About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 610314, 19 pages
http://dx.doi.org/10.1155/2013/610314
Research Article

Application of Fuzzy Fractional Kinetic Equations to Modelling of the Acid Hydrolysis Reaction

1Institute of Advanced Technology, Universiti Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia
2Institute for Mathematical Research, Universiti Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia
3Department of Mathematics, Universiti Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia
4Young Researchers and Elite Club, Mobarakeh Branch, Islamic Azad University, P.O. Box 9189945, Mobarakeh, Iran
5Department of Chemical and Environmental Engineering, Faculty of Engineering, Universiti Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia

Received 20 May 2013; Revised 24 June 2013; Accepted 27 June 2013

Academic Editor: Ali H. Bhrawy

Copyright © 2013 Ferial Ghaemi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. Rodríguez-Chong, J. A. Ramírez, G. Garrote, and M. Vázquez, “Hydrolysis of sugar cane bagasse using nitric acid: a kinetic assessment,” Journal of Food Engineering, vol. 61, no. 2, pp. 143–152, 2004. View at Publisher · View at Google Scholar · View at Scopus
  2. Y. Lu and N. S. Mosier, “Kinetic modeling analysis of maleic acid-catalyzed hemicellulose hydrolysis in corn stover,” Biotechnology and Bioengineering, vol. 101, no. 6, pp. 1170–1181, 2008. View at Publisher · View at Google Scholar · View at Scopus
  3. Y. Sun, X. Lu, S. Zhang, R. Zhang, and X. Wang, “Kinetic study for Fe(NO3)3 catalyzed hemicellulose hydrolysis of different corn stover silages,” Bioresource Technology, vol. 102, no. 3, pp. 2936–2942, 2011. View at Publisher · View at Google Scholar · View at Scopus
  4. J. F. Saeman, “Kinetics of wood Saccharification-Hydrolysis of cellulose and decomposition of sugars in dilute acid at high temperature,” Industrial and Engineering Chemistry, vol. 37, pp. 43–52, 1945. View at Publisher · View at Google Scholar
  5. W. Faith, “Development of the Scholler Process in The United States,” Industrial & Engineering Chemistry, vol. 37, pp. 9–11, 1945.
  6. M. Neureiter, H. Danner, C. Thomasser, B. Saidi, and R. Braun, “Dilute-acid hydrohlysis of sugarcane bagasse at varying conditions,” Applied Biochemistry and Biotechnology A, vol. 98–100, pp. 49–58, 2002. View at Publisher · View at Google Scholar · View at Scopus
  7. J. S. Kim, Y. Y. Lee, and R. W. Torget, “Cellulose hydrolysis under extremely low sulfuric acid and high-temperature conditions,” Applied Biochemistry and Biotechnology A, pp. 91–93, 331–340, 2001. View at Publisher · View at Google Scholar · View at Scopus
  8. M. Caputo, “Linear model of dissipation whose Q is almost frequency independent, II, Geophys,” Journal of the Royal Astronomical Society, vol. 13, pp. 529–539, 1967.
  9. D. D. Clayton, Principles of Stellar Evolution and Nucleosynthesis, The University of Chicago Press, Chicago, Ill, USA, 2nd edition, 1983.
  10. W. G. Glöckle and T. F. Nonnenmacher, “Fox function representation of non-Debye relaxation processes,” Journal of Statistical Physics, vol. 71, no. 3-4, pp. 741–757, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. R. Hilfer, “Fractional time evolution,” in Applications of Fractional Calculus in Physics, pp. 87–130, World Scientific Publishing, River Edge, NJ, USA, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. V. L. Kobelev, E. P. Romanov, L. Y. Kobelev, and Y. L. Kobelev, “Relaxational and diffusive processes in fractal space,” Izvestiya Akademii Nauk. Seriya Fizicheskaya, vol. 62, no. 12, pp. 2401–2408, 1998. View at Scopus
  13. F. Mainardi, “Fractional diffusive waves in viscoelastic solids,” in Nonlinear Waves in Solids, J. L. Wegner and F. R. Norwood, Eds., pp. 93–97, ASME, New York, NY, USA, 1995.
  14. F. Mainardi, “Fractal calculus: some basic problems in continuum and statistical mechanics,” in Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi, Eds., pp. 291–348, Springer, Wien, Austria, 1997.
  15. F. Mainardi and M. Tomirotti, “Seismic pulse propagation with constant Q and stable probability distributions,” Annali di Geofisica, vol. 40, no. 5, pp. 1311–1325, 1997. View at Scopus
  16. R. Metzler, W. G. Glöckle, and T. F. Nonnenmacher, “Fractional model equation for anomalous diffusion,” Physica A, vol. 211, no. 1, pp. 13–24, 1994. View at Scopus
  17. A. I. Saichev and G. M. Zaslavsky, “Fractional kinetic equations: solutions and applications,” Chaos, vol. 7, no. 4, pp. 753–764, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. G. M. Zaslavsky, “Fractional kinetics of Hamiltonian chaotic systems,” in Applications of Fractional Calculus in Physics, pp. 203–239, World Scientific Publishing, River Edge, NJ, USA, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus. Models and Numerical Methods, vol. 3 of Series on Complexity, Nonlinearity and Chaos, World Scientific Publishing, Hackensack, NJ, USA, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  20. D. Baleanu, Z. B. Güvenc, and J. A. Tenreiro Machado, New Trends in Nanotechnology and Fractional Calculus Applications, Springer, New York, NY, USA, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  21. D. Baleanu and J. I. Trujillo, “A new method of finding the fractional Euler-Lagrange and Hamilton equations within Caputo fractional derivatives,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 5, pp. 1111–1115, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. P. L. Butzer and U. Westphal, “An introduction to fractional calculus,” in Applications of Fractional Calculus in Physics, pp. 1–85, World Scientific Publishing, River Edge, NJ, USA, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. M. M. Djrbashian, Harmonic Analysis and Boundary Value Problems in the Complex Domain, vol. 65 of Operator Theory: Advances and Applications, Birkhäuser, Basel, Switzerland, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  24. K. Diethelm, N. J. Ford, A. D. Freed, and Yu. Luchko, “Algorithms for the fractional calculus: a selection of numerical methods,” Computer Methods in Applied Mechanics and Engineering, vol. 194, no. 6–8, pp. 743–773, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  25. K. Diethelm, The Analysis of Fractional Differential Equations. An Application-Oriented Exposition Using Differential Operators of Caputo Type, vol. 2004 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  26. R. Gorenflo and F. Mainardi, “Fractional calculus: integral and differential equations of fractional order,” in Fractals and Fractional Calculus in Continuum Mechanics (Udine, 1996), A. Carpinteri and F. Mainardi, Eds., vol. 378 of CISM Courses and Lectures, pp. 223–276, Springer, Vienna, Austria, 1997. View at MathSciNet
  27. A. Kadem and D. Baleanu, “Fractional radiative transfer equation within Chebyshev spectral approach,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1865–1873, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993. View at MathSciNet
  29. K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974. View at MathSciNet
  30. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. View at MathSciNet
  31. A. N. Kochubeĭ, “The Cauchy problem for evolution equations of fractional order,” Journal of Differential Equations, vol. 25, pp. 967–974, 1989.
  32. H. J. Haubold and A. M. Mathai, “The fractional kinetic equation and thermonuclear functions,” Astrophysics and Space Science, vol. 273, no. 1–4, pp. 53–63, 2000. View at Scopus
  33. R. K. Saxena, A. M. Mathai, and H. J. Haubold, “On fractional kinetic equations,” Astrophysics and Space Science, vol. 282, no. 1, pp. 281–287, 2002. View at Publisher · View at Google Scholar · View at Scopus
  34. R. K. Saxena, A. M. Mathai, and H. J. Haubold, “Unified fractional kinetic equation and a fractional diffusion equation,” Astrophysics and Space Science, vol. 290, no. 3-4, pp. 299–310, 2004. View at Scopus
  35. A. Atangana and A. Kilicman, “Analytical solutions of the space-time fractional derivative of advection dispersion equation,” Mathematical Problems in Engineering, vol. 2013, Article ID 853127, 9 pages, 2013. View at Publisher · View at Google Scholar
  36. A. M. A. El-Sayed, I. L. El-Kalla, and E. A. A. Ziada, “Analytical and numerical solutions of multi-term nonlinear fractional orders differential equations,” Applied Numerical Mathematics, vol. 60, no. 8, pp. 788–797, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  37. H. Jiang, F. Liu, I. Turner, and K. Burrage, “Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain,” Journal of Mathematical Analysis and Applications, vol. 389, no. 2, pp. 1117–1127, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. Z. Odibat, S. Momani, and H. Xu, “A reliable algorithm of homotopy analysis method for solving nonlinear fractional differential equations,” Applied Mathematical Modelling, vol. 34, no. 3, pp. 593–600, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  39. N. H. Sweilam and M. M. Khader, “Exact solutions of some coupled nonlinear partial differential equations using the homotopy perturbation method,” Computers & Mathematics with Applications, vol. 58, no. 11-12, pp. 2134–2141, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  40. M. ur Rehman and R. A. Khan, “A numerical method for solving boundary value problems for fractional differential equations,” Applied Mathematical Modelling, vol. 36, no. 3, pp. 894–907, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  41. A. Pedas and E. Tamme, “Spline collocation methods for linear multi-term fractional differential equations,” Journal of Computational and Applied Mathematics, vol. 236, no. 2, pp. 167–176, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  42. S. Momani and Z. Odibat, “Numerical comparison of methods for solving linear differential equations of fractional order,” Chaos, Solitons & Fractals, vol. 31, no. 5, pp. 1248–1255, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  43. Z. M. Odibat and N. T. Shawagfeh, “Generalized Taylor's formula,” Applied Mathematics and Computation, vol. 186, no. 1, pp. 286–293, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  44. N. T. Shawagfeh, “Analytical approximate solutions for nonlinear fractional differential equations,” Applied Mathematics and Computation, vol. 131, no. 2-3, pp. 517–529, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  45. A. Kilicman and Z. A. A. Al Zhour, “Kronecker operational matrices for fractional calculus and some applications,” Applied Mathematics and Computation, vol. 187, no. 1, pp. 250–265, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  46. A. Kadem, Y. Luchko, and D. Baleanu, “Spectral method for solution of the fractional transport equation,” Reports on Mathematical Physics, vol. 66, no. 1, pp. 103–115, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  47. S. Abbasbandy, “An approximation solution of a nonlinear equation with Riemann-Liouville's fractional derivatives by He's variational iteration method,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 53–58, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  48. S. Abbasbandy and A. Shirzadi, “Homotopy analysis method for multiple solutions of the fractional Sturm-Liouville problems,” Numerical Algorithms, vol. 54, no. 4, pp. 521–532, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  49. C. Hwang and Y. P. Shih, “Laguerre operational matrices for fractional calculus and applications,” International Journal of Control, vol. 34, no. 3, pp. 577–584, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  50. G. Maione, “A digital, noninteger order, differentiator using laguerre orthogonal sequences,” International Journal of Intelligent Systems, vol. 11, pp. 77–81, 2006.
  51. Y. Li and N. Sun, “Numerical solution of fractional differential equations using the generalized block pulse operational matrix,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1046–1054, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  52. S. Kazem, S. Abbasbandy, and S. Kumar, “Fractional-order Legendre functions for solving fractional order differential equations,” Applied Mathematical Modelling, vol. 37, pp. 5498–5510, 2013. View at Publisher · View at Google Scholar
  53. A. Saadatmandi and M. Dehghan, “A new operational matrix for solving fractional-order differential equations,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1326–1336, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  54. E. H. Doha, A. H. Bhrawy, and S. S. Ezz-Eldien, “A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order,” Computers & Mathematics with Applications, vol. 62, no. 5, pp. 2364–2373, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  55. E. H. Doha, A. H. Bhrawy, and S. S. Ezz-Eldien, “Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations,” Applied Mathematical Modelling, vol. 35, no. 12, pp. 5662–5672, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  56. M. M. Khader, T. S. El Danaf, and A. S. Hendy, “A computational matrix method for solving systems of high order fractional differential equations,” Applied Mathematical Modelling, vol. 37, pp. 4035–4050, 2013. View at Publisher · View at Google Scholar
  57. E. H. Doha, A. H. Bhrawy, and S. S. Ezz-Eldien, “A new Jacobi operational matrix: an application for solving fractional differential equations,” Applied Mathematical Modelling, vol. 36, no. 10, pp. 4931–4943, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  58. S. Kazem, “An integral operational matrix based on Jacobi polynomials for solving fractional-order differential equations,” Applied Mathematical Modelling, vol. 37, no. 3, pp. 1126–1136, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  59. A. H. Bhrawy, M. M. Alghamdi, and T. M. Taha, “A new modified generalized Laguerre operational matrix of fractional integration for solving fractional differential equations on the half line,” Advances in Difference Equations, vol. 2012, article 179, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  60. A. H. Bhrawy, A. S. Alofi, and S. S. Ezz-Eldien, “A quadrature tau method for fractional differential equations with variable coefficients,” Applied Mathematics Letters, vol. 24, no. 12, pp. 2146–2152, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  61. S. Esmaeili and M. Shamsi, “A pseudo-spectral scheme for the approximate solution of a family of fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 9, pp. 3646–3654, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  62. S. Abbasbandy, T. A. Viranloo, Ó. López-Pouso, and J. J. Nieto, “Numerical methods for fuzzy differential inclusions,” Computers & Mathematics with Applications, vol. 48, no. 10-11, pp. 1633–1641, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  63. S. Abbasbandy, E. Babolian, and M. Allame, “Numerical solution of fuzzy max-min systems,” Applied Mathematics and Computation, vol. 174, no. 2, pp. 1321–1328, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  64. T. Allahviranloo, N. Ahmady, and E. Ahmady, “Numerical solution of fuzzy differential equations by predictor-corrector method,” Information Sciences, vol. 177, no. 7, pp. 1633–1647, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  65. T. Allahviranloo, S. Abbasbandy, S. Salahshour, and A. Hakimzadeh, “A new method for solving fuzzy linear differential equations,” Computing, vol. 92, no. 2, pp. 181–197, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  66. E. Babolian, H. Sadeghi Goghary, and S. Abbasbandy, “Numerical solution of linear Fredholm fuzzy integral equations of the second kind by Adomian method,” Applied Mathematics and Computation, vol. 161, no. 3, pp. 733–744, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  67. B. Bede and S. G. Gal, “Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations,” Fuzzy Sets and Systems, vol. 151, no. 3, pp. 581–599, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  68. A. Bencsik, B. Bede, J. Tar, and J. Fodor, “Fuzzy differential equations in modeling hydraulic differential servo cylinders,” in Proceedings of the 3rd Romanian-Hungarian Joint Symposium on Applied Computational Intellidence (SACI '06), Timisoara, Romania, 2006.
  69. Y. Chalco-Cano and H. Román-Flores, “On new solutions of fuzzy differential equations,” Chaos, Solitons and Fractals, vol. 38, no. 1, pp. 112–119, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  70. S. Salahshour and T. Allahviranloo, “Applications of fuzzy Laplace transforms,” Soft Computing, pp. 1–14, 2012.
  71. R. P. Agarwal, V. Lakshmikantham, and J. J. Nieto, “On the concept of solution for fractional differential equations with uncertainty,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 6, pp. 2859–2862, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  72. T. Allahviranloo, S. Salahshour, and S. Abbasbandy, “Explicit solutions of fractional differential equations with uncertainty,” Soft Computing, vol. 16, no. 2, pp. 297–302, 2012. View at Publisher · View at Google Scholar · View at Scopus
  73. T. Allahviranloo, Z. Gouyandeh, and A. Armand, “Fuzzy fractional differential equations under generalized fuzzy Caputo derivative,” Journal of Intelligent and Fuzzy Systems, 2013. View at Publisher · View at Google Scholar
  74. S. Salahshour, T. Allahviranloo, S. Abbasbandy, and D. Baleanu, “Existence and uniqueness results for fractional differential equations with uncertainty,” Advances in Difference Equations, vol. 2012, article 112, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  75. S. Salahshour, T. Allahviranloo, and S. Abbasbandy, “Solving fuzzy fractional differential equations by fuzzy Laplace transforms,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 3, pp. 1372–1381, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  76. A. Ahmadian, M. Suleiman, S. Salahshour, and D. Baleanu, “A Jacobi operational matrix for solving a fuzzy linear fractional differential equation,” Advances in Difference Equations, vol. 2013, article 104, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  77. A. Ahmadian, M. Suleiman, and S. Salahshour, “An operational matrix based on legendre polynomials for solving fuzzy fractional-order differential equations,” Abstract and Applied Analysis, vol. 2013, Article ID 505903, 29 pages, 2013. View at Publisher · View at Google Scholar
  78. M. Mazandarani and A. V. Kamyad, “Modified fractional Euler method for solving fuzzy fractional initial value problem,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 1, pp. 12–21, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  79. P. Diamond and P. Kloeden, Metric Spaces of Fuzzy Sets. Theory and Applications, World Scientific Publishing, River Edge, NJ, USA, 1994. View at MathSciNet
  80. D. Dubois and H. Prade, “Towards fuzzy differential calculus part 3: differentiation,” Fuzzy Sets and Systems, vol. 8, no. 3, pp. 225–233, 1982. View at Scopus
  81. R. Goetschel, Jr. and W. Voxman, “Elementary fuzzy calculus,” Fuzzy Sets and Systems, vol. 18, no. 1, pp. 31–43, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  82. H.-J. Zimmermann, Fuzzy Set Theory—And Its Applications, Kluwer Academic Publishers, Boston, Mass, USA, 2nd edition, 1992. View at MathSciNet
  83. G. A. Anastassiou and S. G. Gal, “On a fuzzy trigonometric approximation theorem of Weierstrass-type,” Journal of Fuzzy Mathematics, vol. 9, no. 3, pp. 701–708, 2001. View at Zentralblatt MATH · View at MathSciNet
  84. G. A. Anastassiou, Fuzzy Mathematics: Approximation Theory, vol. 251 of Studies in Fuzziness and Soft Computing, Springer, Berlin, Germany, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  85. O. Kaleva, “Fuzzy differential equations,” Fuzzy Sets and Systems, vol. 24, no. 3, pp. 301–317, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  86. T. Allahviranloo and M. Afshar Kermani, “Solution of a fuzzy system of linear equation,” Applied Mathematics and Computation, vol. 175, no. 1, pp. 519–531, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  87. M. Friedman, M. Ming, and A. Kandel, “Fuzzy linear systems,” Fuzzy Sets and Systems, vol. 96, no. 2, pp. 201–209, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  88. L. F. Wilhelmy, “ber das Gesetz, nach welchem die Einwirkung der Suren auf Rohrzucker stattfindet,” Poggendorff's Annalen der Physik und Chemie, vol. 81, pp. 413–433, 499–526, 1850.
  89. P. Waage and C. M. Guldberg, “Studier over affiniteten forhandlinger,” VIdenskabs-Selskabet I ChrIstIana, vol. 35–40, pp. 111–120, 1864.
  90. K. J. Laidler, The World of Physical Chemistry, Oxford University Press, Oxford, UK, 1993.
  91. A. Cornish-Bowden, Fundamentals of Enzyme Kinetics, John Wiley & Sons, New York, NY, USA, 4th edition, 2012.
  92. M. Robson Wright, An Introduction to Chemical Kinetics, John Wiley & Sons, New York, NY, USA, 2004.
  93. F. Mainardi, “Fractional relaxation-oscillation and fractional diffusion-wave phenomena,” Chaos, Solitons and Fractals, vol. 7, no. 9, pp. 1461–1477, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  94. P. Humbert and R. P. Agarwal, “Sur la fonction de Mittag-Leffler et quelques-unes de ses généralisations,” Bulletin des Sciences Mathématiques. 2e Série, vol. 77, pp. 180–185, 1953. View at Zentralblatt MATH · View at MathSciNet
  95. S. Faridah Salleh, R. Yunus, M. Farid Atan, and D. R. Awg Biak, “Kinetic studies on acid hydrolysis of OPEFB in a batch reactor,” in Proceedings of International Chemical, Biological and Environmental Engineering (IPCBEE '12), vol. 38, IACSIT Press, 2012.
  96. S. Abbasbandy and M. Amirfakhrian, “A new approach to universal approximation of fuzzy functions on a discrete set of points,” Applied Mathematical Modelling, vol. 30, no. 12, pp. 1525–1534, 2006. View at Publisher · View at Google Scholar · View at Scopus
  97. S. Abbasbandy and M. Amirfakhrian, “Best approximation of fuzzy functions,” Journal of Nonlinear Studies, vol. 14, pp. 88–103, 2007.
  98. G. A. Anastassiou, “Fuzzy approximation by fuzzy convolution type operators,” Computers & Mathematics with Applications, vol. 48, no. 9, pp. 1369–1386, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  99. H. Huang and C. Wu, “Approximation of fuzzy functions by regular fuzzy neural networks,” Fuzzy Sets and Systems, vol. 177, pp. 60–79, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  100. M. Dehghan and A. Saadatmandi, “A tau method for the one-dimensional parabolic inverse problem subject to temperature overspecification,” Computers & Mathematics with Applications, vol. 52, no. 6-7, pp. 933–940, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  101. J. H. Freilich and E. L. Ortiz, “Numerical solution of systems of ordinary differential equations with the Tau method: an error analysis,” Mathematics of Computation, vol. 39, no. 160, pp. 467–479, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet