Archive
Special Issues
Thermal Conductivity Equations via the Improved Adomian Decomposition Methods
Applied and Computational Mathematics
Volume 9, Issue 3, June 2020, Pages: 30-55
Received: Apr. 1, 2020; Accepted: May 11, 2020; Published: May 27, 2020
Author
Ashenafi Gizaw Jije, Department of Mathematics, Faculty of Natural and Computational Sciences, Gambella University, Gambella, Ethiopia
Article Tools
Abstract
Several mathematical models that explain natural phenomena are mostly formulated in terms of nonlinear differential equations. Many problems in applied sciences such as nuclear physics, engineering, thermal management, gas dynamics, chemical reaction, studies of atomic structures and atomic calculations lead to singular boundary value problems and often only positive solutions are vital. However, most of the methods developed in mathematics are used in solving linear differential equations. For this reason, this research considered a model problem representing temperature distribution in heat dissipating fins with triangular profiles using MATLAB codes. MADM was used with a computer code in MATLAB to seek solution for the problem involving constant and a power law dependence of thermal conductivity on temperature governed by linear and nonlinear BVPs, respectively, for which considerable results were obtained. A problem formulated dealing with a triangular silicon fin and more examples were solved and analyzed using tables and figures for better elaborations where appreciable agreement between the approximate and exact solutions was observed. All the computations were performed using MATHEMATICA and MATLAB.
Keywords
Fins, Adomian Decomposition, Thermal Conductivity Equation, Nonnlinear
Ashenafi Gizaw Jije, Thermal Conductivity Equations via the Improved Adomian Decomposition Methods, Applied and Computational Mathematics. Vol. 9, No. 3, 2020, pp. 30-55. doi: 10.11648/j.acm.20200903.11
References
[1]
Adomian, G. 1988. A Review of the Decomposition Method in Applied Mathematics. Academic Press, Inc, 135: 501-544.
[2]
Arpaci, S. Vedat 1966. Conduction heat transfer. Addison Wesley Publishing Company, Massachusetts.
[3]
Benabidallah, M. and Cherruault, Y. (2004). Application of the Adomian method for solving a class of boundary problems. Kybernetes, 33 (1): 118-132.
[4]
Cui, M. and Geng, F. 2007. Solving singular two-point boundary value problem in reproducing kernel space. Journal of Computations and Applied Mathematics, 205: 6–15.
[5]
Duan, J.-S, Rach, R., Baleanu, D. and Wazwaz, A.-M. 2012. A review of the Adomian decomposition method and its applications to fractional differential equations. Commun. Frac. Calc., 3 (2): 73-99.
[6]
Ebaid, A. 2010. Modification of Lesnic’s Approach and New Analytic Solutions for Some Nonlinear Second-Order Boundary Value Problems with Dirichlet Boundary Conditions. Zeitschrift f¨ ur Naturforschung, Tubingen, 65a: 692-296.
[7]
Hasan, Y. Q. and Zhu, L. M. 2008. Modified Adomian decomposition method for singular initial value problems in the second order ordinary differential equations. 3 (1): 183–193.
[8]
Hasan, Y. Q. and Zhu, L. M. 2009. A note on the use of modified Adomian decomposition method for solving singular boundary value problems of higher-order ordinary differential equations. Communications in Nonlinear Science and Numerical Simulation, 14 (8): 3261-3265.
[9]
Inc, M. and Evans, D. J. 2003. The decomposition method for solving a class of singular two- point boundary value problems. International Journal of Computational Mathematics. 80 (7): 869–882.
[10]
Kaliyappan, M. and Hariharan, S. 2015. Symbolic computation of Adomian polynomials based on Rach’s Rule. British Journal of Mathematics and Computer Science, 5 (5): 562-570.
[11]
Khuri, S. A. and Sayfy, A. 2010. A novel approach for the solution of a class of singular boundary value problems arising in physiology. Mathematical and Computer Modeling, 52 (3): 626-636.
[12]
Kim, W. and Chun, C. 2010. A modified Adomian decomposition method for solving higher- order singular boundary value problems. Z. Naturforsch, 65a: 1093-1100.
[13]
Kraus, A. D., Aziz, A. and Welty, J. 2001. Extended Surface Heat Transfer, John Wiley and Sons, Inc., New York.
[14]
Kumar, M. 2002. A three-point finite difference method for a class of singular two-point boundary value problems. Journal of Computations and Applied Science, 145: 89–97.
[15]
Lesnic, D. 2001. A computational algebraic investigation of the decomposition method for time-dependent problems. Elsevier Science Inc., 119: 197-206.
[16]
Lin, Y. and Chen, C. K. 2014. Modified Adomian decomposition method for double singular boundary value problems. Rom. Journal of Physics, Bucharest, 59 (5-6): 443-454.
[17]
Mokheimer Esmail, M. A. 2003. Heat transfer from extended surfaces subject to variable heat transfer coefficient. Springer-Verlag, 39: 131-138.
[18]
Noor, M. A. and Mohyud-Din, S. T. 2008. Solution of singular and non-singular initial and boundary value problems by modified variational iteration method. Hindawi Publishing Corporation, 1-23.
[19]
Ravi, A. S. V. and Aruna, K. 2008. Solution of singular two-point boundary value problems using differential transformation method. (A372): 4062-4066.
[20]
Singh, R., Kumar, J., and Nelakanti, G. 2012. New approach for solving a class of doubly singular two-point boundary value problems using Adomian decomposition method. Hindawi Publishing Corporation, 1-23.
[21]
Wazwaz, A.-M. 1999. A Reliable Modification of Adomian Decomposition Method. Applied Mathematics Computation, 102 (1): 77-86.
[22]
Zavalani, G. 2015. Galerkin-finite element method for two point boundary value problems of ordinary differential equations. Science Publishing Group, 4 (2): 64-68.
PUBLICATION SERVICES
RESOURCES
SPECIAL SERVICES
Science Publishing Group
1 Rockefeller Plaza,
10th and 11th Floors,
New York, NY 10020
U.S.A.
Tel: (001)347-983-5186