Haar Wavelet Collocation Method for Solving Riccati and Fractional Riccati Differential Equations

Shiralashetti, S.C.; Deshi, A.B.

doi:10.18052/www.scipress.com/BMSA.17.46

BMSA > Volume 17 > Haar Wavelet Collocation Method for Solving...

< Back to Volume

Haar Wavelet Collocation Method for Solving Riccati and Fractional Riccati Differential Equations

Full Text PDF

Abstract:

In this paper, numerical solutions of Riccati and fractional Riccati differential equations are obtained by the Haar wavelet collocation method. An operational matrix of integration based on the Haar wavelet is established, and the procedure for applying the matrix to solve these equations. The fundamental idea of Haar wavelet method is to convert the proposed differential equations into a group of non-linear algebraic equations. The accuracy of approximate solution can be further improved by increasing the level of resolution and an error analysis is computed. The examples are given to demonstrate the fast and flexibility of the method. The results obtained are in good agreement with the exact in comparison with existing ones and it is shown that the technique introduced here is robust, easy to apply and is not only enough accurate but also quite stable.

Info:

Periodical:

Bulletin of Mathematical Sciences and Applications (Volume 17)

Pages:

46-56

DOI:

https://doi.org/10.18052/www.scipress.com/BMSA.17.46

Citation:

S.C. Shiralashetti and A.B. Deshi, "Haar Wavelet Collocation Method for Solving Riccati and Fractional Riccati Differential Equations", Bulletin of Mathematical Sciences and Applications, Vol. 17, pp. 46-56, 2016

Online since:

November 2016

Authors:

S.C. Shiralashetti, A.B. Deshi

Keywords:

Fractional Riccati Differential Equations, Haar Wavelet Collocation Method, Operational Matrix, Resolution Level, Riccati Differential Equations

Export:

RIS, BibTeX, Text

Distribution:

Open Access

This work is licensed under a
Creative Commons Attribution 4.0 International License

References:

[1] S. Abbasbandy, Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomians decomposition method, Appl. Math. Comput. 172 (2006) 485-490.

DOI: https://doi.org/10.1016/j.amc.2005.02.014

[2] M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform, Cambridge University Press, Cambridge, (1990).

[3] B.D. Anderson, J.B. Moore, Optimal Control-Linear Quadratic Methods, Prentice-Hall, New Jersey, (1990).

[4] M.K. Bouafoura, N.B. Braiek, controller design for integer and fractional plants using piecewise orthogonal functions, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 1267–1278.

[5] N.M. Bujurke, C. S Salimath, S.C. Shiralashetti, Numerical Solution of Stiff Systems from Nonlinear Dynamics Using Single-term Haar Wavelet Series, Nonlinear Dyn. 51 (2008) 595-605.

DOI: https://doi.org/10.1007/s11071-007-9248-8

[6] N.M. Bujurke, C.S. Salimath, S.C. Shiralashetti, Computation of eigenvalues and solutions of regular Sturm-Liouville problems using Haar wavelets, J. Comp. Appl. Math. 219 (2008) 90-101.

DOI: https://doi.org/10.1016/j.cam.2007.07.005

[7] N.M. Bujurke, S.C. Shiralashetti, C. S Salimath, An Application of Single-term Haar Wavelet Series in the Solution of Nonlinear Oscillator Equations, J. Comput. Appl. Math. 227 (2009) 234-244.

DOI: https://doi.org/10.1016/j.cam.2008.03.012

[8] J.F. Carinena et al., Related operators and exact solutions of Schrodinger equations, Int. J. Mod. Phys. A. 13 (1998) 4913–4929.

[9] C.F. Chen, C.H. Hsiao, Haar wavelet method for solving lumped and distributed-parameter systems, IEEE Proc. Pt. D. 144 (1) (1997) 87-94.

DOI: https://doi.org/10.1049/ip-cta:19970702

[10] S. Dhawan, S. Arora, S. Kumar, Wavelet based numerical scheme for differential equations, Int. J. of Differential Equations and Applications. 12(2) (2013) 85-94.

DOI: https://doi.org/10.12732/ijdea.v12i2.932

[11] K. Diethelm et al., Pitfalls in fast numerical solvers for fractional differential equations, J. Comput. Appl. Math. 186 (2006) 482–503.

[12] M.A. El-Tawil, A.A. Bahnasawi, A. Abdel-Naby, Solving Riccati differential equation using Adomians decomposition method, Appl. Math. Comput. 157 (2004) 503-514.

DOI: https://doi.org/10.1016/j.amc.2003.08.049

[13] F. Geng, Y. Lin, M. Cui, A piecewise variational iteration method for Riccati differential equations, Comput. Math. Appl. 58 (2009) 2518-2522.

DOI: https://doi.org/10.1016/j.camwa.2009.03.063

[14] R. Gorenflo, Afterthoughts on interpretation of fractional derivatives and integrals, in: P. Rusev, I. Dimovski, V. Kiryakova (Eds. ), Transform Methods and Special Functions, Varna 96, Bulgarian Academy of Sciences, Institute of Mathematics and Informatics, Sofia, 1998, p.589.

[15] S. Islam, I. Aziz, B. Sarler, The numerical solution of second-order boundary-value problems by collocation method with the Haar wavelets, Math. Comput. Model. 52 (2010) 1577-1590.

DOI: https://doi.org/10.1016/j.mcm.2010.06.023

[16] A. Kilicman, Z. A.A.A. Zhour, Kronecker operational metrics for fractional calculus and some applications, Appl. Math. Comput. 187 (2007) 250–265.

DOI: https://doi.org/10.1016/j.amc.2006.08.122

[17] U. Lepik, Numerical solution of differential equations using Haar wavelets, Math. Comput. Simul. 68 (2005) 127-143.

[18] U. Lepik, Application of the Haar wavelet transform to solving integral and differential Equations, Proc. Estonian Acad Sci. Phys. Math. 56(1) (2007) 28-46.

[19] Y. Li, W. Zhao, Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations, Appl. Math. Comput. 216 (2010) 2276–2285.

DOI: https://doi.org/10.1016/j.amc.2010.03.063

[20] S. Momani, N. Shawagfeh, Decomposition method for solving fractional Riccati differential equations, Appl. Math. Comput. 182 (2006) 1083-1092.

DOI: https://doi.org/10.1016/j.amc.2006.05.008

[21] W.T. Reid, Riccati Differential Equations, Academic Press, NewYork, (1972).

[22] M.R. Scott, Invariant Imbedding and its Applications to Ordinary Differential Equations, Addison-Wesley, (1973).

[23] Z. Shi, Y. Cao, Application of Haar wavelet method to eigenvalue problems of high order differential equations, Appl. Math. Model. 36 (2012) (9) 4020-4026.

DOI: https://doi.org/10.1016/j.apm.2011.11.024

[24] B.Q. Tang, X.F. Li, A new method for determining the solution of Riccati differential equations, Appl. Math. Comput. 194 (2007) 431-440.

Show More Hide

Cited By:

[1] S. Shiralashetti, R. Mundewadi, "Numerical Solution of Nonlinear Volterra-Fredholm Integral Equations Using Haar Wavelet Collocation Method", Bulletin of Mathematical Sciences and Applications, Vol. 18, p. 50, 2017

DOI: https://doi.org/10.18052/www.scipress.com/BMSA.18.50