Solution of the heat equation with variable properties by two-step Adomian decomposition method

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Adomian Decomposition Method [G. Adomian, R. Rach, Equality of partial solutions in the decomposition method for linear or nonlinear partial differential equations, Appl. Math. Comput. 19 (1990) 9–12; G. Adomian, R. Rach, Analytical solution of non linear boundary value problem in several dimension by decomposition, J. Appl. Math. 174 (1993) 118–137; G. Adomian, R. Rach, Modified Adomian polynomials, Math. Comput. Modelling 24 (1996) 39–46] is useful to find the solution of linear and nonlinear equations. There is a renewed interest in the method [A.M. Wazwaz, A reliable modification of Adomian decomposition method, Appl. Math. Comput. 102 (1999) 77–86; A.M. Wazwaz, The modified decomposition method for analytic treatment of differential equations, Appl. Math. Comput. 173 (2006) 165–176; X.G. Luo, A two-step Adomian decomposition method, Appl. Math. Comput. 170 (2005) 570–583; B.Q. Zhang, X.G. Luo, Q.B. Wu, Experimentation with two-step Adomian decomposition method to solve evolution models, Appl. Math. Comput. 175 (2006) 1495–1502; B.Q. Zhang, X.G. Luo, Q.B. Wu, Revisit on partial solutions in the Adomian decomposition method: Solving heat and wave equations, J. Math. Anal. Appl. 321 (2006) 353–363; B.Q. Zhang, X.G. Luo, Q.B. Wu, The restrictions and improvement of the Adomian decomposition method: Solving heat and wave equations, Appl. Math. Comput. 177 (1999) 99–104; Necdet Bildik, Hatice Bayramoglu, The solution of two dimensional nonlinear differential equation by the Adomian decomposition method, Appl. Math. Comput. 163 (1999) 551–567] and a lot of research is being conducted using this method. We attempt to enlarge the scope of its application by presenting the solution of the diffusion equation with variable properties. In this, we present two problems dealing with the heat conduction with variable properties. The compression of the first problem with eigenfunction expansion is also made. The two analytical solution agree exactly with each other. Although, the two methods arrive at the same result, nevertheless, it is of much value to obtain the solution by ADM which provides a powerful method of finding the solution of both linear and nonlinear problems. To apply this method, we have shown that generalized Fourier series is required to build up the solution instead of trigonometric Fourier series.