Abstract
A general approach is presented for proving existence of multiple solutions of the third-order nonlinear differential equation
subject to given proper boundary conditions. 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.
Similar content being viewed by others
References
Shuicai L., Liao S.: An analytic approach to solve multiple solutions of a strongly nonlinear problem. Appl. Math. Comput. 169, 854–865 (2005)
Wazwaz A.: Adomian decomposition method for a reliable treatment of the Bratu-type equations. Appl. Math. Comput. 166, 652–663 (2005)
Sedeek A.M.A.L.: New smoother to enhance multigrid-based methods for Bratu problem. Appl. Math. Comput. 204, 325–339 (2008)
Muhammed I., Hamdan A.: An efficient method for solving Bratu equations. Appl. Math. Comput. 176, 704–713 (2006)
Abbasbandy S., Shivanian E.: Prediction of multiplicity of solutions of nonlinear boundary value problems: novel application of homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul. 15, 3830–3846 (2010)
Chowdhury M., Hashim I.: Analytical solutions to heat transfer equations by homotopy perturbation method revisited. Phys. Lett. A 372, 1240–1243 (2008)
Ganji D.: The application of He’s homotopy perturbation method to nonlinear equations arising in heat transfer. Phys. Lett. A 355, 337–341 (2006)
Tari H.B.H., Ganji D.D.: The application of He’s variational iteration method to nonlinear equations arising in heat transfer. Phys. Lett. A 363, 213–217 (2007)
Abbasbandy S., Shivanian E.: Exact analytical solution of a nonlinear equation arising in heat transfer. Phys. Lett. A 374, 567–574 (2010)
Shivanian E., Abbasbandy S.: Predictor homotopy analysis method: two points second order boundary value problems. Nonlinear Anal. Real. 15, 89–99 (2014)
Abramowitz M., Stegun I.: Handbook of mathematical functions. Dover, New York (1972)
Erdelyi A., Magnus W., Oberhettinger F., Tricomi F.: Higher transcendental functions, vol. 2. McGraw-Hill, New York (1953)
Roberts S., Shipman J.: On the closed form solution of Troesch’s problem. J. Comput. Phys. 21(3), 291–304 (1976)
Hlaváček V., Marek M., Kubíček M.: 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, 1083–1097 (1968)
Seydel, R.: World of bifurcation: Online collection and tutorials of nonlinear phenomena. http://www.bifurcation.de.
William F., James A.: Singular non-linear two-point boundary value problems: existence and uniqueness. Nonlinear Anal. Real. 71, 1059–1072 (2009)
Kumar M., Singh N.: Modified adomian decomposition method and computer implementation for solving singular boundary value problems arising in various physical problems. Comput. Chem. Eng. 34, 1750–1760 (2010)
Makinde O., Mhone P.: Hermite-Pade approximation approach to MHD Jeffery-Hamel flows. Appl. Math. Comput. 181, 966–972 (2006)
Ganji Z., Ganji D., Esmaeilpour M.: Study on nonlinear Jeffery-Hamel flow by He’s semi-analytical methods and comparison with numerical results. Comput. Math. Appl. 58, 2107–2116 (2009)
Domairry G., Mohsenzadeh A., Famouri M.: The application of homotopy analysis method to solve nonlinear differential equation governing Jeffery-Hamel flow. Commun. Nonlinear Sci. Numer. Simul. 14, 85–95 (2008)
Motsa S., Sibanda P., Awad F., Shateyi S.: A new spectral-homotopy analysis method for the MHD Jeffery-Hamel problem. Comput. Fluids. 39, 1219–1225 (2010)
Abbasbandy S., Shivanian E.: Exact analytical solution of the MHD Jeffery-Hamel flow problem. Meccanica 47, 1379–1389 (2012)
Li S.: Positive solutions of nonlinear singular third-order two-point boundary value problem. J. Math. Anal. Appl. 323, 413–425 (2006)
Lepin A., Lepin L., Myshkisb A.: Two-point boundary value problem for nonlinear differential equation of nth order. Nonlinear Anal. Theory Meth. Appl. 40, 397–406 (2000)
Afuwape A.: Frequency domain approach to some third-order nonlinear differential equations. Nonlinear Anal. Theory Meth. Appl. 71, 972–978 (2009)
Boucherif A., Bouguimab S., Malki N., Benbouziane Z.: Third order differential equations with integral boundary conditions. Nonlinear Anal. Theory Meth. Appl. 71, 1736–1743 (2009)
Jankowski T.: Existence of positive solutions to third order differential equations with advanced arguments and nonlocal boundary conditions. Nonlinear Anal. Theory Meth. Appl. 75, 913–923 (2012)
Qian C.: On global stability of third-order nonlinear differential equations. Nonlinear Anal. Theory Meth. Appl. 47, 1379–1389 (2012)
Yao Q.: Solution and positive solution for a semilinear third-order two-point boundary value problem. Nonlinear Anal. Theory Meth. Appl. 17, 1171–1175 (2004)
Yao Q., Feng Y.: The existence of solution for a third-order two-point boundary value problem. Appl. Math. Lett. 15, 227–232 (2002)
Mosconi S., Santra S.: On the existence and non-existence of bounded solutions for a fourth order ODE. J. Differ. Equations. 255, 4149–4168 (2013)
Wu T.: Existence and multiplicity of positive solutions for a class of nonlinear boundary value problems. J. Differ. Equations. 252, 3403–3435 (2012)
Coddington E.: An introduction to ordinary differential equations. Prentice-Hall, Englewood Cliffs (1961)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Shivanian, E., Abdolrazaghi, F. On The Existence of Multiple Solutions of a Class of Third-Order Nonlinear Two-Point Boundary Value Problems. Mediterr. J. Math. 13, 2339–2351 (2016). https://doi.org/10.1007/s00009-015-0627-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00009-015-0627-y