniques. The absolute errors are also estimated and compared which verified that results obtained by Picard's method have least error.
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References
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[27] Kaihong Zhao, Generalized UH-Stability of a Nonlinear Fractional Coupling (p1, p2)- Laplacian System Concerned
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[28] Kaihong Zhao, Solvability and GUH-stability of a Nonlinear CF-Fractional Coupled Laplacian Equations, AIMS Mathematics, 8(6), (2023), 13351-13367
[29] A. Zada, S. Faisal, Y. Li, Hyers-Ulam-Rassias Stability of Nonlinear Delay Differential Equations, Journal of Nonlinear Sciences and Applications, 10, (2017), 504-510.
[30] J. Huang, S.M. Jung, Y. Li, On Hyers-Ulam Stability of Nonlinear Differential Equations, Bulletin of Korean Mathematical Society, 52,(2015), 685-697.
[31] M.N. Qarawani, Hyers-Ulam-Rassias Stability Criteria of Nonlinear Differential Equations of Lane-Emden Type, General Letters in Mathematics, 3, (2017), 57-70.
[32] X. Wang, D. Luo, Q. Zhu, Ulam-Hyers Stability of Caputo Type Fuzzy Fractional Differential Equations with Time-Delays, Chaos, Solitons & Fractals, 156, (2022), 111822.
[33] D. Luo, X. Wang, T. Caraballo, Q. Zhu , Ulam-Hyers Stability of Caputo-Type Fractional Fuzzy Stochastic Differential Equations with Delay, Commun. Nonlinear Sci.Numer. Simul., 121, (2023), 107229.
[34] A.M. Wazwaz, A First Course in Integral Equations, World Scientific Publishing Company, Singapore, 2015.
[35] B.D. Reddy, Introductory Functional Analysis With Application to Boundary Value Problems and Finite Elements, Springer Verlag Incorporation, New York, (1998).
[2] Y.M. Chu, Saif Ullah, M. Ali, G.F. Tuzzahrah, T. Munir, Numerical Investigation of Volterra Integral Equations of Second Kind using Optimal Homotopy Asymptotic Method, Applied Mathematics and Computation, 430, 2022), 127304.
[3] Saif Ullah, F. Amin, M. Ali, Numerical Investigation with Convergence and Stability Analyses of Integro-Differential Equations of Second Kind, International Journal of Computational Methods, 21(3), (2024), 2350036.
[4] A. Mohsin, A Simple Solution of Bratu Problem, Computers and Mathematics with Applications, 67, (2014), 26-33.
[5] I.M. Gelfand, Some Problems in the Theory of Quasi-linear Equations, Transaction of the American Mathematical Society, 29, (1963), 295-381.
[6] Y.Q.Wan, Q. Guo, N. Pan, Thermo Electro Hydrodynamic Model for Electrospinning Process, International Journal of Nonlinear Sciences and Numerical Simulations, 5(1), (2004), 5-8.
[7] J. Jacobsen, K. Schmitt, The Liouville-Bratu-Gelfand Problem for Radial operators, Journal of Differential Equations, 184, (2002), 283-298.
[8] H.Q. Kafri, S.A. Khuri, A Novel Approach Using Fixed Point Iterations and Green’s Functions, Computer Physics Communications, 198, (2016), 97-104.
[9] M. Abukhaled, S. Khuri, A. Sayfy, Spline-Based Numerical Treatments of Bratu-Type Equations, Palestine Journal of Mathematics, 1, (2012), 63-70.
[10] J.P. Boyd, Chebyshev Polynomial Expansions for Simultaneous Approximation of Two Branches of a Function with Application to the One-Dimensional Bratu Equation, Applied Mathematics and Computation, 142, (2003), 189-200.
[11] A.M. Wazwaz, Adomian decomposition method for a reliable treatment of the Bratu-type equations, Applied Mathematics and Computation, 166, (2005), 652-663.
[12] B.S.H. Kashkari, S.Z. Abbas, Solution of Initial Value Prpblem of Bratu Type Equation using Modifications of Homotopy Perturbation Method, International Journal of Computer Applications, 162, (2017), 44-49.
[13] X. Feng, Y. He, Y. Meng, Application of Homotopy Perturbation Method to the Bratu Type Problems, Topological Methods in Nonlinear Analysis: Journal of the Juliusz Schauder Center, 31, (2008), 243-252.
[14] Y. Aksoy, M. Pakdemirli, New Perturbation Iteration Solutions for Bratu-Type Equations, Computers and Mathematics with Applications, 59, (2010), 2802-2808.
[15] M.A. Darwish, B.S. Kashkari, Numerical Solutions of Second Order Initial Value Problems of Bratu Type via Optimal Homotopy Asymptotic Method, American Journal of Computational Mathematics, 4, (2014), 47-54.
[16] S.A. Khuri, A New Approach to Bratu’s Problem, Applied Mathematics and Computation, 147, (2004), 131-136.
[17] B. Batiha, Numerical Solutions of Bratu Type Equations by the Variational Iteration Method, Hacettepe Journal of Mathematics and Statistics, 391 (2010), 23-29.
[18] S.A. Salem, T.Y. Thanoon, On Solving Bratu’s Type Equation by Perturbation Method, International Journal of Nonlinear Analysis and Applications, 13(1), (2022), 2755-2763.
[19] H. Caglar, N. Caglar, M. Ozer, A. Valarystos, B Spline Method for Solving Bratu’s Problem, International Journal of Computer Mathematics, 3987 (2010), 1885-1891.
[20] J.M. Gutierrez, M.A.H. Veron, A Picard-Type Iterative Scheme for Fredholm Integral Equations of the Second Kind, Mathematics, 9(83), (2021), 1-15.
[21] W. Robin, Iterative Solutions to Classical Second-Order Ordinary Differential Equations, Journal of Innovative Technology and Education, 6, (2019), 1-12.
[22] M. Rahman, Integral Equations and their Applications, WIT Press, Southhampton, 2007.
[23] Y.Wang, G. Ni, Y. Liu, Multi Step Newton-Picard Method for Nonlinear Differential Equations, Journal of Guidance Control and Dynamics, 43(11), (2020) 2148-2155.
[24] A. M. Yang, C. Zhang, H. Jafari, C. Cattani, Y. Jiao, Picard Successive Approximation Method for Solving Differential Equations arising in Fractal Heat Transfer with Local Fractional Derivative, Abstract and Applied Analysis, 2014, (2014), 1-6.
[25] S. M. Ulam, A Collection of Mathematical Problems, Interscience, New York, 1960.
[26] D. H. Hyers, On the Stability of the Linear Functional Equation, Proc. Nat. Acad. Sci. USA, 27, (1941), 222-224.
[27] Kaihong Zhao, Generalized UH-Stability of a Nonlinear Fractional Coupling (p1, p2)- Laplacian System Concerned
with Nonsingular Atangana-Baleanu Fractional Calculus, Journal of Inequalities and Applications, 2023, (2023), 1-16.
[28] Kaihong Zhao, Solvability and GUH-stability of a Nonlinear CF-Fractional Coupled Laplacian Equations, AIMS Mathematics, 8(6), (2023), 13351-13367
[29] A. Zada, S. Faisal, Y. Li, Hyers-Ulam-Rassias Stability of Nonlinear Delay Differential Equations, Journal of Nonlinear Sciences and Applications, 10, (2017), 504-510.
[30] J. Huang, S.M. Jung, Y. Li, On Hyers-Ulam Stability of Nonlinear Differential Equations, Bulletin of Korean Mathematical Society, 52,(2015), 685-697.
[31] M.N. Qarawani, Hyers-Ulam-Rassias Stability Criteria of Nonlinear Differential Equations of Lane-Emden Type, General Letters in Mathematics, 3, (2017), 57-70.
[32] X. Wang, D. Luo, Q. Zhu, Ulam-Hyers Stability of Caputo Type Fuzzy Fractional Differential Equations with Time-Delays, Chaos, Solitons & Fractals, 156, (2022), 111822.
[33] D. Luo, X. Wang, T. Caraballo, Q. Zhu , Ulam-Hyers Stability of Caputo-Type Fractional Fuzzy Stochastic Differential Equations with Delay, Commun. Nonlinear Sci.Numer. Simul., 121, (2023), 107229.
[34] A.M. Wazwaz, A First Course in Integral Equations, World Scientific Publishing Company, Singapore, 2015.
[35] B.D. Reddy, Introductory Functional Analysis With Application to Boundary Value Problems and Finite Elements, Springer Verlag Incorporation, New York, (1998).
Published
2025-10-15
How to Cite
1.
Ullah S, Ali M, Bajwa S, Bilal A. Convergence and Ulam-Hyers-Rassias Stability Analyses with Error Estimation of Numerical Solutions of Bratu Type Equations. Sci Inquiry Rev [Internet]. 2025Oct.15 [cited 2025Oct.26];9(2):1-17. Available from: https://journals.umt.edu.pk/index.php/SIR/article/view/7164
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Mathematics
Copyright (c) 2025 Saif Ullah, Muzaher Ali, Sana Bajwa, Ahsan Bilal
This work is licensed under a Creative Commons Attribution 4.0 International License.
