On the Existence of Multiple Solutions of a Class of Second-order Nonlinear Two-point Boundary Value Problems
-
2290
Downloads
-
3701
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.
Share and Cite
ISRP Style
E. Shivanian, F. Abdolrazaghi, On the Existence of Multiple Solutions of a Class of Second-order Nonlinear Two-point Boundary Value Problems, Journal of Mathematics and Computer Science, 14 (2015), no. 2, 97-107
AMA Style
Shivanian E., Abdolrazaghi F., On the Existence of Multiple Solutions of a Class of Second-order Nonlinear Two-point Boundary Value Problems. J Math Comput SCI-JM. (2015); 14(2):97-107
Chicago/Turabian Style
Shivanian, E., Abdolrazaghi, F.. "On the Existence of Multiple Solutions of a Class of Second-order Nonlinear Two-point Boundary Value Problems." Journal of Mathematics and Computer Science, 14, no. 2 (2015): 97-107
Keywords
- Closed-form solution
- exact analytical solution
- special function
- unique solution
- multiple solutions.
MSC
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)