## An application for linear and nonlinear heat equations by Adomian’s decomposition method.(English)Zbl 1060.65653

Summary: The particular exact solutions of a linear heat equation, and a nonlinear heat equation that usually arises in mathematical biology are obtained in both $$x$$ and $$t$$ directions using Adomian’s decomposition method. In addition, numerical comparison of particular solutions in the decomposition method for linear and nonlinear problems indicates that there is a good agreement between the numerical solutions and particular exact solutions in terms of accuracy and efficiency.

### MSC:

 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs 35K57 Reaction-diffusion equations 35K05 Heat equation 92E20 Classical flows, reactions, etc. in chemistry
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### References:

 [1] Adomian, G., Solving Frontier Problems of Physics: the Decomposition Method (1994), Kluwer Academic Publishers: Kluwer Academic Publishers Boston · Zbl 0802.65122 [2] Cherruault, Y.; Adomian, G., Decomposition methods: a new proof of convergence, Math. Comp. Model., 18, 103-106 (1993) · Zbl 0805.65057 [3] Kaya, D.; Yokus, A., A numerical comparison of partial solutions in the decomposition method for linear and nonlinear partial differential equations, Math. Comput. Simulat., 60, 507-512 (2002) · Zbl 1007.65078 [4] Kaya, D.; Aassila, M., An application for a generalized KdV equation by the decomposition method, Phys. Lett. A., 299, 201-206 (2002) · Zbl 0996.35061 [5] Cherniha, R. M., New Ansätze and exact solutions for nonlinear reaction-diffusion equations arising in mathematical biology, Sym. Nonlinear Math. Phys., 1, 138-146 (1997) · Zbl 0954.35038 [6] Fushchych, W.; Zhdanov, R., Antireduction and exact solutions of nonlinear heat equations, Nonlinear Math. Phys., 1, 1, 60-64 (1994) · Zbl 0954.35037 [7] Murray, J. D., Mathematical Biology (1993), Springer: Springer Berlin · Zbl 0779.92001 [8] Euler, M.; Euler, N., Symmetries for a class of explicitly space and time dependent (1+1)-dimensional wave equations, Sym. Nonlinear Math. Phys., 1, 70-78 (1997) · Zbl 0948.35081 [9] Parnuk, S., Qualitative analysis of a mathematical model for capillary formation in tumor angiogenesis, Math. Models Meth. Appl. Sci., 13, 1, 19-33 (2003) [10] Levine, H. A.; Pamuk, S.; Sleeman, B. D.; Nilsen-Hamilton, M., Mathematical modeling of capillary formation and development in tumor angiogenesis: Penetration into the stroma, Bull. Math. Biol., 63, 5, 801-863 (2001) · Zbl 1323.92029
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