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Tau method for the numerical solution of a fuzzy fractional kinetic model and its application to the oil palm frond as a promising source of xylose. (English) Zbl 1349.65207

Summary: The Oil Palm Frond (a lignocellulosic material) is a high-yielding energy crop that can be utilized as a promising source of xylose. It holds the potential as a feedstock for bioethanol production due to being free and inexpensive in terms of collection, storage and cropping practices. The aim of the paper is to calculate the concentration and yield of xylose from the acid hydrolysis of the Oil Palm Frond through a fuzzy fractional kinetic model. The approximate solution of the derived fuzzy fractional model is achieved by using a tau method based on the fuzzy operational matrix of the generalized Laguerre polynomials. The results validate the effectiveness and applicability of the proposed solution method for solving this type of fuzzy kinetic model.

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
34A07 Fuzzy ordinary differential equations
34A08 Fractional ordinary differential equations
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[1] Himmel, M.; Ding, S. Y.; Johnson, D. K.; Adney, W. S.; Nimlos, M. R.; Brady, J. W.; Foust, T. D., Biomass recalcitrance: engineering plants and enzymes for biofuels production, Science, 315, 804-807 (2007)
[2] Salamatinia, B.; Kamaruddi, A. H.; Abdullah, A. Z., Modeling of the continuous copper and zinc removal by sorption onto sodium hydroxide-modified oil palm frond in a fixed-bed column, Chem. Eng. J., 145, 259-266 (2008)
[3] Chew, T. L.; Bhatia, S., Effect of catalyst additives on the production of biofuels from palm oil cracking in a transport riser reactor, Bioresour. Technol., 100, 2540-2545 (2009)
[4] Taherzadeh, M. J.; Eklund, R.; Gustafsson, L.; Niklasson, C.; Lidén, G., Characterization and fermentation of dilute-acid hydrolyzates from wood, Ind. Eng. Chem. Res., 36, 4659-4665 (1997)
[5] Palmarola-Adrados, B.; Chotěborská, P.; Galbe, M.; Zacchi, G., Ethanol production from non-starch carbohydrates of wheat bran, Bioresour. Technol., 96, 843-850 (2005)
[6] Saeman, J. F., Kinetics of wood saccharification-hydrolysis of cellulose and decomposition of sugars in dilute acid at high temperature, Ind. Eng. Chem., 37, 43-52 (1945)
[7] Liu, X.; Lu, M.; Ai, N.; Yu, F.; Ji, J., Kinetic model analysis of dilute sulfuric acid-catalyzed hemicellulose hydrolysis in sweet sorghum bagasse for xylose production, Ind. Crop. Prod., 38, 81-86 (2012)
[8] Valério, D.; Tenreiro Machado, J.; Kiryakova, V., Historical survey: some pioneers of the applications of fractional calculus, Fract. Calc. Appl. Anal., 17, 552-578 (2014) · Zbl 1305.26008
[9] Rivero, M.; Trujillo, J. J.; Vázquez, L.; Velasco, M. P., Fractional dynamics of populations, Appl. Comput. Math., 218, 1089-1095 (2011) · Zbl 1226.92060
[10] Velasco, M. P.; Vázquezb, L., On the fractional Newton and wave equation in one space dimension, Appl. Math. Model., 38, 3314-3324 (2014) · Zbl 1449.35451
[11] Benson, D. A.; Wheatcraft, S. W.; Meerschaert, M. M., Application of a fractional advection-dispersion equation, Water Resour. Res., 36, 1403-1412 (2000)
[12] Hall, M. G.; Barrick, T. R., From diffusion-weighted MRI to anomalous diffusion imaging, Magn. Reson. Med., 59, 447-455 (2008)
[13] Bazhlekova, E.; Bazhlekov, I., Viscoelastic flows with fractional derivative model: computational approach by convolutional calculus of Dimovski, Fract. Calc. Appl. Anal., 17, 954-976 (2014) · Zbl 1314.76007
[14] Orsingher, E.; Beghin, L., Time-fractional telegraph equations and telegraph processes with Brownian time, Probab. Theory Relat. Fields, 128, 141-160 (2004) · Zbl 1049.60062
[15] Povstenko, Y. Z., Signaling problem for time-fractional diffusion-wave equation in a half-space in the case of angular symmetry, Nonlinear Dyn., 55, 593-605 (2010) · Zbl 1189.35169
[16] Meerschaert, M. M.; Tadjeran, C., Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math., 56, 80-90 (2006) · Zbl 1086.65087
[17] Zayernouri, M.; Karniadakis, G. E., Fractional spectral collocation methods for linear and nonlinear variable order FPDEs, J. Comput. Phys., 293, 312-338 (2015) · Zbl 1349.65531
[18] Inc, M., The approximate and exact solutions of the space- and time-fractional Burger’s equations with initial conditions by VIM, J. Math. Anal. Appl., 345, 476-484 (2008) · Zbl 1146.35304
[19] Sweilam, N. H.; Khader, M. M., Exact solutions of some coupled nonlinear partial differential equations using the homotopy perturbation method, Comput. Math. Appl., 58, 2134-2141 (2009) · Zbl 1189.65259
[20] Ding, X. L.; Jiang, Y. L., Waveform relaxation methods for fractional differential-algebraic equations with the Caputo derivatives, Fract. Calc. Appl. Anal., 17, 585-604 (2014) · Zbl 1305.26017
[21] Sweilam, N. H.; Khader, M. M., A Chebyshev pseudo-spectral method for solving fractional integro-differential equations, ANZIAM J., 51, 464-475 (2010) · Zbl 1216.65187
[22] Ford, N. J.; Morgado, M. L.; Rebelo, M., Nonpolynomial collocation approximation of solutions of fractional differential equations, Fract. Calc. Appl. Anal., 16, 874-891 (2013) · Zbl 1312.65124
[23] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral Methods (2006), Springer Verlag: Springer Verlag New York · Zbl 1093.76002
[24] Saadatmandi, A.; Dehghan, M., A new operational matrix for solving fractional-order differential equations, Comput. Math. Appl., 59, 1326-1336 (2010) · Zbl 1189.65151
[25] Kazem, S.; Abbasbandy, S.; Kumar, S., Fractional-order Legendre functions for solving fractional-order differential equations, Appl. Math. Model., 37, 5498-5510 (2013) · Zbl 1449.33012
[26] Doha, E. H.; Bhrawy, A. H.; Ezz-Eldien, S. S., A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order, Comput. Math. Appl., 62, 2364-2373 (2011) · Zbl 1231.65126
[27] Bhrawy, A. H.; Doha, E. H.; Baleanu, D.; Ezz-Eldien, S. S., A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations, J. Comput. Phys., 293, 142-156 (2015) · Zbl 1349.65504
[28] Bhrawy, A. H.; Alhamed, Y. A.; Baleanu, D.; Al-Zahrani, A. A., New spectral techniques for systems of fractional differential equations using fractional-order generalized Laguerre orthogonal functions, Fract. Calc. Appl. Anal., 17, 1137-1157 (2014) · Zbl 1312.65166
[29] Chang, S. S.L.; Zadeh, L. A., On fuzzy mapping and control, IEEE Trans. Syst. Man Cybern., SMC-2, 30-34 (1972) · Zbl 0305.94001
[30] Hüllermeier, E., An approach to modelling and simulation of uncertain dynamical systems, Int. J. Uncertain. Fuzziness Knowl.-Based Syst., 5, 117-137 (1997) · Zbl 1232.68131
[31] Khastan, A.; Nieto, J. J., A boundary value problem for second order fuzzy differential equations, Nonlinear Anal., 72, 3583-3593 (2010) · Zbl 1193.34004
[32] Bede, B.; Rudas, I. J.; Bencsik, A. L., First order linear fuzzy differential equations under generalized differentiability, Inf. Sci., 177, 1648-1662 (2007) · Zbl 1119.34003
[33] Agarwal, R. P.; Lakshmikantham, V.; Nieto, J. J., On the concept of solution for fractional differential equations with uncertainty, Nonlinear Anal., 72, 2859-2862 (2010) · Zbl 1188.34005
[34] Ahmadian, A.; Suleiman, M.; Salahshour, S.; Baleanu, D., A Jacobi operational matrix for solving fuzzy linear fractional differential equation, Adv. Differ. Equ., 2013, 104 (2013) · Zbl 1380.34004
[35] Ahmadian, A.; Suleiman, M.; Salahshour, S., An operational matrix based on Legendre polynomials for solving fuzzy fractional-order differential equations, Abstr. Appl. Anal., 2013 (2013), 29 pp., Art. ID 505903 · Zbl 1294.65074
[36] Allahviranloo, T.; Salahshour, S.; Abbasbandy, S., Explicit solutions of fractional differential equations with uncertainty, Soft Comput., 16, 297-302 (2012) · Zbl 1259.34009
[37] Salahshour, S.; Allahviranloo, T.; Abbasbandy, S.; Baleanu, D., Existence and uniqueness results for fractional differential equations with uncertainty, Adv. Differ. Equ., 2012, 112 (2012) · Zbl 1350.34011
[38] Salahshour, S.; Allahviranloo, T.; Abbasbandy, S., Solving fuzzy fractional differential equations by fuzzy Laplace transforms, Commun. Nonlinear Sci. Numer. Simul., 17, 1372-1381 (2012) · Zbl 1245.35146
[39] Mazandarani, M.; Vahidian Kamyad, A., Modified fractional Euler method for solving fuzzy fractional initial value problem, Commun. Nonlinear Sci. Numer. Simul., 18, 12-21 (2013) · Zbl 1253.35208
[40] Malinowski, M. T., Random fuzzy fractional integral equations - theoretical foundations, Fuzzy Sets Syst., 265, 39-62 (2015) · Zbl 1361.45010
[41] Alikhani, R.; Bahrami, F., Global solutions for nonlinear fuzzy fractional integral and integrodifferential equations, Nonlinear Sci. Numer. Simul., 18, 2007-2017 (2013) · Zbl 1281.45005
[42] Dubios, D.; Prade, H., Towards fuzzy differential calculus - part 3, Fuzzy Sets Syst., 8, 225-234 (1982) · Zbl 0499.28009
[43] Bhrawy, A. H.; Alghamdi, M. A.; Taha, T. M., A new modified generalized Laguerre operational matrix of fractional integration for solving fractional differential equations on the half line, Adv. Differ. Equ., 179 (2012) · Zbl 1380.34008
[44] Bhrawy, A. H.; Alghamdi, M. A., The operational matrix of Caputo fractional derivatives of modified generalized Laguerre polynomials and its applications, Adv. Differ. Equ., 2013, 307 (2013) · Zbl 1444.65023
[45] Baleanu, D.; Bhrawy, A. H.; Taha, T. M., Two efficient generalized Laguerre spectral algorithms for fractional initial value problems, Abstr. Appl. Anal., 2013 (2013), 10 pp., Art. ID 546502 · Zbl 1291.65240
[46] Goetschel, R.; Voxman, W., Elementary calculus, Fuzzy Sets Syst., 18, 31-43 (1986) · Zbl 0626.26014
[47] Anastassiou, G. A.; Gal, S. G., On a fuzzy trigonometric approximation theorem of Weierstrass-type, J. Fuzzy Math., 9, 701-708 (2001) · Zbl 1004.42005
[48] Anastassiou, G. A., Fuzzy Mathematics: Approximation Theory (2010), Springer: Springer Heidelberg · Zbl 1476.41001
[49] Kaleva, O., Fuzzy differential equations, Fuzzy Sets Syst., 24, 301-317 (1987) · Zbl 0646.34019
[50] Chalco-Cano, Y.; Román-Flores, H., On new solutions of fuzzy differential equations, Chaos Solitons Fractals, 38, 112-119 (2008) · Zbl 1142.34309
[51] Allahviranloo, T.; Afshar Kermani, M., Solution of a fuzzy system of linear equation, Appl. Math. Comput., 175, 519-531 (2006) · Zbl 1095.65036
[52] Diethelm, K.; Ford, N. J.; Freed, A. D.; Luchko, Yu., Algorithms for the fractional calculus: a selection of numerical methods, Comput. Methods Appl. Mech. Eng., 194, 743-773 (2005) · Zbl 1119.65352
[53] Baleanu, D.; Diethelm, K.; Scalas, E.; Trujillo, J. J., Fractional Calculus: Models and Numerical Methods (2012), World Scientific · Zbl 1248.26011
[54] Sun, R.; Song, X.; Sun, R.; Jiang, J., Effect of lignin content on enzymatic hydrolysis of furfural residue, BioResour., 1, 317-328 (2011)
[55] Funaro, D., Polynomial Approximations of Differential Equations (1992), Springer-Verlag · Zbl 0785.65087
[56] Szegö, G., Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23 (1975), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · JFM 65.0278.03
[57] Dimitrov, D. K.; Marcellán, F.; Rafaeli, F. R., Monotonicity of zeros of Laguerre-Sobolev-type orthogonal polynomials, Math. Anal. Appl., 368, 80-89 (2010) · Zbl 1202.33015
[58] Dehghan, M.; Saadatmandi, A., A tau method for one-dimensional parabolic inverse problem subject to temperature overspecification, Comput. Math. Appl., 52, 933-940 (2006) · Zbl 1125.65340
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