Document Type : Research Paper

**Authors**

Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, Iran.

**Abstract**

This work presents a new approximation approach to solve the linear/nonlinear distributed order fractional differential equations using the Chebyshev polynomials. Here, we use the Chebyshev polynomials combined with the idea of the collocation method for converting the distributed order fractional differential equation into a system of linear/nonlinear algebraic equations. Also, fractional differential equations with initial conditions can be solved by the present method. We also give the error bound of the modified equation for the present method. Moreover, four numerical tests are included to show the effectiveness and applicability of the suggested method.

**Keywords**

[1] M. Abbaszadeh and M. Dehghan, An improved meshless method for solving two-dimensional distributed order time fractional diffusion-wave equation with error estimate, Numer. Algor., 75 (2017), 173-211.

[2] O. Abdulaziz, I. Hashim, and S. Momani, Application of homotopy perturbation method to fractional IVPs, J. Comput. Appl. Math., 216 (2008), 574-584.

[3] M. Ahmadi Darani and A. Saadatmandi, The operational matrix of fractional derivative of the fractional order Chebyshev functions and its applications, Comput. Methods Differ. Equ., 5 (2017), 67-87.

[4] H. Aminikhah, A. H. Refahi Sheikhani, T. Houlari, and H. Rezazadeh, Numerical Solution of the Distributed-Order Fractional Bagley-Torvik Equation, IEEE/CAA J. Autom. Sin., 6 (2019), 760-765.

[5] T. M. Atanackovic, A generalized model for the uniaxial isothermal deformation of a viscoelastic body, Acta Mech., 159 (2002), 77-86.

[6] R. L. Bagley and P. J. Torvik, On the existence of the order domain and the solution of distributed order equations-part I, Int. J. Appl. Math., 2 (2000), 865-882.

[7] R. L. Bagley and P. J. Torvik, On the existence of the order domain and the solution of distributed order equations-part II, Int. J. Appl. Math.,2 (2000), 965-987.

[8] A. Baseri, S. Abbasbandy, and E. Babolian, A collocation method for fractional diffusion equation in a long time with Chebyshev functions, Appl. Math. Comput., 322 (2018), 55–65.

[9] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer, New York, 1988.

[10] M. Caputo, Mean fractional-order-derivative differential equation and filters, Ann. Univ. Ferrara Sez. VII (N.S.), 41 (1995), 73-84.

[11] A. V. Chechkin, R. Gorenflo, I. M. Sokolov, and V. Y. Gonchar, Distributed order time fractional diffusion equation, Fract. Calc. Appl. Anal., 6 (2003) 259-279.

[12] A. V. Chechkin, J. Klafter, and I. M. Sokolov, Fractional fokker-Planck equation for ultraslow kinetics, Europhys. Lett., 63 (2003), 326-332.

[13] M. Dehghan and M. Abbaszadeh, A finite difference/finite element technique with error estimate for space fractional tempered diffusion-wave equation, Comput. Math. Appl., 75 (2018), 2903-2914.

[14] M. Dehghan and M. Abbaszadeh, An efficient technique based on finite difference/finite element method for solution of two-dimensional space/multi-time fractional Bloch-Torrey equations, Appl. Numer. Math., 131 (2018) 190-206.

[15] J. Deng and L. Ma, Existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations, Appl. Math. Lett., 23 (2010), 676-680.

[16] K. Diethelm and N. J. Ford, Numerical analysis for distributed order differential equations, J. Comput. Appl. Math., 225 (2009), 96-104.

[17] 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, Comput. Math. Appl., 62 (2011), 2364-2373.

[18] N. J. Ford, M. L. Morgado, and M. Rebelo, An implicit finite difference approximation for the solution of the diffusion equation with distributed order in time, Electron. Trans. Numer. Anal., 44 (2015), 289-305.

[19] F. B. Hildebrand, Introduction to Numerical Analysis, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1956.

[20] Z. Jiao, Y. Q. Chen, and I. Podlubny, Distributed-Order Dynamic System Stability, Simulation, Applications and and Perspective, Springer, London, 2012.

[2] O. Abdulaziz, I. Hashim, and S. Momani, Application of homotopy perturbation method to fractional IVPs, J. Comput. Appl. Math., 216 (2008), 574-584.

[3] M. Ahmadi Darani and A. Saadatmandi, The operational matrix of fractional derivative of the fractional order Chebyshev functions and its applications, Comput. Methods Differ. Equ., 5 (2017), 67-87.

[4] H. Aminikhah, A. H. Refahi Sheikhani, T. Houlari, and H. Rezazadeh, Numerical Solution of the Distributed-Order Fractional Bagley-Torvik Equation, IEEE/CAA J. Autom. Sin., 6 (2019), 760-765.

[5] T. M. Atanackovic, A generalized model for the uniaxial isothermal deformation of a viscoelastic body, Acta Mech., 159 (2002), 77-86.

[6] R. L. Bagley and P. J. Torvik, On the existence of the order domain and the solution of distributed order equations-part I, Int. J. Appl. Math., 2 (2000), 865-882.

[7] R. L. Bagley and P. J. Torvik, On the existence of the order domain and the solution of distributed order equations-part II, Int. J. Appl. Math.,2 (2000), 965-987.

[8] A. Baseri, S. Abbasbandy, and E. Babolian, A collocation method for fractional diffusion equation in a long time with Chebyshev functions, Appl. Math. Comput., 322 (2018), 55–65.

[9] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer, New York, 1988.

[10] M. Caputo, Mean fractional-order-derivative differential equation and filters, Ann. Univ. Ferrara Sez. VII (N.S.), 41 (1995), 73-84.

[11] A. V. Chechkin, R. Gorenflo, I. M. Sokolov, and V. Y. Gonchar, Distributed order time fractional diffusion equation, Fract. Calc. Appl. Anal., 6 (2003) 259-279.

[12] A. V. Chechkin, J. Klafter, and I. M. Sokolov, Fractional fokker-Planck equation for ultraslow kinetics, Europhys. Lett., 63 (2003), 326-332.

[13] M. Dehghan and M. Abbaszadeh, A finite difference/finite element technique with error estimate for space fractional tempered diffusion-wave equation, Comput. Math. Appl., 75 (2018), 2903-2914.

[14] M. Dehghan and M. Abbaszadeh, An efficient technique based on finite difference/finite element method for solution of two-dimensional space/multi-time fractional Bloch-Torrey equations, Appl. Numer. Math., 131 (2018) 190-206.

[15] J. Deng and L. Ma, Existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations, Appl. Math. Lett., 23 (2010), 676-680.

[16] K. Diethelm and N. J. Ford, Numerical analysis for distributed order differential equations, J. Comput. Appl. Math., 225 (2009), 96-104.

[17] 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, Comput. Math. Appl., 62 (2011), 2364-2373.

[18] N. J. Ford, M. L. Morgado, and M. Rebelo, An implicit finite difference approximation for the solution of the diffusion equation with distributed order in time, Electron. Trans. Numer. Anal., 44 (2015), 289-305.

[19] F. B. Hildebrand, Introduction to Numerical Analysis, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1956.

[20] Z. Jiao, Y. Q. Chen, and I. Podlubny, Distributed-Order Dynamic System Stability, Simulation, Applications and and Perspective, Springer, London, 2012.

[21] N. Jibenja, B. Yuttanan, and M. Razzaghi, An efficient method for numerical solutions of distributed-order fractional differential equations, J. Comput. Nonlinear Dynam., 13 (2018), 111003.

[22] J. T. Katsikadelis, Numerical solution of distributed order fractional differential equations, J. Comput. Phys., 259 (2014), 11-22.

[23] J. T. Katsikadelis, The fractional distributed order oscillator: A numerical solution, J. Serb. Soc. Comput. Mech., 6 (2012), 148-159.

[24] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science B.V., Amsterdam, 2006.

[25] Y. Li, H. Sheng, and Y. Q. Chen, On distributed order integrator/differentiator, signal processing, 91 (2011), 1079-1084.

[26] S. Mashayekhi and M. Razzaghi, Numerical solution of distributed order fractional differential equations by hybrid functions, J. Comput. Phys., 315 (2016), 169-181.

[27] S. Mashayekhi and M. Razzaghi, Numerical solution of the fractional Bagley-Torvik equation by using hybrid functions approximation, Math. Meth. Appl. Sci., 39 (2016), 353-365.

[28] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. New York: Wiley 1993.

[29] A. Mohebbi, Analysis of a numerical method for the solution of time fractional Burgers equation, Bull. Iranian Math. Soc., 44 (2018), 457-480.

[30] A. Mohebbi, On the split-step method for the solution of nonlinear Schr¨odinger equation with the Riesz space fractional derivative, Comput. Methods Differ. Eq., 4 (2016) 54-69.

[31] K. B. Oldham and J. Spanier, The Fractional Calculus. New York: Academic Press 1974.

[32] I. Podlubny, Fractional differential equations, Academic Press, New York, 1999.

[33] I. Podlubny, T. Skovranek, B. M. Vinagre Jara, I. Petras, V. Verbitsky, and Y. Q. Chen, Matrix approach to discrete fractional calculus III: non-equidistant grids, variable step length and distributed orders, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 371 (2013).

[34] M. Pourbabaee and A. Saadatmandi, A novel Legendre operational matrix for distributed order fractional differential equations, Appl. Math. Comput., 361 (2019), 215-231.

[35] A. Saadatmandi and M. Dehghan, A new operational matrix for solving fractional order differential equations, Comput. Math. Appl., 59 (2010), 1326-1336.

[36] A. Saadatmandi, A. Khani, and M. R. Azizi, A sinc-Gauss-Jacobi collocation method for solving Volterra’s population growth model with fractional order, Tbilisi Math. J., 11 (2018), 123-137.

[37] J. Shen, T. Tang, and L. L. Wang, Spectral Methods Algorithms, Analysis and Applications, Springer-Verlag Berlin Heidelberg 2011.

[38] P. L. Trung Duong, E. Kwok, and M. Lee, Deterministic analysis of distributed order systems using operational matrix, Appl. Math. Model., 40 (2016), 1929-1940.

[39] Y. Yang, Y. Ma, and L. Wang, Legendre polynomials operational matrix method for solving fractional partial differential equations with variable coefficients, Math. Probl. Eng. (2015), Art. ID 915195.

[40] B. Yuttanan and M. Razzaghi, Legendre wavelets approach for numerical solutions of distributed order fractional differential equations, Appl. Math. Model., 70 (2019), 350–364.

[41] M. A. Zaky and J. A. Tenreiro Machado, On the formulation and numerical simulation of distributed order fractional optimal control, Commun. Nonlinear Sci. Numer. Simul., 52 (2017), 177-189.

[42] F. Zhou, Y. Zhao, Y. Li, and Y. Q. Chen, Design, implementation and application of distributed order PI control, ISA Trans., 52 (2013), 429-437.

[22] J. T. Katsikadelis, Numerical solution of distributed order fractional differential equations, J. Comput. Phys., 259 (2014), 11-22.

[23] J. T. Katsikadelis, The fractional distributed order oscillator: A numerical solution, J. Serb. Soc. Comput. Mech., 6 (2012), 148-159.

[24] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science B.V., Amsterdam, 2006.

[25] Y. Li, H. Sheng, and Y. Q. Chen, On distributed order integrator/differentiator, signal processing, 91 (2011), 1079-1084.

[26] S. Mashayekhi and M. Razzaghi, Numerical solution of distributed order fractional differential equations by hybrid functions, J. Comput. Phys., 315 (2016), 169-181.

[27] S. Mashayekhi and M. Razzaghi, Numerical solution of the fractional Bagley-Torvik equation by using hybrid functions approximation, Math. Meth. Appl. Sci., 39 (2016), 353-365.

[28] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. New York: Wiley 1993.

[29] A. Mohebbi, Analysis of a numerical method for the solution of time fractional Burgers equation, Bull. Iranian Math. Soc., 44 (2018), 457-480.

[30] A. Mohebbi, On the split-step method for the solution of nonlinear Schr¨odinger equation with the Riesz space fractional derivative, Comput. Methods Differ. Eq., 4 (2016) 54-69.

[31] K. B. Oldham and J. Spanier, The Fractional Calculus. New York: Academic Press 1974.

[32] I. Podlubny, Fractional differential equations, Academic Press, New York, 1999.

[33] I. Podlubny, T. Skovranek, B. M. Vinagre Jara, I. Petras, V. Verbitsky, and Y. Q. Chen, Matrix approach to discrete fractional calculus III: non-equidistant grids, variable step length and distributed orders, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 371 (2013).

[34] M. Pourbabaee and A. Saadatmandi, A novel Legendre operational matrix for distributed order fractional differential equations, Appl. Math. Comput., 361 (2019), 215-231.

[35] A. Saadatmandi and M. Dehghan, A new operational matrix for solving fractional order differential equations, Comput. Math. Appl., 59 (2010), 1326-1336.

[36] A. Saadatmandi, A. Khani, and M. R. Azizi, A sinc-Gauss-Jacobi collocation method for solving Volterra’s population growth model with fractional order, Tbilisi Math. J., 11 (2018), 123-137.

[37] J. Shen, T. Tang, and L. L. Wang, Spectral Methods Algorithms, Analysis and Applications, Springer-Verlag Berlin Heidelberg 2011.

[38] P. L. Trung Duong, E. Kwok, and M. Lee, Deterministic analysis of distributed order systems using operational matrix, Appl. Math. Model., 40 (2016), 1929-1940.

[39] Y. Yang, Y. Ma, and L. Wang, Legendre polynomials operational matrix method for solving fractional partial differential equations with variable coefficients, Math. Probl. Eng. (2015), Art. ID 915195.

[40] B. Yuttanan and M. Razzaghi, Legendre wavelets approach for numerical solutions of distributed order fractional differential equations, Appl. Math. Model., 70 (2019), 350–364.

[41] M. A. Zaky and J. A. Tenreiro Machado, On the formulation and numerical simulation of distributed order fractional optimal control, Commun. Nonlinear Sci. Numer. Simul., 52 (2017), 177-189.

[42] F. Zhou, Y. Zhao, Y. Li, and Y. Q. Chen, Design, implementation and application of distributed order PI control, ISA Trans., 52 (2013), 429-437.

July 2021

Pages 858-873

**Receive Date:**21 February 2020**Revise Date:**11 June 2020**Accept Date:**12 June 2020**First Publish Date:**01 July 2021