Semi-Numerical Analytical Solution to Linear and Nonlinear Heat Equations via the Reduced Differential Transformation Method


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Abstract


In this study, the reduced differential transform method (RDTM) is employed to obtain approximate analytical solution to different kinds of heat equations. These kinds of heat equations include linear, nonlinear, homogeneous and non-homogeneous types. The solution is easily and accurately calculated in the form of convergent power series. The very small relative error occurs after few numbers of steps shows the powerful, efficiency, and effectiveness of the method. Hence, the proposed technique is a hopeful tool to solve these kinds of heat equations
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Keywords


Reduced Differential Transform Method; Heat Equation; Exact Solution; Error

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References


K. Andersen, H. Madsen, H. Hansen, Modelling the heat dynamics of a building using stochastic differential equations, Energy and Buildings, vol. 31 n. 1, January 2000, pp. 13-24.
http://dx.doi.org/10.1016/s0378-7788(98)00069-3

H. Lienhard IV, H. Lienhard V, A Heat Transfer TextBook ( Third Edition, Cambridge, Massachusettes, Inc., 2001).

A.J. Frijins, G.M.J. Van Leeuwen, A.A. van Steenhoven, Modelling Heat Transfer in Humans, International Journal of Rotating Machinery, vol. 68, 2006, pp. 43-47.

M.J Chadam, H.M. Yin, A Diffusion Equation With Localized Chemical Reactions, Proceedings of the Edinburgh Mathematical Society, vol. 37, 1993, pp. 101-118.
http://dx.doi.org/10.1017/s0013091500018721

H. Sekimoto, Nuclear Reactor Theory (Tokyo Institute of Technology, The 21st Century Center of Excellence Program, Inc., 2007).

Y. Pinchover, J. Rubinstein, An Introduction to Partial Differential Equations (Cambridge University Press, Inc., 2005).
http://dx.doi.org/10.1017/cbo9780511801228.003

V.R. Cabanillas, S.B. de Menezes, E. Zuazua, Null Controllability in Unbounded Domains for the Semilinear Heat Equation with Nonlinearities Involving Gradient Terms, Journal of Optimization Theory and Applications, vol. 110 n. 2, 2001, pp. 245-264.

R. Abazari, M. Abazari, Numerical simulation of generalized Hirota--Satsuma coupled KdV equation by RDTM and comparison with DTM, Commun Nonlinear Sci Numer Simulat, vol. 17, 2012, pp. 619--629.
http://dx.doi.org/10.1016/j.cnsns.2011.05.022

J.K. Zhou, Differential Transformation and Its Applications for Electrical Circuits (Huazhong University Press, Inc., 1986).

P.K. Gupta, Approximate analytical solutions of fractional Benney--Lin equation by reduced differential transform method and the homotopy perturbation method, Computers and Mathematics with Applications, vol. 61, 2011, pp. 2829--2842.
http://dx.doi.org/10.1016/j.camwa.2011.03.057

J. Biazar, M. Eslami, Differential Transform Method for Nonlinear Parabolic-hyperbolic Partial Differential Equations, Applications and Applied Mathematics: An International Journal (AAM), vol. 5 n. 10, 2010, pp. 1493-1503.

S.A. El-Wakilai1, M.A. Abdou, On the generalized differential transform method and its applications, Mathematics Scientific Journal, vol. 6 n. 1, 2010, pp. 17-32.

M.F. Patricio, P.M. Rosa, The Differential Transform Method for Advection-Diffusion Problems, World Academy of Science, Engineering and Technology, vol. 1 n. 4, 2007, pp. 218-222.

F. Ayaz, Solutions of the system of differential equations by differential transform method, Applied Mathematics and Computation, vol. 147, 2004, pp. 547--567.
http://dx.doi.org/10.1016/s0096-3003(02)00794-4

Y. Keskin, G. Oturanc, Reduced Differential Transform Method for Partial Differential Equations, International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10 n. 6, 2009, pp. 741-749.
http://dx.doi.org/10.1515/ijnsns.2009.10.6.741

R. Abazari, M. Abazari, Numerical simulation of generalized Hirota--Satsuma coupled KdV equation by RDTM and comparison with DTM, Communications in nonlinear science and numerical simulation, vol. 17, 2012, pp. 619--629.
http://dx.doi.org/10.1016/j.cnsns.2011.05.022

R. Abazari and Ra. Abazari, Numerical Study of Some Coupled PDEs by using differential transformation method, World academy of Science, Engineering and Technology, vol. 1 n. 42, 2010, pp. 52-59.

Y. Keskin, G. Oturanc, Numerical Solution of Regularized Long Wave Equation by Reduced Differential Transform Method, Applied Mathematical Sciences, vol. 4 n. 25, 2010, pp. 1221 -- 1231.

Y. Keskin, G. Oturanc, Reduced Differential Transform Method For Solving Linear and Nonlinear Wave Equations, Iranian Journal of Science & Technology, vol. 34, n. 2, 2010, pp. 113--122.

Y. Keskin, G. Oturanc, Application of Reduced Differential Transformation Method for Solving Gas Dynamics Equation, Int. J. Contemp. Math. Sciences, vol. 5 n. 22, 2010, pp. 1091 -- 1096.

Y. Keskin, S. Servi, G. Oturanç, Reduced Differential Transform Method for Solving Klein Gordon Equations, Proceedings of the World Congress on Engineering, vol. 1, 2011, July 2011, pp. 267-271.

Y. Keskin, G. Oturanc, Reduced differential transform method for fractional partial differential equations, Nonlinear Science Letters, vol. 1, 2010, pp. 61-72.
http://dx.doi.org/10.1515/ijnsns.2009.10.6.741

Y. Keskin, G. Oturanc, Numerical solution of Regularized Long Wave equation by reduced differential transform method, Applied Mathematical Sciences, vol. 4, 2010, pp. 1221- 1231.

Y. Keskin, G. Oturanc, Reduced differential transform method for generalized KdV equations, Mathematical and Computational Applications, vol. 15, 2010, pp. 382-393.

N. Taghizadeh, M. Akbari, M. Shahidi, Application of Reduced Differential Transform Method to the Wu-Zhang Equation, Australian Journal of Basic and Applied Sciences, vol. 5 n. 5, 2011, pp. 565-571.

Miersemann, Lecture notes in partial differential equations (Leipzig university, Inc., 2012).

A. Cheniguel, Numerical Method for Solving Non-Homogeneous Heat Equation with Derivative Boundary Conditions, International Mathematical Forum, vol. 14, 2011, pp. 651-658.

W. Hereman, Exact Solutions of Nonlinear Partial Differential Equations The Tanh/Sech Method (Wolfram Research, Inc., 2000).

B. Neta, Partial Differential Equations 3132Lecture Notes (MANet-A-Sof, Inc., 2002).


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