Abstract
We consider the boundary value problem
where \(f: [0, 1] \times \mathbb {R}^{4} \rightarrow \mathbb {R}^{+},\ a: [0, 1] \rightarrow \mathbb {R}^{+}\) are continuous functions. For f = f(u(t)), very recently in Benaicha and Haddouchi (An. Univ. Vest Timis. Ser. Mat.-Inform. 1(54): 73–86, 2016) the existence of positive solutions was studied by employing the fixed point theory on cones. In this paper, by the method of reducing the boundary value problem to an operator equation for the right-hand sides we establish the existence, uniqueness, and positivity of solution and propose an iterative method on both continuous and discrete levels for finding the solution. We also give error analysis of the discrete approximate solution. Some examples demonstrate the validity of the obtained theoretical results and the efficiency of the iterative method.
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References
Agarwal, R. P., Chow, Y. M.: Iterative methods for a fourth order boundary value problem. J. Comput. Appl. Math. 10, 203–217 (1984)
Azarnavid, B., Parand, K., Abbasbandy, S.: An iterative kernel based method for fourth order nonlinear equation with nonlinear boundary condition. Commun. Nonlinear Sci. Numer. Simulat. 59, 544–552 (2018)
Benaicha, S., Haddouchi, F.: Positive solutions of a nonlinear fourth-order integral boundary value problem. An. Univ. Vest Timis. Ser. Mat.-Inform. 1(54), 73–86 (2016)
Chai, G.: Positive solution of fourth-order integral boundary value problem with two parameters, abstract and applied analysis 2011, Article ID 859497. https://doi.org/10.1155/2011/859497
Dang, Q. A., Ngo, T. K. Q.: Existence results and iterative method for solving the cantilever beam equation with fully nonlinear term. Nonlinear Anal. Real World Appl. 36, 56–68 (2017)
Dang, Q. A., Dang, Q. L., Ngo, T. K. Q.: A novel efficient method for nonlinear boundary value problems. Numer. Algor. 76, 427–439 (2017)
Dang, Q. A., Ngo, T. K. Q.: New fixed point approach for a fully nonlinear fourth order boundary value problem. Bol. Soc. Paran. Mat. 36(4), 209–223 (2018)
Dang, Q. A., Nguyen, T. H.: The unique solvability and approximation of BVP for a nonlinear fourth order kirchhoff type equation, East Asian. J. Appl. Math. 8 (2), 323–335 (2018)
Dang, Q. A., Nguyen, T. H.: Existence results and iterative method for solving a nonlinear biharmonic equation of Kirchhoff type. Comput. Math. Appl. 76, 11–22 (2018)
Dang, Q. A., Dang, Q. L.: A simple efficient method for solving sixth-order nonlinear boundary value problems, Comp. Appl. Math 37(1). https://doi.org/10.1007/s40314-018-0643-1 (2018)
Dang, Q. A., Nguyen, T. H.: Existence results and numerical method for a fourth order nonlinear problem. Int. J. Appl. Comput. Math. 4, 148 (2018)
Dang, Q. A., Nguyen, T. H.: Solving the Dirichlet problem for fully fourth order nonlinear differential equation, Afrika Matematika. https://doi.org/10.1007/s13370-019-00671-6 (2019)
Dang Q.A., Vu, T.L.: Iterative method for solving a nonlinear fourth order boundary value problem. Comput. Math. Appl. 60, 112–121 (2010)
Geng F.: A new reproducing kernel Hilbert space method for solving nonlinear fourth-order boundary value problems. Appl. Math. Comput. 213, 163–169 (2009)
Hajji, S. M. A., Al-Khaled, K.: Numerical methods for nonlinear fourth-order boundary value problems with applications. Int. J. Comput. Math. 85(1), 83–104 (2008)
Kang, P., Wei, Z., Xu, J.: Positive solutions to fourth order singular boundary value problems with integral boundary conditions in abstract spaces. Appl. Math. Comput. 206(1), 245–256 (2008)
Li, H., Wang, L., Pei, M.: Solvability of a fourth-order boundary value problem with integral boundary conditions, Journal of Applied Mathematics 2013, Article ID 782363. https://doi.org/10.1155/2013/782363
Lv, X., Wang, L., Pei, M.: Monotone positive solution of a fourth-order BVP with integral boundary conditions. Boundary Value Problems 2015, 172 (2015)
Ma, R., Chen, T.: Existence of positive solutions of fourth-order problems with integral boundary conditions, boundary value problems 2011, Article ID 297578. https://doi.org/10.1155/2011/297578
Mohanty, R.K.: A fourth-order finite difference method for the general one-dimensional nonlinear biharmonic problems of first kind. J. Comput. Appl. Math. 114, 275–290 (2000)
Noor M.A., Mohyud-Din S.T.: An efficient method for fourth-order boundary value problems. Comput. Math. Appl. 54, 1101–1111 (2007)
Shen J., Tang T., Wang L.-L.: Spectral Methods: Algorithms, Analysis and Applications. Springer, Berlin (2011)
Siddiqi, S.S., Akram, G.: Solution of the system of fourth-order boundary value problems using non-polynomial spline technique. Appl. Math. Comput. 185, 128–135 (2007)
Singh, R., Kumar, J., Nelakanti, G.: Approximate series solution of fourth-order boundary value problems using decomposition method with Green’s function. J. Math. Chem. 12(52), 1099–1118 (2014)
Srivastava, P.K., Kumar, M., Mohapatra, R.N.: Solution of fourth order boundary value problems by numerical algorithms based on nonpolynomial quintic splines. Journal of Numerical Mathematics and Stochastics 4(1), 13–25 (2012)
Yang, Y., Chen, Y., Huang, Y., Yang, W.: Convergence analysis of Legendre-collocation methods for nonlinear Volterra type integral Equations. Adv. Appl. Math. Mech. 7, 74–88 (2015)
Yang, Y., Chen, Y.: Jacobi spectral Galerkin and iterated methods for nonlinear Volterra integral equation. J. Comput. Nonlinear Dyn. 11(4), 041027 (2016)
Zhang, X., Feng, M., Ge, W.: Existence results for nonlinear boundary-value problems with integral boundary conditions in Banach spaces. Nonlinear Anal. 69, 3310–3321 (2008)
Zhang, X., Ge, W.: Positive solutions for a class of boundary value problems with integral boundary conditions. Comput. Math. Appl. 58(2), 203–215 (2009)
Wei, Y., Song, Q., Bai, Z.: Existence and iterative method for some fourth order nonlinear boundary value problems. Appl. Math. Lett. 87, 101–107 (2019)
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This work is supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under the grant number 102.01-2017.306.
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Dang, Q.A., Dang, Q.L. Existence results and iterative method for a fully fourth-order nonlinear integral boundary value problem. Numer Algor 85, 887–907 (2020). https://doi.org/10.1007/s11075-019-00842-3
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DOI: https://doi.org/10.1007/s11075-019-00842-3