Numerical modeling of generalized nonlinear system arising in thermoelasticity

[1.] J.A. Burns, Z. Liu, S. Zheng, On the energy decay of a linear thermoelastic
bar, J. Math. Anal. Appl. 179 (1993).591–574
[2.] C.M. Dafermos, On the existence and the asymptotic stability of solutions to
the equations of linear thermoelasticity, Arch. Rational Mech. Anal. 29
(1969) 241–271.
[3.] C.M. Dafermos, L. Hsiao, Global smooth thermomechanical processes in
one-dimensional nonlinear thermoviscoelasticity, Nonlinear Anal. 6 (1982)
435–454.
[4.] L.H. Fatori, J.E.M. Rivera, Energy decay for hyperbolic thermoelastic
systems of memory type, Quart. Appl.Math., 59 (2001) 441-458.
[5.] J.S. Gibson, I.G. Rosen, G. Tao, Approximation in control of thermoelastic
systems, J. Control Optim. 30 (1992)1189–1163
[6.] W.J. Hrusa, S. Messaoudi, On formation of singularities in one-dimensional
nonlinear thermoelasticity, Arch. Rational Mech. Anal. 111 (1990) 135–151.
[7.] S. Jiang, R. Racke, Evolution Equations in Thermoelasticity, in: Monogr.
Surveys Pure Appl.Math., Vol. 112, Chapman and Hall/CRC, Boca Raton,
2000.
[8.] R. Racke, Y. Shibata, Global smooth solutions and asymptotic stability in
one-dimensional nonlinear thermoelasticity,Arch. Rational Mech. Anal. 116
(1991) 1–34.
[9.] R. Racke , Y. Shibata, S. Zheng, Global solvability and exponential stability
in one-dimensional nonlinear thermoelasticity, Quart. Appl. Math. 51 (1993)
751–763.
[10.] J.E.M. Rivera, Energy decay rates in linear thermoelasticity, Funkcial. Ekvac.
35 (1992) 19–35.
Numerical modeling of generalized nonlinear system arising in thermoelasticity 525
[11.] J.E.M. Rivera, Y. Qin, Global existence and exponential stability in onedimensional
nonlinear thermoelasticity with thermal memory, Nonlinear
Anal. 51 (2002) 11–32.
[12.] M. Slemrod, Global existence, uniqueness, and asymptotic stability of
classical smooth solutions in one dimensional nonlinear thermoelasticity,
Arch. Rational Mech. Anal. 76 (1981) 97–133.
[13.] S. Zheng, W. Shen, Global solutions to the Cauchy problem of a class of
hyperbolic–parabolic coupled systems. Sci. Sinica Ser. A 30 (1987), pp.
1133–1149.
[14.] A.N. Abd-Alla, A.F. Ghaleb, G.A. Maugin, Harmonic wave generation in
nonlinear thermoelasticity, Int. J. Eng. Sci., 32 (1994) 113-1116.
[15.] G. Adomain, Kluwer A academic publisher, Boston,1994.
[16.] G. Adomain, Math. Anal. Appl. 135(1988) 501-544.
[17.] J. D. Munkhammar, Fractional calculus and the Taylor–Riemann series.
Undergrad J Math 2005:6.
[18.] R. L. Magin, Fractional calculus in bioengineering, Crit. Rev. Biomed. Eng.,
32 (1) (2004) 1-104.
[19.] R. L. Magin, Fractional calculus in bioengineering-part 2, Crit. Rev. Biomed.
Eng., 32(2) (2004) 105-193.
[20.] R. L. Magin, Fractional calculus in bioengineering-part 3, Crit. Rev. Biomed.
Eng., 32(3/4) (2004) 194-377.
[21.] K. B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New
York, 1974.
[22.] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and
Derivatives-Theory and Applications, Gordon and Breach Science
Publishers, Longhorne, PA, 1993.
[23.] S. Z. Rida, A.M.A. El-Sayed, A.A.M. Arafa, Effect of bacterial memory
dependent growth by using fractional derivatives reaction-diffusion
chemotactic model, J. Statistical Physics, 140: (2010) 797–811.
[24.] A.M.A. El-Sayed, S. Z. Rida, A.A.M. Arafa, On the Solutions of the
generalized reaction-diffusion model for bacteria growth, Acta Appl. Math.,
110 (2010) 1501–1511.
[25.] S. Z. Rida, A.M.A. El-Sayed, A.A.M. Arafa, On the solutions of timefractional
reaction–diffusion equations, Commun Nonlinear Sci Numer
Simulat 15 (2010) 3847–3854.
[26.] S. Z. Rida, A.M.A. El-Sayed, A.A.M. Arafa, A Fractional Model for
Bacterial Chemoattractant in a Liquid Medium, Nonlinear Sci. Lett. A, Vol.
1, No.4, (, 2010) 415-420.
[27.] A.M.A. El-Sayed, S. Z. Rida, A.A.M. Arafa, Exact solutions of the
fractional-order biological Population model, Comm. In Theor. Phys, Vol.52
(2009) 992-996.
[28.] A.M.A. El-Sayed ,S. Z. Rida, A.A.M. Arafa, On the Solutions of Timefractional
Bacterial Chemotaxis in a Diffusion Gradient Chamber, Int. J. of
Nonlinear Science, Vol.7 (2009) 485-492.
526 A.A.M. Arafa
[29.] S. Z. Rida, H. M. El-Sherbiny, and A. A. M. Arafa, "On solution of nonlinear
Schrödinger equation of fractional order, Physics Letters A 372 (2008) 553–
558.
[30.] Y. Cherruault, Convergence od Adomian’s method, Kybernetes 18 (1989)
31-38.
[31.] Y. Cherruault, G. Adomian, Decomposition methods: a new proof of
convergence, Math. Comput. Modell. 18 (1993) 103-106
[32.] N. Nagarhasta, B. Some, K. Abbaoui, Y. Cherruault, New numerical study of
Adomian method applied to a diffusion model, Kybernetes 31 (2002) 61.
[33.] Podlubny I. Fractional differential equations: an introduction to fractional
derivatives, fractional differential equations, to methods of their solution and
some of their applications. New York: Academic Press; 1999.