Numerical solution of nonlinear fractional Riccati differential equations using compact finite difference method

Iranian Journal of Numerical Analysis and Optimization

مقاله 6، دوره 12، Issue 3 (Special Issue) - شماره پیاپی 23، بهمن 2022، صفحه 585-606   

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.2022.76489.1129

نویسندگان

H. Porki؛ M. Arabameri^* ؛ R. Gharechahi

Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran.

چکیده

This paper aims to apply and investigate the compact finite difference methods for solving integer-order and fractional-order Riccati differential equations. The fractional derivative in the fractional case is described in the Caputo sense. In solving the Riccati equation, we first approximate first-order derivatives using the approach of compact finite difference. In this way, the system of nonlinear equations is obtained, which solves the Riccati equation. In addition, we examine the convergence analysis of the proposed approach for the fractional and nonfractional cases and prove that the methods are convergent under some suitable conditions. Examples are also given to illustrate the efficiency of our method compared to other methods.
 

کلیدواژه‌ها

Fractional Riccati equation؛ Caputo fractional derivative؛ Com-pact finite difference methods

مراجع

1. Abbasbandy, S. Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian’s decomposition method, Appl. Math. Comput. 172(1), (2006) 91–102.
2. Agheli, B. Approximate solution for solving fractional Riccati differential equations via trigonometric basic functions, Trans. A. Razmadze Math. Inst. 172, (2018) 299–308.
3. Aminikhah, H., Sheikhani, A.H.R. and Rezazadeh, H. Approximate analytical solutions of distributed order fractional Riccati differential equation, Ain Shams Eng. J. 9 (4) (2018) 581–588.
4. Anderson, B.D.O. and Moore, J.B. Optimal filtering, Englewood, Cliffs. 1979.
5. Anderson, B.D. and Moore, J.B. Optimal control: Linear quadratic methods, Prentice-Hall, New Jersey, 2007.

6. Azin, H., Mohammadi, F. and Tenreiro Machado, J.A. A piecewise spectral-collocation method for solving fractional Riccati differential equation in large domains, Comp. Appl. Math. 38(3) (2019), Paper No. 96, 13 pp.
7. Biazar, J. and Eslami, M. Differential transform method for quadratic Riccati differential equation, Int. J. Nonlinear Sci. 9(4), (2010) 444–447.
8. Bota, C. and Căruntu, B. Analytical approximate solutions for quadratic Riccati differential equation of fractional order using the Polynomial Least Squares Method, Chaos Solitons Fractals, 102 (2017) 339–345.
9. Boyle, P.P., Tian, W. and Guan, F. The Riccati equation in mathematical finance, J. Symbolic Comput. 33(3), (2002) 343–355.
10. Chen, C.M., Liu, F., Turner, I. and Anh, V. A Fourier method for the fractional diffusion equation describing sub-diffusion, J. Comput. Phys. 227, (2007) 886–897.
11. Chen, C.M., Liu, F., Turner, I. and Anh, V. Numerical methods with fourth-order spatial accuracy for variable-order nonlinear Stokes’ first problem for a heated generalized second grade fluid, Comput. Math. Appl. 62, (2011) 971–986.
12. Chen, S., Liu, F., Zhuang, P. and Anh, V. Finite difference approximations for the fractional Fokker-Planck equation, Appl. Math. Model. 33, (2009) 256–273.
13. Cui, M. Compact finite difference method for the fractional diffusion equation, J. Comput. Phys. 228, (2009) 7792–7804.
14. Du, R., Cao, W.R. and Sun, Z.Z. A compact difference scheme for the fractional diffusion-wave equation, Appl. Math. Model. 34 (2010) 2998–3007.
15. Einicke, G.A., White, L.B. and Bitmead, R.R. The use of fake algebraic Riccati equations for co-channel demodulation, IEEE Trans. Signal Process. 51(9), (2003) 2288–2293.
16. Esmaeili, S. and Shamsi, M. A pseudo-spectral scheme for the approximate solution of a family of fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. 16, (2011) 3646–3654.
17. Gao, G. and Sun, Z.Z. A compact finite difference scheme for the fractional sub-diffusion equations, J. Comput. Phys. 230, (2011) 586–595.
18. Geng, F. A modified variational iteration method for solving Riccati differential equations, Comput. Math. Appl. 60(7), (2010) 1868–1872.

19. Gerber, M., Hasselblatt, B. and Keesing, D. The Riccati equation: pinching of forcing and solutions, Experiment. Math. 12(2), (2003) 129–134.
20. Langlands, T.A.M. and Henry, B.I. The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comput. Phys. 205, (2005) 719–736.
21. Lasiecka, I. and Triggiani, R. Differential and algebraic Riccati equations with application to boundary/point control problems: continuous theory and approximation theory, Lecture Notes in Control and Information Sciences, 164. Springer-Verlag, Berlin, 1991.
22. Liu, F., Anh, V. and Turner, I. Numerical solution of the space fractional Fokker-Planck equation, J. Comput. Appl. Math. 166, (2004) 209–219.
23. Liu, F., Yang, C. and Burrage, K. Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term, J. Comput. Appl. Math. 231, (2009) 160–176.
24. Liu, F., Zhuang, P., Anh, V., Turner, I. and Burrage, K.Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation, Appl. Math. Comput. 191, (2007) 12–20.
25. Maleknejad, K. and Torkzadeh, L. Hybrid functions approach for the fractional Riccati differential equation, Filomat 30(9) (2016) 2453–2463.
26. Miller, K.S. and Ross, B. An introduction to the fractional calculus and fractional differential equations., A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1993.
27. Neamaty, A., Agheli, B. and Darzi, R. The shifted Jacobi polynomial integral operational matrix for solving Riccati differential equation of fractional order, Appl. Appl. Math. 10(2) (2015) 878–892.
28. Ntogramatzidis, L. and Ferrante, A. On the solution of the Riccati differential equation arising from the LQ optimal control problem, Systems. Control. Lett. 59(2), (2010) 114–121.
29. Odibat, Z.M. Computational algorithms for computing the fractional derivatives of functions, Math. Comput. Simul. 79 , (2009) 2013–2020.
30. Odibat, Z. A Riccati equation approach and travelling wave solutions for nonlinear evolution equations, Int. J. Appl. Comput. Math. 3(1), (2017) 1–13.
31. Podlubny, I. Fractional differential equations, Academic Press, San Diego, 1999.
32. Reid, W.T. Riccati differential equations, Mathematics in Science and Engineering, Vol. 86. Academic Press, New York-London, 1972.

33. Saadatmandi, A. and Dehghan, M. A new operational matrix for solving fractional-order differential equations, Comput. Math. Appl. (2010) 59, 1326–1336.
34. Saadatmandi, A., Dehghan, M. and Azizi, M.R. The sinc-Legendre collocation method for a class of fractional convection-diffusion equation with variable coefficients, Commun. Nonlinear Sci. Numer. Simul. 17, (2012) 4125–4136.
35. Sun, Z.Z. and Wu, X.N.A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math. 56, (2006) 193–209.
36. Tadjeran, C., Meerschaert, M.M. and Scheffler, H.P. A second-order accurate numerical approximation for the fractional diffusion equation, J. Comput. Phys. 213, (2006) 205–213.
37. Tan, Y. and Abbasbandy, S. Homotopy analysis method for quadratic Riccati differential equation, Commun. Nonlinear Sci. Numer. Simul. 13(3), (2008) 539–546.
38. Wess, W.The fractional diffusion equation, J. Math. Phys. 27, (1996) 2782–2785.
39. Yuste, S.B. Weighted average finite difference methods for fractional diffusion equations, J. Comput. Phys. 216, (2006) 264–274.
40. Zhang, P.G. and J. P. Wang, A predictor–corrector compact finite difference scheme for Burgers’ equation, Appl. Math. Comput. 219(3), (2012) 892–898.
41. Zhuang, P., Liu, F., Anh, V. and Turner, I. New solution and analytical techniques of the implicit numerical methods for the anomalous sub-diffusion equation, SIAM J. Numer. Anal. 46, (2008) 1079–1095.