The inverse solution of the coupled nonlinear reaction–diffusion equations by the Haar wavelets

References

• O.M. Alifanov, Inverse Heat Transfer Problems, Springer, New York, 1994.
   Google Scholar
• I. Aziz, S.U. Islam, and F. Khan, A new method based on Haar wavelet for the numerical solution of two-dimensional nonlinear integral equations, J. Comput. Appl. Math. 272 (2014), pp. 70–80. doi: 10.1016/j.cam.2014.04.027
   Web of Science ®Google Scholar
• J.V. Beck, B. Blackwell, and C.R.St. Clair, Inverse Heat Conduction: IllPosed Problems, Wiley-Interscience, New York, 1985.
   Google Scholar
• J.V. Beck and D.A. Murio, Combined function specification-regularization procedure for solution of inverse heat conduction problem, AIAA J. 24 (1986), pp. 180–185. doi: 10.2514/3.9240
   Google Scholar
• R.E. Bellman and R.E. Kalaba, Quasilinearization and Nonlinear Boundary-Value Problems, American Elsevier Publishing Company, Inc, New York, 1965.
   Google Scholar
• J.M.G. Cabeza, J.A.M. Garcia, and A.C. Rodriguez, A sequential algorithm of inverse heat conduction problems using singular value decomposition, Int. J. Thermal Sci. 44 (2005), pp. 235–244. doi: 10.1016/j.ijthermalsci.2004.06.009
   Web of Science ®Google Scholar
• L.M.S. Castro, A.J.M. Ferreira, S. Bertoluzza, R.C. Batra, and J.N. Reddy, A wavelet collocation method for the static analysis of sandwich plates using a layerwise theory, Compos. Struct. 92 (2010), pp. 1786–1792. doi: 10.1016/j.compstruct.2010.01.021
   Web of Science ®Google Scholar
• I. Celik, Haar wavelet approximation for magneto-hydrodynamic flow equations, Appl. Math. Model. 37 (2013), pp. 3894–3902. doi: 10.1016/j.apm.2012.07.048
   Web of Science ®Google Scholar
• C.F. Chen and C.H. Hsiao, Haar wavelet method for solving lumped and distributed-parameter systems, IEE Proc. Control Theory Appl. 144 (1997), pp. 87–94. doi: 10.1049/ip-cta:19970702
   Google Scholar
• S. Foadian, R. Pourgholi, and S.H. Tabasi, Cubic B-spline method for the solution of an inverse parabolic system, Appl. Anal. (2017), pp. 1–28. Available at http://dx.doi.org/10.1080/00036811.2016.1272102.
   Web of Science ®Google Scholar
• P.C. Hansen, Analysis of discrete ill-posed problems by means of the L-curve, SIAM Rev. 34 (1992), pp. 561–580. doi: 10.1137/1034115
   Web of Science ®Google Scholar
• P.C. Hansen, Rank-deficient and Discrete Ill-posed Problems, SIAM, Philadelphia, 1998.
   Google Scholar
• G. Hariharan and K. Kannan, A comparison of Haar wavelet and Adomain decomposition method for solving one-dimensional reaction–diffusion equations, Int. J. Appl. Math. Comput. Sci. 2 (2010), pp. 50–61.
   Google Scholar
• G. Hariharan and K. Kannan, Haar wavelet method for solving some nonlinear Parabolic equations, J. Math. Chem. 48 (2010), pp. 1044–1061. doi: 10.1007/s10910-010-9724-0
   Web of Science ®Google Scholar
• G. Hariharan and K. Kannan, A comparative study of Haar wavelet method and homotopy perturbation method for solving one-dimensional reaction–diffusion equations, Int. J. Appl. Math. Comput. Sci. 3 (2011), pp. 21–34.
   Google Scholar
• G. Hariharan and K. Kannan, Review of wavelet methods for the solution of reaction–diffusion problems in science and engineering, Appl. Math. Model. 38 (2014), pp. 799–813. doi: 10.1016/j.apm.2013.08.003
   Web of Science ®Google Scholar
• Y.C. Hon and T. Wei, A fundamental solution method for inverse heat conduction problem, Eng. Anal. Bound. Elem. 28 (2004), pp. 489–495. doi: 10.1016/S0955-7997(03)00102-4
   Web of Science ®Google Scholar
• A.H. Khater, W. Malfliet, D.K. Callebaut, and E.S. Kamel, The tanh method, a simple transformation and exact analytical solutions for nonlinear reaction–diffusion equations, Chaos Solitons Fractals 14 (2002), pp. 513–522. doi: 10.1016/S0960-0779(01)00247-8
   Web of Science ®Google Scholar
• A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer, New York, 1996.
   Google Scholar
• M. Kumar and S. Pandit, A composite numerical scheme for the numerical simulation of coupled Burgers equation, Comput. Phys. Commun. 185 (2014), pp. 809–817. doi: 10.1016/j.cpc.2013.11.012
   Web of Science ®Google Scholar
• Ü. Lepik and H. Hein, Haar Wavelets: With Applications, Springer International Publishing, Cham, Switzerland, 2014.
   Google Scholar
• J. Majak, B.S. Shvartsman, M. Kirs, M. Pohlak, and H. Herranen, Convergence theorem for the Haar wavelet based discretization method, Compos. Struct. 126 (2015), pp. 227–232. doi: 10.1016/j.compstruct.2015.02.050
   Web of Science ®Google Scholar
• M.A. Mohamed and M.S. Torky, Numerical solution of nonlinear system of partial differential equations by the Laplace decomposition method and the Pade approximation, Amer. J. Comput. Math. 3 (2013), pp. 175–184. doi: 10.4236/ajcm.2013.33026
   Google Scholar
• D.A. Murio, The Mollification Method and the Numerical Solution of Ill-Posed Problems, Wiley-Interscience, New York, 1993.
   Google Scholar
• D.C. Murio and J.R. Paloschi, Combined mollification-future temperature procedure for solution of inverse heat conduction problem, J. Comput. Appl. Math. 23 (1988), pp. 235–244. doi: 10.1016/0377-0427(88)90282-8
   Web of Science ®Google Scholar
• A. Patra and S.S. Ray, Two-dimensional Haar wavelet collocation method for the solution of stationary neutron transport equation in a homogeneous isotropic medium, Ann. Nucl. Energy 70 (2014), pp. 30–35. doi: 10.1016/j.anucene.2014.01.046
   Web of Science ®Google Scholar
• A. Patra and S.S. Ray, Numerical simulation based on Haar wavelet operational method to solve neutron point kinetics equation involving sinusoidal and pulse reactivity, Ann. Nucl. Energy 73 (2014), pp. 408–412. doi: 10.1016/j.anucene.2014.07.025
   Web of Science ®Google Scholar
• R. Pourgholi, H. Dana, and S.H. Tabasi, Solving an inverse heat conduction problem using genetic algorithm: Sequential and multi-core parallelization approach, Appl. Math. Model. 38 (2014), pp. 1948–1958. doi: 10.1016/j.apm.2013.10.019
   Web of Science ®Google Scholar
• R. Pourgholi, S. Foadian, and A. Esfahani, Haar basis method to solve some inverse problems for two-dimensional parabolic and hyperbolic equations, TWMS J. Appl. Eng. Math. 3 (2013), pp. 10–32.
   Google Scholar
• R. Pourgholi, M. Rostamian, and M. Emamjome, A numerical method for solving a nonlinear inverse parabolic problem, Inverse Probl. Sci. Eng. 18 (2010), pp. 1151–1164. doi: 10.1080/17415977.2010.518287
   Web of Science ®Google Scholar
• R. Pourgholi, N. Tavallaie, and S. Foadian, Applications of Haar basis method for solving some ill-posed inverse problems, J. Math. Chem. 50 (2012), pp. 2317–2337. doi: 10.1007/s10910-012-0036-4
   Web of Science ®Google Scholar
• R. Pourgholi, A. Esfahani, S. Foadian, and S. Parehkar, Resolution of an inverse problem by Haar basis and Legendre wavelet methods, Int. J. Wavelets Multiresolut. Inf. Process. 11 (2013), p. 1350034 (21 pages). doi: 10.1142/S0219691313500343
   Web of Science ®Google Scholar
• S.S. Ray, On Haar wavelet operational matrix of general order and its application for the numerical solution of fractional Bagley–Torvik equation, Appl. Math. Comput. 218 (2012), pp. 5239–5248.
   Web of Science ®Google Scholar
• S.S. Ray and A.K. Gupta, Comparative analysis of variational iteration method and Haar wavelet method for the numerical solutions of Burgers–Huxley and Huxley equations, J. Math. Chem. 52 (2014), pp. 1066–1080. doi: 10.1007/s10910-014-0327-z
   Web of Science ®Google Scholar
• S.S. Ray and A. Patra, Haar wavelet operational methods for the numerical solutions of fractional order nonlinear oscillatory van der Pol system, Appl. Math. Comput. 220 (2013), pp. 659–667.
   Web of Science ®Google Scholar
• A.A. Soliman and M.A. Abdou, Numerical solutions of nonlinear evolution equations using variational iteration method, J. Comput. Appl. Math. 207 (2007), pp. 111–120. doi: 10.1016/j.cam.2006.07.016
   Web of Science ®Google Scholar
• M. Tadi, Inverse heat conduction based on boundary measurement, Inverse Probl. 13 (1997), pp. 1585–1605. doi: 10.1088/0266-5611/13/6/012
   Web of Science ®Google Scholar
• J. Talwar, R.K. Mohanty, and S. Singh, A new spline in compression approximation for one space dimensional quasilinear parabolic equations on a variable mesh, Appl. Math. Comput. 260 (2015), pp. 82–96.
   Web of Science ®Google Scholar
• A.N. Tikhonov and V.Y. Arsenin, On the Solution of Ill-Posed Problems, Wiley, New York, 1977.
   Google Scholar
• A.N. Tikhonov and V.Y. Arsenin, Methods for Solving Ill-Posed Problems, Nauka, Moscow, 1986.
   Google Scholar