Collocation method for solving systems of fractional differential equations; a case study of HIV infection by using Muntz wavelets basis
محورهای موضوعی : فصلنامه ریاضی
1 - Islamic Azad University, Isfahan(Khorasgan) Branch
کلید واژه: Muntz wavelels, Fractional differential equations systems, Collocation method, Jacobi polynomials, HIV infection,
چکیده مقاله :
This work was written with the aim of solving the fractional differential equation system using the Muntz wavelets. Muntz wavelets are modified using polynomials, respectively. The error of the proposed method is evaluated. This method applies to the fractional version of the HIV infection model. Numerical results confirm the accuracy of the proposed method.
In the present work, in the first stage the Muntz wavelets are introduced. These wawelets are definitely faster and more accurate than the Muntz Legendre polynomials.In the second stage the M\"{u}ntz Legendre polynomials in the interval [0,1] are introduced.In the continuation of the stages the definitions and properties of wavelets will be discussed.The Muntz wavelets are then presented in the range [0, T]. Subsequently, with the help of Jacobi polynomials, a more stable formula for Muntz wavelets is obtained. Finally the fractional differential equations are solved and the error analysis is reviewed. Also, to investigate of the accuracy of the presented method, mathematical and practical examples are presented.
This work was written with the aim of solving the fractional differential equation system using the Muntz wavelets. Muntz wavelets are modified using polynomials, respectively. The error of the proposed method is evaluated. This method applies to the fractional version of the HIV infection model. Numerical results confirm the accuracy of the proposed method.
In the present work, in the first stage the Muntz wavelets are introduced. These wawelets are definitely faster and more accurate than the Muntz Legendre polynomials.In the second stage the M\"{u}ntz Legendre polynomials in the interval [0,1] are introduced.In the continuation of the stages the definitions and properties of wavelets will be discussed.The Muntz wavelets are then presented in the range [0, T]. Subsequently, with the help of Jacobi polynomials, a more stable formula for Muntz wavelets is obtained. Finally the fractional differential equations are solved and the error analysis is reviewed. Also, to investigate of the accuracy of the presented method, mathematical and practical examples are presented.
[1] B. Ross, The development of fractional calculus 1695-1900, Hist. Math. 4 (1) (1977) 75-89.
[2] R. Hilfer (Ed.), Applications of Fractional Calculus in Physics, World Scientific Publishing Company, Singapore, (2000).
[3] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA, (1999).
[4] Y.A. Rossikhin, M.V. Shitikova, Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent result, Appl. Mech.Rev. 63 (2010) 52 pages.
[5] K. Diethelm, G. Walz, Numerical solution of fractional order differential equations by extrapolation, Numer. Algorithms 16 (1997) 231-253.
[6] K. Diethelm, N. J. Ford, A. D. Freed, A Predictor-Corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam, 16 (2002) 3-22.
[7] K. Diethelm, The analysis of fractional differential equations, Springer-Verlag, Berlin, Heidelberg and New York, (2010).
[8] A. El-sayed, M. Gaber, The Adomian decomposition method for solving partial differential
equations of fractional order in finite domains, Phys. Lett. A, 359 (2006) 175-182.
[9] L. Galeone, R. Garrappa, On multistep methods for differential equations of fractional order, Mediterr. J. Math. 3 (2006) 565-580.
[10] R. Garrappa and M. Popolizio, On accurate product integration rules for linear fractional
differential equations, J. Comput. Appl. Math. 235 (2011) 1085-1097.
[11] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, 204 (2006).
[12] M. Lakestani, M. Dehghan, S. Irandoust-Pakchin, The construction of operational matrix
of fractional derivatives using B-spline functions, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 1149-1162.
[13] K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential
equations, A Wiley-Interscience Publication, John Wiley and Sons, New York, Chichester,
Brisbane and Toronto, (1993).
[14] I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, Academic Press, New York, London and SanDiego,198 (1999).
[15] A. Babaaghaie and K. Maleknejad, Numerical solutions of nonlinear two-dimensional partial Volterra integro-differential equations by Haar wavelet, Journal of Computational and Applied Mathematics, 317 (2017) 643-651.
[16] Yi-Ming Chen, Yan-Qiao Wei , Da-Yan Liub, Hao Yua , Numerical solution for a class of nonlinear variable order fractional differential equations with Legendre wavelets, Applied Mathematics Letters, 46 (2015) 83-88
[17] M. Bahmanpour, M.A.Fariborzi Araghi, Numerical Solution of Fredholm and Volterra Integral Equations of the First Kind Using Wavelets bases, The Journal of Mathematics and Computer Science, 5(4) (2012) 337-345
[18] M. Rasouli Gandomani, M. Tavassoli Kajani, Numerical Solution of a Fractional Order Model of HIV Infection of CD4+T Cells Using Müntz-Legendre Polynomials, INT.J.BIOAUTOMATION,20(2) (2016) 193-204.
[19] M. Rasouli Gandomani1, M. Tavassoli Kajani ,Application of shifted M\"{u}ntz-legendre polynomials for solving fractional differential equations , International Journal of Pure and Applied Mathematics, 103 (2) (2015) 263-279.
[20] M. Behroozifar , A. Sazmand, An approximate solution based on Jacobi polynomials for time-fractional convection-diffusion equation, Applied Mathematics and Computation,296(1) (2017) 1-17.
[21] Fractal Functions, Fractal Surfaces, and Wavelets (Second Edition) , Academic Press, (2016) Pages 261-327.
[22] E.Chorukova, I. Simeonov , A Simple Mathematical Model of the Anaerobic Digestion of Wasted Fruits and Vegetables in Mesophilic Conditions, International Journal Bioautomation, 19 (2015) S69-S80.
[23] N. V.Kumar , L. Mathew, E. G. Wesely , Computational Modelling of Pisum
sativum L. Superoxide Dismutase and Prediction of Mutational Variations through in silico Methods, International Journal Bioautomation, 18(2) (2014) 75-88.
[24] L. Wang , M. Y. Li , Mathematical Analysis of the Global Dynamics of a Model for
HIV Infection of CD4+T Cells, Mathematical Biosciences. 200(1) (2006) 44-57.
[25] Merdan M., A. Gokdogan, A. Yildirim, On the Numerical Solution of the Model
for HIV Infection of CD4+T Cells, Computers and Mathematics with Applications, 62(1) (2011)
118-123.
[26] M. Y.Ongun ,The Laplace Adomian Decomposition Method for Solving a Model
for HIV Infection of CD4+T Cells, Mathematical and Computer Modelling, 53(6) (2011) 597-
603.
[27] S.Yuzbasi , A numerical approach to solve the model for HIV infection of CD4+T
cells, Applied Mathematical Modelling, 36(12) (2012) 5876-5890.
[28] M.A Friborzi Araghi, M. Bahmanpour, Numerical Solution of Fredholm Integral Equation of the First Kind using Legendre, Chebyshev and CAS wavelets, International Journal Of Mathematical Sciences And Engineering Applications, 2(4)(2008) 1-9.
[29] M. Bahmanpour, M.A. Fariborzi Araghi , A Method for Solving Fredholm Integral Equation of the First Kind Based on Chebyshev wavelets, Analysis in Theory and Applications
Anal. Theory Appl., 29(3) (2013) 197-207.
