×

Numerical solution of Riccati equation using the cubic B-spline scaling functions and Chebyshev cardinal functions. (English) Zbl 1205.65206

Summary: Two numerical techniques are presented for solving the solution of Riccati differential equation. These methods use the cubic B-spline scaling functions and Chebyshev cardinal functions. The methods consist of expanding the required approximate solution as the elements of cubic B-spline scaling function or Chebyshev cardinal functions. Using the operational matrix of derivative, we reduce the problem to a set of algebraic equations. Some numerical examples are included to demonstrate the validity and applicability of the new techniques. The methods are easy to implement and produce very accurate results.

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abbasbandy, S., Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian’s decomposition method, Appl. Math. Comput., 172, 485-490 (2006) · Zbl 1088.65063
[2] Abbasbandy, S., A new application of He’s variational iteration method for quadratic Riccati differential equation by using Adomian’s polynomials, J. Comput. Appl. Math., 207, 59-63 (2007) · Zbl 1120.65083
[3] Abbasbandy, S., Iterated He’s homotopy perturbation method for quadratic Riccati differential equation, Appl. Math. Comput., 175, 581-589 (2006) · Zbl 1089.65072
[4] Boyd, John P., Chebyshev and Fourier Spectral Methods (2000), DOVER Publications, Inc. · Zbl 0994.65128
[5] Carinena, J. F.; Marmo, G.; Perelomov, A. M.; Ranada, M. F., Related operators and exact solutions of Schrödinger equations, Int. J. Mod. Phys. A, 13, 4913-4929 (1998) · Zbl 0927.34065
[6] Chui, C. K., An Introduction to Wavelets (1992), Academic Press: Academic Press San Diego, Calif · Zbl 0925.42016
[7] de Boor, C., A Practical Guide to Spline (1978), Springer-Verlag: Springer-Verlag New York · Zbl 0406.41003
[8] Dehghan, M.; Taleei, A., A compact split-step finite difference method for solving the nonlinear Schrödinger equations with constant and variable coefficients, Comput. Phys. Commun., 181, 43-51 (2010) · Zbl 1206.65207
[9] Dehghan, M., Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices, Math. Comput. Simul., 71, 16-30 (2006) · Zbl 1089.65085
[10] Dehghan, M.; Shakeri, F., Approximate solution of a differential equation arising in astrophysics using the variational iteration method, New Astronomy, 13, 53-59 (2008)
[11] Tatari, M.; Dehghan, M., On the convergence of He’s variational iteration method, J. Comput. Appl. Math., 207, 121-128 (2007) · Zbl 1120.65112
[12] Dehghan, M.; Shakourifar, M.; Hamidi, A., The solution of linear and nonlinear systems of Volterra functional equations using Adomian-Pade technique, Chaos Solitons Fractals, 39, 2509-2521 (2009) · Zbl 1197.65223
[13] Dehghan, M.; Shakeri, F., The use of the decomposition procedure of Adomian for solving a delay differential equation arising in electrodynamics, Phys. Scripta, 78, 065004 (2008), (11 pp.) · Zbl 1159.78319
[14] Saadatmandi, A.; Dehghan, M.; Eftekhari, A., Application of He’s homotopy perturbation method for non-linear system of second-order boundary value problems, Nonlinear Anal.: Real World Appl., 10, 1912-1922 (2009) · Zbl 1162.34307
[15] Dehghan, M.; Shakeri, F., Solution of an integro-differential equation arising in oscillating magnetic fields using He’s homotopy perturbation method, Progress in Electromagnetic Research, PIER, 78, 361-376 (2008)
[16] Genga, F.; Lin, Y.; Cui, M., A piecewise variational iteration method for Riccati differential equations, Comput. Math. Appl., 58, 2518-2522 (2009) · Zbl 1189.65164
[17] Goswami, J. C.; Chan, A. K., Fundamentals of Wavelets, Theory, Algorithms, and Applications (1999), John Wiley & Sons, Inc. · Zbl 1209.65156
[18] Meyer, Y., Ondelettes et Opérateurs, I: Ondelettes, II: Opérateurs de Calderón-Zygmond, III: Opérateurs multilinéaires (with R. Coifman) (1990), Hermann: Hermann Paris
[19] Reid, W. T., Riccati Differential Equations (1972), Academic Press: Academic Press New York · Zbl 0209.11601
[20] Scott, M. R., Invariant Imbedding and Its Applications to Ordinary Differential Equations (1973), Addison-Wesley
[21] Aminikhah, H.; Hemmatnezhad, M., An efficient method for quadratic Riccati differential equation, Commun. Nonlinear Sci. Numer. Simul., 15, 835-839 (2010) · Zbl 1221.65193
[22] Bulut, Hasan; Evans, D. J., On the solution of the Riccati equation by the decomposition method, Int. J. Comput. Math., 79, 103-109 (2002) · Zbl 0995.65073
[23] El-Tawil, M. A.; Bahnasawi, A. A.; Abdel-Naby, A., Solving Riccati differential equation using Adomian’s decomposition method, Appl. Math. Comput., 157, 503-514 (2004) · Zbl 1054.65071
[24] Dehghan, M.; Shakeri, F., The numerical solution of the second Painlev equation, Numer. Methods Partial Differential Equations, 25, 1238-1259 (2009) · Zbl 1172.65037
[25] M. Dehghan, R. Salehi, The use of variational iteration method and Adomian decomposition method to solve the Eikonal equation and its application in the reconstruction problem, Comm. Numer. Methods in Engrg. (2009), in press, doi:10.1002/cnm.1315; M. Dehghan, R. Salehi, The use of variational iteration method and Adomian decomposition method to solve the Eikonal equation and its application in the reconstruction problem, Comm. Numer. Methods in Engrg. (2009), in press, doi:10.1002/cnm.1315 · Zbl 1218.65112
[26] Dehghan, M.; Saadatmandi, A., Variational iteration method for solving the wave equation subject to an integral conservation condition, Chaos Solitons Fractals, 41, 1448-1453 (2009) · Zbl 1198.65202
[27] Tatari, M.; Dehghan, M., Improvement of He’s variational iteration method for solving systems of differential equations, Comput. Math. Appl., 58, 2160-2166 (2009) · Zbl 1189.65178
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.