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The Riccati Equation, Differential Transform, Rational Solutions and Applications
Department of Mathematics and Computer Sciences, Marist College, Poughkeepsie, NY, USA
- 1 Department of Mathematics and Computer Sciences, Marist College, Poughkeepsie, NY, USA
Applied Mathematics·Volume 13 (2022)·Pages 774–792·Published 1 September 2022·DOI10.4236/am.2022.139049
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Abstract
In this article, the Riccati Equation is considered. Various techniques of finding analytical solutions are explored. Those techniques consist mainly of making a change of variable or the use of Differential Transform. It is shown that the nonconstant rational functions whose numerator and denominator are of degree 1, cannot be solutions to the Riccati equation. Two applications of the Riccati equation are discussed. The first one deals with Quantum Mechanics and the second one deal with Physics.
KeywordsRiccati EquationDifferential TransformRational Solutions
- Jacopo Francesco Riccati. https://mathshistory.st-andrews.ac.uk/Biographies/Riccati
- Anderson, B.D. and Moore, J.B. (1999) Optimal Control-Linear Quadratic Methods. Prentice-Hall, Hoboken.
- Lasiecka, I. and Triggiani, R. (1991) Differential and Algebraic Riccati Equations with Application to Boundary/Point Control Problems: Continuous Theory and Approximation Theory (Lecture Notes in Control and Information Sciences). Springer, Berlin, 164. https://doi.org/10.1007/BFb0006880
- Schwabl, F. (1992) Quantum Mechanics. Springer, Berlin. https://doi.org/10.1007/978-3-662-02703-5
- Fraga, E.S. (1999) The Schrodinger and Riccati Equations. Lecture Notes in Chemistry Vol. 70. Springer, Berlin. https://doi.org/10.1007/978-3-642-51458-6
- Dieter, S. (2014) Nonlinear Riccati Equations as a Unifying Link between Linear Quantum Mechanics and Other Fields of Physics. Journal of Physics: Conference Series, 538, Article ID: 012019. https://doi.org/10.1088/1742-6596/538/1/012019
- Al Bastami, A., Belic, M.R., Milovic, D. and Petrovic, N.Z. (2011) Analytical Chirped Solutions to the (3+1)-Dimensional Gross-Pitaevskii Equation for Various Diffraction and Potential Functions. Physical Review E, 84, Article ID: 016606. https://doi.org/10.1103/PhysRevE.84.016606
- Rosu, H.C. and Aceves de la Cruz, F. (2002) One-Parameter Darboux-Transformed Quantum Actions in Thermodynamics. Physica Scripta, 65, 377-382. https://doi.org/10.1238/Physica.Regular.065a00377
- Kajiwara, K. and Kimura, K. (2003) On a q-Difference Painlevé III Equation. I. Derivation, Symmetry and Riccati Type Solutions. Journal of Nonlinear Mathematical Physics, 10, 86-102. https://doi.org/10.2991/jnmp.2003.10.1.7
- Boyle, P.P., Tian, W. and Guan, F. (2002) The Riccati Equation in Mathematical Finance. Journal of Symbolic Computation, 33, 343-355. https://doi.org/10.1006/jsco.2001.0508
- Reid, W.T. (1980) Riccati Differential Equations. Academic Press, New York.
- Dubois, F. and Saidi, A. (2000) Unconditionally Stable Scheme for Ricatti Equation. ESAIM Proceedings, 8, 39-52. https://doi.org/10.1051/proc:2000003
- Abbasbandy, S. (2007) A New Application of He’s Variational Iteration Method for Quadratic Riccati Differential Equation Using Adomian’s Polynomials. Journal of Computational and Applied Mathematics, 207, 59-63. https://doi.org/10.1016/j.cam.2006.07.012