Esmaeili, S. (2017). Numerical solution of gas solution in a fluid: fractional derivative model. Iranian Journal of Mathematical Chemistry, 8(4), 425-437. doi: 10.22052/ijmc.2017.54560.1203

S. Esmaeili. "Numerical solution of gas solution in a fluid: fractional derivative model". Iranian Journal of Mathematical Chemistry, 8, 4, 2017, 425-437. doi: 10.22052/ijmc.2017.54560.1203

Esmaeili, S. (2017). 'Numerical solution of gas solution in a fluid: fractional derivative model', Iranian Journal of Mathematical Chemistry, 8(4), pp. 425-437. doi: 10.22052/ijmc.2017.54560.1203

Esmaeili, S. Numerical solution of gas solution in a fluid: fractional derivative model. Iranian Journal of Mathematical Chemistry, 2017; 8(4): 425-437. doi: 10.22052/ijmc.2017.54560.1203

Numerical solution of gas solution in a fluid: fractional derivative model

^{}Department of Applied Mathematics, University of Kurdistan

Abstract

A computational technique for solution of mathematical model of gas solution in a fluid is presented. This model describes the change of mass of the gas volume due to diffusion through the contact surface. An appropriate representation of the solution based on the M"{u}ntz polynomials reduces its numerical treatment to the solution of a linear system of algebraic equations. Numerical examples are given and discussed to illustrate the effectiveness of the proposed approach.

1. A. Ansari and M. Ahmadi Darani, On the generalized mass transfer with a chemical reaction: Fractional derivative model, Iranian J. Math. Chem. 7 (2016) 77–88 2. Y. I. Babenko, Heat and Mass Transfer: The Method of Calculation for the Heat and Diffusion Flows (in Russian), Khimiya, Leningrad, 1986.

3. Y. I. Babenko, The Method of Fractional Differentiation in Applied Problems of Heat and Mass Transfer Theory (in Russian), NPO Professional Publ, St Petersburg, 2009. 4. D. Baleanu, K. Diethelm, E. Scalas and J.J. Trujillo, Fractional Calculus: Models and Numerical Methods 2nd ed., World Scientific, Singapore, 2016. 5. P. Borwein, T. Erdélyi and J. Zhang, Müntz systems and orthogonal Müntz–Legendre polynomials, Trans. Amer. Math. Soc.342 (1994) 523–542. 6. D. W. Brzeziński, Accuracy problems of numerical calculation of fractional order derivatives and integrals applying the Riemann–Liouville/Caputo formulas, Appl. Math. Nonlinear Sci. 1 (2016) 23–44. 7. M. Caputo, Linear models of dissipation whose Q is almost frequency independent – II, Geophys. J. Roy. Astron. Soc. 13 (1967) 529–539. 8. A. S. Cvetković, G. V. Milovanović, The Mathematica package orthogonal polynomials, Facta. Univ. Ser. Math. Inform. 19 (2004) 17–36. 9. K. Diethelm, The Analysis of Fractional Differential Equations, Springer, Berlin, 2010. 10. K. Diethelm and Y. Luchko, Numerical solution of linear multi–term initial value problems of fractional order, J. Comput. Anal. Appl. 6 (2004) 243–263. 11. S. Esmaeili and G. V. Milovanović, Nonstandard Gauss–Lobatto quadrature approximation to fractional derivatives, Fract. Calc. Appl. Anal. 17 (2014) 1075–1099. 12. S. Esmaeili, M. Shamsi and Y. Luchko, Numerical solution of fractional differential equations with a collocation method based on Müntz polynomials, Comput. Math. Appl. 62 (2011) 918–929. 13. B. A. Finlayson, The Method of Weighted Residuals and Variational Principles, with Application in Fluid Mechanics, Heat and Mass Transfer, Academic Press, New York, 1972. 14. W. Gautschi, Orthogonal Polynomials in MATLAB: Exercises and Solutions, SIAM, Philadelphia, 2016. 15. G. H. Golub and J. H. Welsch, Calculation of Gauss quadrature rules, Math. Comp. 23 (1969) 221–230. 16. Y. Luchko and R. Gorenflo, An operational method for solving fractional differential equations with the Caputo derivatives, Acta Math. Vietnam. 24 (1999) 207–233. 17. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, 2010. 18. G. V. Milovanović, Müntz orthogonal polynomials and their numerical evaluation, in: W. Gautschi, G. H. Golub and G. Opfer (Eds.), Applications and Computation of Orthogonal Polynomials, Birkhäuser, Basel 131 (1999) 179–194. 19. G. V. Milovanović and A. S. Cvetković, Special classes of orthogonal polynomials and corresponding quadratures of Gaussian type, Math. Balkanica. 26 (2012) 169–184.

20. P. Mokhtary, F. Ghoreishi and H. M. Srivastava, The Müntz–Legendre Tau method for fractional differential equations, Appl. Math. Model. 40 (2016) 671–684. 21. F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, NIST Handbook of Mathematical Functions, Cambridge University Press, Cambridge, 2010. 22. A. Pedas and E. Tamme, Spline collocation methods for linear multi–term fractional differential equations, J. Comput. Appl. Math. 236 (2011) 167–176. 23. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA, 1999. 24. J. M. Prausnitz, R. N. Lichtenthaler and E. Gomes de Azevedo, Molecular Thermodynamics of Fluid–Phase Equilibria, 3th ed., Prentice Hall, Englewood Cliffs, 1999. 25. L. N. Trefethen, Approximation Theory and Approximation Practice, SIAM, Philadelphia, PA, 2013.