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Technical Brief

Fisher–Kolmogorov–Petrovsky– Piscounov Reaction and n-Diffusion Cattaneo Telegraph Equation

[+] Author and Article Information
Ulrich Olivier Dangui-Mbani, Jize Sui

School of Energy and Environmental Engineering,
University of Science and Technology Beijing,
Beijing 100083, China;
School of Mathematics and Physics,
University of Science and Technology Beijing,
Beijing 100083, China

Liancun Zheng

School of Mathematics and Physics,
University of Science and Technology Beijing,
Beijing 100083, China
e-mail: liancunzheng@ustb.edu.cn

Bandar Bin-Mohsin

Department of Mathematics,
College of Science,
King Saud University,
Riyadh 14451, Saudi Arabia

Goong Chen

Department of Mathematics and
Institute for Quantum Science and Engineering,
Texas A&M University,
College Station, TX 77843;
Science Program,
Texas A & M University at Qatar,
Education City Student Center,
Doha, Qatar

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received August 17, 2016; final manuscript received February 9, 2017; published online March 21, 2017. Assoc. Editor: Alan McGaughey.

J. Heat Transfer 139(7), 074502 (Mar 21, 2017) (5 pages) Paper No: HT-16-1516; doi: 10.1115/1.4036005 History: Received August 17, 2016; Revised February 09, 2017

This paper presents research for a class of recombination reaction and diffusion problems in which the Cattaneo relaxation, n-diffusion flux, and p-Fisher–Kolmogorov–Petrovsky–Piscounov (KPP) reaction are considered. Approximate analytical solutions are obtained by Adomian decomposition method (ADM) and shown graphically. Some interesting results for spatial variable and temporal variable evolution are obtained. For specified spatial variable, the temperature profiles decrease with respect to the increase of relaxation parameter and power-law index n but decrease with respect to Fisher–KPP reaction parameter p. For specified temporal variable, the temperature profiles are seem oscillating with values of the relaxation parameter and power-law index n.

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References

Özisik, M. N. , 1993, Heat Conduction, Wiley, Hoboken, NJ.
Yang, Y. W. , 1972, “ Periodic Heat Transfer in Straight Fins,” ASME J. Heat Transfer, 94(3), pp. 310–314. [CrossRef]
Eslinger, R. G. , and Chung, B. T. F. , 1979, “ Periodic Heat Transfer in Radiating and Convecting Fins or Fin Arrays,” AIAA J., 17(10), pp. 1134–1140. [CrossRef]
Aziz, A. , and Na, T. Y. , 1981, “ Periodic Heat Transfer in Fins With Variable Thermal Parameters,” Int. J. Heat Mass Transfer, 24(8), pp. 1397–1404. [CrossRef]
Al-Sanea, S. A. , and Mujahid, A. A. , 1993, “ A Numerical Study of the Thermal Performance of Fins With Time-Dependent Boundary Conditions, Including Initial Transient Effects,” Warme Stoffubertragung, 28(7), pp. 417–424. [CrossRef]
Joseph, D. D. , and Prezioso, L. , 1989, “ Heat Weaves,” Rev. Mod. Phys., 61(1), pp. 41–73. [CrossRef]
Cattaneo, C. , 1948, “ On the Conduction of Heat,” Atti Semin. Mat. Fis. Univ. Modena, 3, pp. 3–21.
Vernotte, P. M. , 1958, “ Paradoxes in the Continuous Theory of the Heat Equation,” C. R. Acad. Sci. Paris, 246(22), pp. 3154–3155.
Qi, H. , and Guo, X. , 2014, “ Transient Fractional Heat Conduction With Generalized Cattaneo Model,” Int. J. Heat Mass Transfer, 76, pp. 535–539. [CrossRef]
Xu, G.-Y. , Wang, J.-B. , and Han, Z. , 2016, “ Notes on “The Cattaneo-Type Time Fractional Heat Conduction Equation for Laser Heating” [Comput. Math. Appl. 66 (2013) 824-831],” Comput. Math. Appl., 71(10), pp. 2132–2137. [CrossRef]
Ronney, P. , 1995, “ Some Open Issues in Premixed Turbulent Combustion,” Mathematical Modelling in Combustion Science (Lecture Notes in Physics), J. Buckmaster and T. Takeno , eds., Springer-Verlag, Berlin, pp. 3–22.
Williams, F. , 1983, Combustion Theory, Addison-Wesley, Reading, MA.
Zeldovich, Y. B. , Barenblatt, G. I. , Librovich, V. B. , and Makhvi-Ladze, G. M. , 1985, The Mathematical Theory of Combustion and Explosions, Consultants Bureau, New York.
Peters, N. , 2000, Turbulent Combustion, Cambridge University Press, Cambridge, UK.
Tretyakov, M. V. , and Fedotov, S. , 2001, “ On the FKPP Equation With Gaussian Shear Advection,” Phys. D: Nonlinear Phenomena, 159(3–4), pp. 190–201. [CrossRef]
Philip, J. R. , 1961, “ n-Diffusion,” Aust. J. Phys., 14(1), pp. 1–13. [CrossRef]
Wu, Z. , 1985, “ A Free Boundary Problem for Degenerate Quasilinear Parabolic Equations,” Nonlinear Anal., 9(9), pp. 937–951. [CrossRef]
Zheng, L. , Zhang, X. , and He, J. , 2002, “ Free Boundary Value Problems for a Class of Generalized Diffusion Equation,” J. Univ. Sci. Technol. Beijing, 9(6), pp. 422–425.
Adomian, G. , 1994, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Boston, MA.
Adomian, G. , 1988, “ A Review of the Decomposition Method in Applied Mathematics,” J. Math. Anal. Appl., 135(2), pp. 501–544. [CrossRef]
Adomian, G. , 1994, “ On the Solution of Partial Differential Equations With Specified Boundary Conditions,” Math. Comput. Modell., 140(2), pp. 569–581.
Wazwaz, A. M. , 1997, A First Course in Integral Equations, World Scientific, Singapore.
Wazwaz, A. M. , 1999, “ The Modified Decomposition Method and Padé Approximants for Solving the Thomas–Fermi Equation,” Appl. Math. Comput., 105(1), pp. 11–29.
Wazwaz, A. M. , 2000, “ A New Algorithm for Calculating Adomian Polynomials for Nonlinear Polynomials,” Appl. Math. Comput., 111(1), pp. 53–69.
Mickens, R. E. , and Jordan, P. M. , 2004, “ A Positivity-Preserving Nonstandard Finite Difference Scheme for the Damped Wave Equation,” Numer. Methods Partial Differ. Equations, 20(5), pp. 639–649. [CrossRef]
Cherrauault, Y. , Saccomandi, G. , and Some, B. , 1992, “ New Results for Convergence of Adomian's Method Applied to Integral Equations,” Math. Comput. Modell., 16(2), pp. 85–93. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Comparison between ADM and exact temperature profiles when n=1,p=0,ζ=τ0=0.02 at t=0.1

Grahic Jump Location
Fig. 2

The effects of relaxation parameter ζ on the temperature profile at x=0.1 when n=0.5,p=3

Grahic Jump Location
Fig. 3

The effects of power-law exponent n on the temperature profile at x=0.1 when ζ=0.1,p=2

Grahic Jump Location
Fig. 4

The effects of Fisher–KPP reaction parameter p on temperature at x=0.1 when n=0.5,ζ=0.2

Grahic Jump Location
Fig. 5

The effects of relaxation parameter ζ on temperature at t=0.1 when n=2,p=0.2

Grahic Jump Location
Fig. 6

The effects of Fisher–KPP reaction parameter p on temperature at t=0.1 when n=2,ζ=0.2

Grahic Jump Location
Fig. 7

The effects of n on temperature at t=0.1 when p=0.8,ζ=0.2

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