On the Existence of Multiple Solutions of a Class of Second-order Nonlinear Two-point Boundary Value Problems

Volume 14, Issue 2, pp 97-107
• 1052 Views

Authors

E. Shivanian - Department of Mathematics, Imam Khomeini International University, Ghazvin, 34149-16818, Iran F. Abdolrazaghi - Department of Mathematics, Imam Khomeini International University, Ghazvin, 34149-16818, Iran

Abstract

A general approach is presented for proving existence of multiple solutions of the second-order nonlinear differential equation $u'' (x) + f (u(x)) = 0,\quad x\in [0,1],$ subject to given boundary conditions: $u(0) = B_1, u(1) = B_2$ or $u'(0) = B'_1, u(1)=B_2$. The proof is constructive in nature, and could be used for numerical generation of the solution or closed-form analytical solution by introducing some special functions. The only restriction is about $f(u)$ , where it is supposed to be differentiable function with continuous derivative. It is proved the problem may admit no solution, may admit unique solution or may admit multiple solutions.

Keywords

• Closed-form solution
• exact analytical solution
• special function
• unique solution
• multiple solutions.

•  65L12
•  65L10
•  65N85

References

• [1] L. Shuicai, S. Liao, An analytic approach to solve multiple solutions of a strongly nonlinear problem, Appl. Math. Comput., 169 (2005), 854–865.

• [2] A. Wazwaz, Adomian decomposition method for a reliable treatment of the Bratu-type equations, Appl. Math. Comput., 166 (2005), 652–663.

• [3] A. Mohsen, L. Sedeek, New smoother to enhance multigrid-based methods for Bratu problem, Appl. Math. Comput., 204 (2008), 325–339.

• [4] I. Muhammed, A. Hamdan, An efficient method for solving Bratu equations, Appl. Math. Comput. , 176 (2006), 704–713.

• [5] S. Abbasbandy, E. Shivanian, Prediction of multiplicity of solutions of nonlinear boundary value problems: Novel application of homotopy analysis method, Commun. Nonlinear Sci. Numer. Simulat. , 15 (2010), 3830–3846.

• [6] M. Chowdhury, I. Hashim, Analytical solutions to heat transfer equations by homotopy perturbation method revisited, Phys. Lett. A , 372 (2008), 1240–1243.

• [7] D. Ganji, The application of He’s homotopy perturbation method to nonlinear equations arising in heat transfer, Phys. Lett. A , 355 (2006), 337–341.

• [8] H. B. H. Tari, D. D. Ganji, The application of He’s variational iteration method to nonlinear equations arising in heat transfer, Phys. Lett. A, 363 (2007), 213–217.

• [9] S. Abbasbandy, E. Shivanian, Exact analytical solution of a nonlinear equation arising in heat transfer, Phys. Lett. A , 374 (2010), 567–574.

• [10] E. Shivanian, S. Abbasbandy, Predictor homotopy analysis method: Two points second order boundary value problems, Nonlinear Anal. Real. , 15 (2014), 89–99.

• [11] M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, Dover, New York (1972)

• [12] A. Erdelyi, W. Magnus, F. Oberhettinger, F. Tricomi, Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York. (1953)

• [13] S. Roberts, J. Shipman, On the Closed form Solution of Troesch’s Problem, J. Comput. Phys., 21(3) (1976), 291–304.

• [14] V. Hlavácek, M. Marek, M. Kubícek, Modelling of chemical reactors- X Multiple solutions of enthalpy and mass balances for a catalytic reaction within a porous catalyst particle, Chem. Eng. Sci. , 23 (1968), 1083–1097.

• [15] R. Seydel, World of bifurcation: Online collection and tutorials of nonlinear phenomena, See http://www.bifurcation.de. , (),

• [16] F. William, A. James, Singular non-linear two-point boundary value problems: Existence and uniqueness, Nonlinear Anal. Real. , 71 (2009), 1059–1072.

• [17] M. Kumar, N. Singh, Modified Adomian Decomposition Method and computer implementation for solving singular boundary value problems arising in various physical problems, Comput. Chem. Eng. , 34 (2010), 1750–1760.

• [18] M. Aslefallah, E. Shivanian, A nonlinear partial integro-differential equation arising in population dynamic via radial basis functions and theta-method, J. Math. Computer Sci., TJMCS, 13 (2014), 14-25.

• [19] S. Abbasbandy, E. Shivanian, Application of variational iteration method for nth-order integro-differential equations, Zeitschrift für Naturforschung A , 64 (2014), 439-444.

• [20] S. Abbasbandy, E. Shivanian, I. Hashim, Exact analytical solution of forced convection in a porous-saturated duct, Communications in Nonlinear Science and Numerical Simulation, 16 (10) (), 3981-3989.

• [21] E. Shivanian, S. Abbasbandy, Predictor homotopy analysis method: Two points second order boundary value problems, Nonlinear Analysis: Real World Applications , 15 (2014), 89-99.

• [22] E. Shivanian, S. Abbasbandy, M. S. Alhuthali, Exact analytical solution to the Poisson- Boltzmann equation for semiconductor devices, The European Physical Journal Plus, 129 (6) (2014), 1- 8.

• [23] E. Shivanian, On the multiplicity of solutions of the nonlinear reactive transport model, Ain Shams Engineering Journal, 5 (2014), 637–645.

• [24] E. Shivanian, Existence results for nano boundary layer flows with nonlinear Navier boundary condition, Physics Letters A, 377 (41) (2013), 2950-2954.

• [25] S. Li, Positive solutions of nonlinear singular third-order two-point boundary value problem, J. Math. Anal. Appl., 323 (2006), 413–425.

• [26] A. Lepin, L. Lepin, A. Myshkisb, Two-point boundary value problem for nonlinear differential equation of n’th order, Nonlinear Anal. Theory Meth. Appl., 40 (2000), 397–406.

• [27] A. Afuwape, Frequency domain approach to some third-order nonlinear differential equations, Nonlinear Anal. Theory Meth. Appl. , 71 (2009), 972–978.

• [28] A. Boucherif, S. Bouguimab, N. Malki, Z. Benbouziane, Third order differential equations with integral boundary conditions, Nonlinear Anal. Theory Meth. Appl., 71 (2009), 1736–1743.

• [29] T. Jankowski, Existence of positive solutions to third order differential equations with advanced arguments and nonlocal boundary conditions, Nonlinear Anal. Theory Meth. Appl. , 75 (2012), 913–923.

• [30] C. Qian, On global stability of third-order nonlinear differential equations, Nonlinear Anal. Theory Meth. Appl. , 47 (2012), 1379–1389.

• [31] Q. Yao, Solution and Positive Solution for a Semilinear Third-Order Two-Point Boundary Value Problem, Nonlinear Anal. Theory Meth. Appl. , 17 (2004), 1171–1175.

• [32] Q. Yao, Y. Feng, The Existence of Solution for a Third-Order Two-Point Boundary Value Problem, Appl. Math. Lett., 15 (2002), 227–232.

• [33] S. Mosconia, S. Santra, On the existence and non-existence of bounded solutions for a fourth order ODE, J. Differential Equations, 255 (2013), 4149–4168.

• [34] T. Wu, Existence and multiplicity of positive solutions for a class of nonlinear boundary value problems, J. Differential Equations, 252 (2012), 3403–3435.

• [35] S. H. Rasouli, G. A. Afrouzi, J. Vahidi, On positive weak solutions for some nonlinear elliptic boundary value problems involving the p-Laplacian, J. Math. Computer Sci., TJMCS, 3 (2011), 94 - 101.

• [36] N. Nyamoradi, Existence of positive solutions for third-order boundary value problems, J. Math. Computer Sci., TJMCS, 4 (2012), 8 - 18.

• [37] E. Coddington, An Introduction to Ordinary Differential Equations, Prentice-Hall , Englewood Cliffs, N.J. (1961)