Solving of partial differential equations with distributed order in time using fractional-order Bernoulli-Legendre functions

Document Type : Research Paper

Authors

Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran.

Abstract

In this paper, an efficient numerical method is used to provide the approximate solution of distributed-order fractional partial differential equations (DFPDEs). The proposed method is based on the fractional integral operator of fractional-order Bernoulli-Legendre functions and the collocation scheme. In our technique, by approximating functions that appear in the DFPDEs by fractional-order Bernoulli functions in space and fractional-order Legendre functions in time using Gauss-Legendre numerical integration, the under study problem is converted to a system of algebraic equations. This system is solved by using Newton's iterative scheme, and the numerical solution of DFPDEs is obtained. Finally, some numerical experiments are included to show the accuracy, efficiency, and applicability of the proposed method.

Keywords


[1] T. M. Atanackovic, A generalized model for the uniaxial isothermal deformation of a viscoelastic body, Acta Mech., 159 (2002), 77– 86.
[2] T. M. Atanackovic, M. Budincevic, and S. Pilipovic, On a fractional distributed-order oscillator, J. Phys. A, Math. Gen., 38 (2005), 6703–6713.
[3] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods: Fundamentals in Single Domains, New York, Springer, 2006.
[4] M. Caputo and F. Mainardi, Linear models of dissipation in anelastic solids, Rivista. del. Nuovo. Cimento., 1 (1971), 1971–1977.
[5] M. Dehghan, J. Manafian, and A. Saadatmandi, Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numer. Methods Partial Differential Eq., 26 (2) (2009), 448–479.
[6] C. H. Eab and S. C. Lim, Fractional Langevin equations of distributed order, Physical Review E., 83 (2011), 031136.
[7] N. J. Ford, M. L. Morgado, and M. Rebelo, An implicitfinite difference approximation for the solution of the diffusion equation with distributed order in time, Electron. Trans. Numer. Anal., 44 (2015), 289–305.
[8] L. Gaul, P. Klein and S. Kemple, Damping description involving fractional operators, Mech. Syst. Signal. Process., 5 (1991), 81–88.
[9] R. Gorenflo, Y. Luchko, and M. Stojanovic, Fundamental solution of a distributed order timefractional diffusion-wave equation as probability density, Fract. Calc. Appl. Anal., 16(2) (2013), 297–316.
[10] J. H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Methods Appl. Mech. Eng., 167 (1998), 57–68.
[11] J. H. He, Homotopy perturbation technique , Comput. Methods Appl. Mech. Eng., 178(1999), 257–62.
[12] J. H. He, Nonlinear oscillation with fractional derivative and its applications, In: International Conference on Vibrating Engg98, Dalian, (1998), 288–291.
[13] E. Kharazmi, M. Zayernouri, and G. E. Karniadakis, Petrov-Galerkin and spectral collocation methods for distributed order differential equations, SIAM J. Sci. Comput., 39(3) (2017), A1003– A1037.
[14] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
[15] A. Lischke, M. Zayernouri, and G. E. Karniadakis, A Petrov-Galerkin spectral method of linear complexity for fractional multiterm ODEs on the half line, SIAM J. Sci. Comput., 39(3) (2017), A922–A946.
[16] C. F. Lorenzo and T. T. Hartley, Variable order and distributed order fractional operators, Nonlinear Dyn., 29 (2002), 57–98.
[17] Y. Luchko, Boundary value problems for the generalized time-fractional diffusion equation of distributed order, Fract. Calc. Appl. Anal., 12 (2009), 409–422.
[18] S. Mashayekhi and M. Razzaghi, Numerical solution of distributed order fractional differential equations by hybrid functions, J. Comput. Phys., 315 (2016), 169–181.
[19] J. C. Mason and D. C. Handscomb, Chebyshev Polynomials, CRC Press LLC, 2003.
[20] M. M. Meerschaert, E. Nane, and P. Vellaisamy, Distributed-order fractional diffusions on bounded domains, J. Math. Anal. Appl., 379 (2011), 216–228.
[21] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
[22] F. Mirzaee and S. Alipour, Fractional-order orthogonal Bernstein polynomials for numerical solution of nonlinear fractional partial Volterra integro-differential equations, Math. Meth. Appl. Sci., 42(6) (2019), 1870–1893.
[23] F. Mirzaee and S. Alipour, Numerical solution of nonlinear partial quadratic integrodifferential equations of fractional order via hybrid of blockpulse and parabolic functions, Numer. Methods Partial Differential Eq., 35(3) (2019), 1134–1151.
[24] F. Mirzaee, S. Alipour, and N. Samadyar, A numerical approach for solving weakly singular partial integrodifferential equations via two-dimensional-orthonormal Bernstein polynomials with the convergence analysis, Numer. Methods Partial Differential Eq., 35(2) (2018), 615–637.
[25] S. Momani and K. Al-Khaled, Numerical solutions for systems of fractional differential equations by the decomposition method, Appl. Math. Comput., 162 (2005), 1351–1365.
[26] M. L. Morgado and M. Rebelo, Numerical approximation of distributed order reaction-diffusion equations, J. Comput. Appl. Math., 275 (2015), 216–227.
[27] M. Morgado, M. Rebelo, L. Ferrs, and N. Ford, Numerical solution for diffusion equations with distributed order in time using a Chebyshev collocation method, Appl. Numer. Math., 114 (2017), 108–123.
[28] M. Naber, Distributed order fractional sub-diffusion, Fractals, 12 (2004), 23–32.
[29] S. Nemati, P. M. Lima, and Y. Ordokhani, Numerical solution of a class of two-dimensional nonlinear Volterra integral equations using Legendre polynomials, J. Comput. Appl. Math., 242 (2013), 53–69.
[30] Z. Odibat and S. Momani, Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonl. Sci. Numer. Simul., 7 (2006), 27–34.
[31] Z. Odibat and N. Shawagfeh, Generalized Taylors formula, Appl. Math. Comput., 186(1) (2007), 286–293.
[32] P. Rahimkhani and R. Moeti, Numerical solution of the fractional order Duffing-van der Pol oscillator equation by using Bernoulli wavelets collocation method, Int. J. Appl. Comput. Math., 4:59 (2018). doi: 10.1007/s40819-018-0494-x.
[33] P. Rahimkhani and Y. Ordokhani, Generalized fractional-order Bernoulli-Legendre functions: an effective tool for solving two-dimensional fractional optimal control problems, IMA J. Math. Control Inform., 36(1) (2019), 185–212.
[34] P. Rahimkhani and Y. Ordokhani, Numerical solution a class of 2D fractional optimal control problems by using 2D M¨untz-Legendre wavelets, Optim. Control. Appl. Meth., 39(6) (2018), 1916–1934.
[35] P. Rahimkhani, Y. Ordokhani, and E. Babolian, Fractional-order Bernoulli functions and their applications in solving fractional Fredholem-Volterra integro-differential equations, Appl. Numer. Math., 122 (2017), 66–81.
[36] P. Rahimkhani, Y. Ordokhani, and E. Babolian, Fractional-order Legendre wavelets and their applications for solving fractional-order differential equations with initial/boundary conditions, Comput. Methods Differ. Equ., 5(2) (2017), 117-140.
[37] P. Rahimkhani, Y. Ordokhani, and E. Babolian, M¨untz-Legendre wavelet operational matrix of fractional-order integration and its applications for solving the fractional pantograph differential equations, Numer. Algor., 77(4) (2018), 1283–1305.
[38] N. Samadyar and F. Mirzaee, Numerical scheme for solving singular fractional partial integrodifferential equation via orthonormal Bernoulli polynomials, Int. J. Numer. Model., 32(6) (2019), e2652.
[39] L. Suarez and A. Shokooh, An eigenvector expansion method for the solution of motion containing fractional derivatives, J. Appl. Mech., 64 (1997), 629–735.
[40] M. A. Zaky, A Legendre collocation method for distributed-order fractional optimal control problems, Nonlinear Dyn., 91(4) (2018), 2667–2681.
[41] M. A. Zaky and J. A. Tenreiro Machado, On the formulation and numerical simulation of distributed-order fractional optimal control problems, Commun. Nonlinear Sci. Numer. Simulat., 52 (2017), 177–189.