Abstract
In recent years of the literature, fixed point iteration schemes have been studied theoretically for many nonlinear problems, including differential and integral equations; however, numerically, there has been a little work on these iterative schemes for finding numerical solutions of differential equations. The main purpose of this paper is to propose a new numerically efficient iterative approach based on fixed point theory and Green’s function. First, we compute a Green’s function for a wider class fourth order BVPs and express the sought solution as a fixed point of a certain integral operator. We show by Banach Contraction Principle that this new integral operator admits a unique fixed point which is the solution for a given BVP. We embed the integral operator into our iterative scheme and obtain a generalized iterative scheme. We prove the convergence of our new scheme under possible mild assumptions. Instead of stability, we prove a weak \(w^{2}\)-stability result for our scheme which is a natural and weak notion of the stability for any given iterative scheme. To support numerically the convergence of our scheme, we provide two different examples. Using these examples, we compare the high accuracy of our new scheme with the classical approaches of the literature. As an application of our scheme, we show that the scheme can be used for finding solutions for the Bratu’s problem in a Banach space setting. At the end of the paper, we include some interesting discussion about the main results of the paper.
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References
Banach, S.: Sur les operations dans les ensembles abstraits et leurs applications aux equations integrales. Fund. Math. 3, 133–181 (1922)
Kannan, R.: Some results on fixed points. Bull. Cal. Math. Soc. 60, 71–76 (1968)
Chatterjea, S.K.: Fixed-point theorems. C. R. Acad. Bulgare Sci. 25, 15–18 (1972)
Browder, F.E.: Nonexpansive nonlinear operators in a Banach space. Proc. Natl. Acad. Sci. U.S.A. 54, 1041–1044 (1965)
Gohde, D.: Zum Prinzip der Kontraktiven Abbildung. Math. Nachr. 30, 251–258 (1965)
Kirk, W.A.: A fixed point theorem for mappings which do not increase distance. Am. Math. Mon. 72, 1004–1006 (1965)
Picard, E.M.: Memorie sur la theorie des equations aux derivees partielles et la methode des approximation ssuccessives. J. Math. Pure Appl. 6, 145–210 (1890)
Mann, W.R.: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506–510 (1953)
Ishikawa, S.: Fixed points by a new iteration method. Proc. Am. Math. Soc. 44, 147–150 (1974)
Kumar, V., Latif, A., Rafiq, A., Hussain, N.: S-iteration process for quasi-contractive mappings. J. Inequal. Appl. 2013, 206 (2013)
Karakaya, V., Atalan, Y., Dogan, K., Bouzara, N.E.H.: Some fixed point results for a new three steps iteration process in Banach spaces. Fixed Point Theory 18(2), 625–640 (2017)
Hajipour, M., Jajarmi, A., Baleanu, D.: On the accurate discretization of a highly nonlinear boundary value problem. Numer. Algorithms 79(3), 679–695 (2018)
Jajarmi, A., Baleanu, D.: A new iterative method for the numerical solution of high-order nonlinear fractional boundary value problems. Front. Phys. 8, 220 (2020)
Khuri, S.A., Sayfy, A.: Numerical solution of functional differential equations: a Green’s function based iterative approach. Int. J. Comput. Math. 95(10), 1937–1949 (2018)
Ali, F., Ali, J., Uddin, I.: A novel approach for the solution of BVPs via Green’s function and fixed point iterative method. J. Appl. Math. Comput. 66, 167–181 (2021)
Kafri, H.Q., Khuri, S.A., Sayfy, A.: A new approach based on embedding green’s functions into fixed point iterations for highly accurate solution to Troesch’s problem. Int. J. Comput. Methods Eng. Sci. Mech. 17(2), 93–105 (2016)
Khuri, S.A., Louhichi, I.: A novel Ishikawa–Green’s fixed point scheme for the solution of BVPs. Appl. Math. Lett. 82, 50–57 (2018)
Ahmad, J., Arshad, M., Hussain, A., Al-Sulami, H.: A Green’s function based iterative approach for solutions of BVPs in symmetric spaces. Symmetry 15(10), 1838 (2023)
Ahmad, J., Arshad, M., George, R.: A fixed point iterative scheme based on Green’s function for numerical solutions of singular BVPs. AIMS Math. 8, 29517–29534 (2023)
Ahmad, J., Arshad, M., Ma, Z.: Numerical solution of Bratu’s boundary value problem based on Green’s function and a novel iterative scheme. Bound. Value Probl. 2023(1), 102 (2023)
Ahmad, J., Arshad, M., Ma, Z.: Numerical solutions of Troesch’s problem based on a faster iterative scheme with an application. AIMS Math. 9, 9164–9183 (2024)
Akgun, F.A., Rasulov, Z.: Generalized iteration method for the solution of fourth order BVP via Green’s function. Eur. J. Pure Appl. Math. 14(3), 969–979 (2021)
Thenmozhi, S., Marudai, M.: Solution of nonlinear boundary value problem by S-iteration. J. Appl. Math. Comput. 68, 1047–1068 (2022)
Bayin, S.: Mathematical Methods in Science and Engineering. Wiley, Hoboken (2006)
Cardinali, T., Rubbioni, P.: A generalization of the Caristi fixed point theorem in metric spaces. Fixed Point Theory 11, 3–10 (2010)
Harder, A.M., Hicks, T.L.: Stability results for fixed point iteration procedures. Math. Japonica 33, 693–706 (1988)
Timis, I.: On the weak stability of Picard iteration for some contractive type mappings. Annal. Uni. Craiova Math. Comput. Sci. Ser. 37, 106–114 (2010)
Bratu, G.: Sur les equation integrals non-lineaires. Bull. Math. Soc. France 42, 113–142 (1914)
Osilike, M.O.: Stability of the Mann and Ishikawa Iteration procedures for \(\phi \)-strong pseudocontractions and nonlinear equations of the \(\phi \)-strongly accretive type. J. Math. Anal. Appl. 227, 319–334 (1998)
Sahin, A.: Some new results of M-iteration process in hyperbolic spaces. Carpathian J. Math. 35, 221–232 (2019)
Sahin, A.: Some results of the Picard–Krasnoselskii hybrid iterative process. Filomat 33, 359–365 (2019)
Urabe, M.: Convergence of numerical iteration in solution of equations. J. Sci. Hiroshima Univ. A 19, 479–489 (1956)
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Junaid Ahmad and Muhammad Arshad extend their appreciation to the reviewer for providing useful suggestions for improvement of the paper.
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Muhamamd Arahad gave the idea of the paper as a supervisor. Junaid Ahamd wrote the first draft of the paper. Muhamamd Arshad contributed to the Bratu’s problem in the paper and edited the final version. Both authors agree to publish this final version.
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Ahmad, J., Arshad, M. A novel fixed point approach based on Green’s function for solution of fourth order BVPs. J. Appl. Math. Comput. (2024). https://doi.org/10.1007/s12190-024-02071-x
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DOI: https://doi.org/10.1007/s12190-024-02071-x