Solving Riccati type q-Difference Equations via Difference Transform Method
Keywords:
quantum calculus, Riccati type difference equations, time scale, difference transform method.
Abstract
In this paper, we deal on the time scale that its delta derivative of graininess function is a nonzero positive constant. Based on the Taylor formula for this time scale, we investigate the difference transform method (DTM). This method has been applied successfully to solve Riccati type difference equations in quantum calculus. To demonstrate the ability and efficacy of this method, some examples have been provided.
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References
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[2] M. Dehghan and A. Taleei, “A compact split-step finite difference method for solving the nonlinear schr dinger equations with constant and variable coefficients,” Computer Physics Communications, vol. 181, pp. 43–51, jan 2010.
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[34] G.-C. Wu, “Variational iteration method for q-difference equations of second order,” Journal of Applied Mathematics, vol. 2012, pp. 1–5, 2012.
[35] M. S. Semary and H. N. Hassan, “The homotopy analysis method for q-difference equations,” Ain Shams Engineering Journal, vol. 9, pp. 415–421, sep 2018.
[36] S. Georgiev, Fractional dynamic calculus and fractional dynamic equations on time scales. Cham, Switzerland: Springer, 2018.
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[38] M. Bohner and A. Peterson, Dynamic equations on time scales: An introduction with applications. Springer Science Business Media, 2001.
[39] H.-K. Liu, “Application of a differential transformation method to strongly nonlinear damped q-difference equations,” Computers Mathematics with Applications, vol. 61, no. 9, pp. 2555–2561, 2011.
[40] H.-K. Liu, “The formula for the multiplicity of two generalized polynomials on time scales,” Applied Mathematics Letters, vol. 25, no. 10, pp. 1420–1425, 2012.
[41] J. Zhou, Differential Transformation and Its Applications for Electrical Circuits. Huazhong Univ. Press, Wuhan, China, 1986.
[42] S.-C. Jing and H.-Y. Fan, “q -taylor’s formula with its q -remainder,” Communications in Theoretical Physics, vol. 23, pp. 117–120, jan 1995.
[43] H.-K. Liu, “Application of a differential transformation method to strongly nonlinear damped q-difference equations,” Computers Mathematics with Applications, vol. 61, pp. 2555–2561, may 2011.
[1] W. Reid, Riccati differential equations. New York: Academic Press, 1972.
[2] M. Dehghan and A. Taleei, “A compact split-step finite difference method for solving the nonlinear schr dinger equations with constant and variable coefficients,” Computer Physics Communications, vol. 181, pp. 43–51, jan 2010.
[3] M. Nowakowski and H. C. Rosu, “Newton’s laws of motion in the form of a riccati equation,” Physical Review E, vol. 65, mar 2002.
[4] A. Neirameh, “Exact analytical solutions for 3d- gross–pitaevskii equation with periodic potential by using the kudryashov method,” Journal of the Egyptian Mathematical Society, vol. 24, pp. 49–53, jan 2016.
[5] H. C. Rosu, S. C. Mancas, and P. Chen, “Barotropic FRW cosmologies with chiellini damping,” Physics Letters A, vol. 379, pp. 882–887, may 2015.
[6] D. D. Alessandro, “Invariant manifolds and projective combinations of solutions of the riccati differential equation,” Linear Algebra and its Applications, vol. 279, pp. 181–193, aug 1998.
[7] M. I. Zelikin, “Riccati equation in the classical calculus of variations,” in Control Theory and Optimization I, pp. 60–79, Springer Berlin Heidelberg, 2000.
[8] L. Ntogramatzidis and A. Ferrante, “On the solution of the riccati differential equation arising from the LQ optimal control problem,” Systems Control Letters, vol. 59, pp. 114–121, feb 2010.
[9] R. Bellman and R. Vasudevan, “Dynamic programming and solution of wave equations,” in Wave Propagation, pp. 259–309, Springer Netherlands, 1986.
[10] K. Parand, S. A. Hossayni, and J. Rad, “Operation matrix method based on bernstein polynomials for the riccati differential equation and volterra population model,” Applied Mathematical Modelling, vol. 40, pp. 993–1011, jan 2016.
[11] F. Geng, “A modified variational iteration method for solving riccati differential equations,” Computers Mathematics with Applications, vol. 60, pp. 1868–1872, oct 2010.
[12] C. Bota and B. C runtu, “Analytical approximate solutions for quadratic riccati differential equation of fractional order using the polynomial least squares method,” Chaos, Solitons Fractals, vol. 102, pp. 339–345, sep 2017.
[13] S. Abbasbandy, “Iterated he’s homotopy perturbation method for quadratic riccati differential equation,” Applied Mathematics and Computation, vol. 175, pp. 581–589, apr 2006.
[14] S. Balaji, “Legendre wavelet operational matrix method for solution of fractional order riccati differential equation,” Journal of the Egyptian Mathematical Society, vol. 23, pp. 263–270, jul 2015.
[15] M. A. El-Tawil, A. A. Bahnasawi, and A. Abdel-Naby, “Solving riccati differential equation using adomian’s decomposition method,” Applied Mathematics and Computation, vol. 157, pp. 503–514, oct 2004.
[16] G. D. Nicolao, “On the time-varying riccati difference equation of optimal filtering,” SIAM Journal on Control and Optimization, vol. 30, pp. 1251–1269, nov 1992.
[17] A. Beghi and D. D’alessandro, “Discrete-time optimal control with control-dependent noise and generalized riccati difference equations,” Automatica, vol. 34, pp. 1031–1034, aug 1998.
[18] G. Freiling, G. Jank, and H. Abou-Kandil, “Generalized riccati difference and differential equations,” Linear Algebra and its Applications, vol. 241-243, pp. 291–303, jul 1996.
[19] D. D’Alessandro, “Geometric aspects of the riccati difference equation in the nonsymmetric case,” Linear Algebra and its Applications, vol. 255, pp. 1–18, apr 1997.
[20] K. Balla, “Asymptotic behavior of certain riccati difference equations,” Computers Mathematics with Applications, vol. 36, pp. 243–250, nov 1998.
[21] J. Zou and Y. P. Gupta, “Robust stabilizing solution of the riccati difference equation,” European Journal of Control, vol. 6, pp. 384–391, jan 2000.
[22] L. Peled-Eitan and I. Rusnak, “State dependent difference riccati equation based estimation for corkscrew maneuver,” IFAC Proceedings Volumes, vol. 47, no. 3, pp. 2347–2352, 2014.
[23] H. Zhang and P. M. Dower, “Max-plus fundamental solution semigroups for a class of difference riccati equations,” Automatica, vol. 52, pp. 103–110, feb 2015.
[24] J. Sugie, “Nonoscillation theorems for second-order linear difference equations via the riccati-type transformation, II,” Applied Mathematics and Computation, vol. 304, pp. 142–152, jul 2017.
[25] S. Nishioka, “Differential transcendence of solutions of difference riccati equations and tietze’s treatment,” Journal of Algebra, vol. 511, pp. 16–40, oct 2018.
[26] H. Tietze, “ ber funktionalgleichungen, deren l sungen keiner algebraischen differentialgleichung gen gen k nnen,” Monatshefte f r Mathematik und Physik, vol. 16, no. 1, pp. 329–364, 1905.
[27] T. Ernst, “The history of q-calculus and a new method,” 2000.
[28] H. L. Saad and A. A. Sukhi, “Another homogeneous q-difference operator,” Applied Mathematics and Computation, vol. 215, pp. 4332–4339, feb 2010.
[29] A. P. Magnus, “Special nonuniform lattice (snul) orthogonal polynomials on discrete dense sets of points,” Journal of Computational and Applied Mathematics, vol. 65, pp. 253–265, dec 1995.
[30] M. Jambu, “Quantum calculus an introduction,” in New Trends in Algebras and Combinatorics, WORLD SCIENTIFIC, feb 2020.
[31] A. Lavagno, “Basic-deformed quantum mechanics,” Reports on Mathematical Physics, vol. 64, pp. 79–91, aug 2009.
[32] M. El-Shahed and M. Gaber, “Two-dimensional q-differential transformation and its application,” Applied Mathematics and Computation, vol. 217, pp. 9165–9172, jul 2011.
[33] M. El-Shahed, M. Gaber, and M. Al-Yami, “The fractional q-differential transformation and its application,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, pp. 42–55, jan 2013.
[34] G.-C. Wu, “Variational iteration method for q-difference equations of second order,” Journal of Applied Mathematics, vol. 2012, pp. 1–5, 2012.
[35] M. S. Semary and H. N. Hassan, “The homotopy analysis method for q-difference equations,” Ain Shams Engineering Journal, vol. 9, pp. 415–421, sep 2018.
[36] S. Georgiev, Fractional dynamic calculus and fractional dynamic equations on time scales. Cham, Switzerland: Springer, 2018.
[37] M. Bohner and A. C. Peterson, Advances in dynamic equations on time scales. Springer Science & Business Media, 2002.
[38] M. Bohner and A. Peterson, Dynamic equations on time scales: An introduction with applications. Springer Science Business Media, 2001.
[39] H.-K. Liu, “Application of a differential transformation method to strongly nonlinear damped q-difference equations,” Computers Mathematics with Applications, vol. 61, no. 9, pp. 2555–2561, 2011.
[40] H.-K. Liu, “The formula for the multiplicity of two generalized polynomials on time scales,” Applied Mathematics Letters, vol. 25, no. 10, pp. 1420–1425, 2012.
[41] J. Zhou, Differential Transformation and Its Applications for Electrical Circuits. Huazhong Univ. Press, Wuhan, China, 1986.
[42] S.-C. Jing and H.-Y. Fan, “q -taylor’s formula with its q -remainder,” Communications in Theoretical Physics, vol. 23, pp. 117–120, jan 1995.
[43] H.-K. Liu, “Application of a differential transformation method to strongly nonlinear damped q-difference equations,” Computers Mathematics with Applications, vol. 61, pp. 2555–2561, may 2011.
Published
2021-07-22
How to Cite
Ahmed Y. Abdulmajeed, & Ayad R. Khudair. (2021). Solving Riccati type q-Difference Equations via Difference Transform Method. Al-Qadisiyah Journal of Pure Science, 26(4), 181–192. https://doi.org/10.29350/qjps.2021.26.4.1318
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Special Issue (Silver Jubilee)
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