Numerical method for simulation of quadratic Riccati differential equations

Adam Ajimoti Ishaq; Folashade Mistura Jimoh; Kazeem Issa

doi:10.61298/rans.2026.4.1.159

Authors

Adam Ajimoti Ishaq
Department of Physical Sciences Al-Hikmah University, Ilorin, Nigeria
Folashade Mistura Jimoh
Department of Physical Sciences Al-Hikmah University, Ilorin, Nigeria
Kazeem Issa
[email protected]

Department of Mathematics and Statistics, Kwara State University, Malete, Nigeria

Keywords:

Collocation method, Computational method, Numerical simulations, Nonlinear equations

Abstract

Differential equations widely applied in various fields of engineering and mathematical studies, particularly in control theory, optimal control, and stochastic realization. Numerous methods have been proposed for their solution. Although QRDEs pose significant analytical challenges, several approaches such as the Variational Iteration Method, Differential Transform Method, Runge-Kutta methods, and Non-Standard Finite Difference Methods have been developed to address them. In this paper, we present a computational method for solving QRDEs by employing the collocation technique in combination with a power series approximation. Fundamental properties of the method, including order, consistency, zero-stability, and convergence, are analyzed. Numerical simulations demonstrate that the proposed method provides accurate and stable results, showing strong agreement with existing methods when applied to different QRDE models. Furthermore, the findings highlight the efficiency and applicability of the approach in solving nonlinear equations of considerable complexity.

Dimensions

REFERENCES

[1] B. Batiha, ‘‘A new efficient method for solving quadratic Riccati differential equation’’, Int Appl. Math. Res. 4 (2015) 24. https://doi.org/10.14419/IJAMR.V4I1.4113.

[2] I. D. E. Nasr Al-Din, ‘‘Comparison of numerical methods for approximating solution of the Quadratic Riccati Differential Equation’’, European Journal of Applied Sciences 12 (2020) 32. https://doi.org/10.5829/idosi.ejas.2020.32.35.

[3] I. D. E. Nasr Al-Din, ‘‘Comparison of Newton-Raphson based modified Laplace Adomian Decomposition method and Newton’s Interpolation and Aitken’s Method for solving Quadratic Riccati Differential Equations’’, Middle-East Journal of Scientific Research 28 (2020) 235. https://doi.org/10.5829/idosi.mejsr.2020.235.239.

[4] M. Vinod & R. Dimple, ‘‘Newton-Raphson based modified laplace adomian decomposition method for solving quadratic riccati differential equations’’, MATEC Web of Conferences 57 (2016) 05001. http://dx.doi.org/10.1051/matecconf/2016570ICAET2016.

[5] K. Wase, S. Alemayehu & G. Solomon, ‘‘Eighth order Predictor-Corrector method to solve quadratic riccati differential equations’’, Momona Ethiopian Journal of Science 13 (2021) 213. http://dx.doi.org/10.4314/mejs.v13i2.2.

[6] A. A. Opanuga, O. E. Sunday, I. O. Hilary & O. A. Grace, ‘‘A novel approach for solving quadratic Riccati differential equations’’, International Journal of Applied Engineering Research 10 (2015) 29121. https://tinyurl.com/mr2t5y3p.

[7] A. Misir, ‘‘A new analytic solution method for a class of generalized riccati differential equations’’, Universal Journal of Mathematics and Applications 6 (2023) 1. https://doi.org/10.32323/ujma.1143751.

[8] B. Batiha, M. S. M. Noorani, I. Hashim & F S. Ismail, ‘‘The multistage variationaliterationmethodforaclassofnonlinearsystemofODEs’’, Phys. Scr. 76 (2017) 388. https://doi.org/10.1088/0031-8949/76/4/018.

[9] A. Ghorbani & S. Momani, ‘‘An effective variational iteration algorithm for solving Riccati differential equations”,. Appl. Math. Letters 23 (2010) 922. https://doi.org/10.1016/j.aml.2010.04.012.

[10] I. Hashim, M. S. M. Noorani, R. Ahmad, S. A. Bakar, E. S. Ismail & A. M. Zakari, ‘‘Accuracy of the Adomian decomposition method applied to the Lorenz system”, Chaos, Soliton and Fractals 28 (2006) 1149. https://doi.org/10.1016/j.chaos.2005.08.135.

[11] N. Kamoh, G. Kumleng & J. Sunday, ‘‘Matrix approach to the direct computation method for the solution of fredholm integro- differential equations ofthesecondkindwithdegeneratekernels”, CAUCHY–JurnalMatematika Murni dan Aplikasi 6 (2020) 100. https://doi.org/10.18860/ca.v6i3.8960.

[12] G. M. Kumleng, J. P. Chollom & S. Omagwu, ‘‘A class of new block generalized adams implicit runge-kutta collocation methods”, International Journal of Scientific and Engineering Research 6 (2015) 10. https://tinyurl.com/4msma7xr.

[13] K. Issa, K., R. A. Bello & U. J. Abubakar, ‘‘Approximate analytical solution of fractional-order generalized integro-differential equations via fractional derivative of shifted Vieta-Lucas polynomial’’, J. Nig. Phys. Sci. 6 (2024) 1821. https://doi.org/10.46481/jnsps.2024.1821.

[14] A. R. Vahidi & M. Didgar, ‘‘Improving the accuracy of the solutions of Riccati equations’’, Int J Ind Math. 4 (2018) 11. https://tinyurl.com/26hsxzb5.

[15] Y. Skwame, J. Sabo & T. Y. Kyagya, ‘‘The constructions of implicit onestep block hybrid methods with multiple off-grid points for the solution of stiff differential equations’’, Journal of Scientific Research and Report 16 (2017) 1. https://doi.org/10.9734/JSRR/2017/36187.

[16] J. Sunday & J. Philip, ‘‘On the derivation and analysis of a highly efficient method for the approximation of quadratic riccati equations’’, Computer Reviews Journal 2 (2018) 1. http://purkh.com/index.php/tocomp.

[17] F. M. Jimoh, A. A. Ishaq & K. Issa, ‘‘Approximate solution of higher-order oscillatory differential equations via modified linear block techniques’’, African Scientific Reports 4 (2025) 275. https://doi.org/10.46481/asr.2025.4.2.275.

[18] T. A. Driscoll & R. J. Braun, ‘‘Fundamentals of numerical computation’’, SIAM, Philadelphia, USA, 2022, pp. 227 – 278. https://doi.org/10.1137/1.9781611975086.ch6.

[19] G. File & S. Aga, ‘‘Numerical solution of quadratic Riccati differential equations’’, Egyptian Journal of Basic and Applied Sciences 3 (2016) 392. https://doi.org/10.1016/j.ejbas.2016.08.006.

[20] J. Sunday, ‘‘Riccati Differential Equations: a computational approach’’, Archives of Current Research International 9 (2017) 1. https://doi.org/10.9734/ACRI/2017/36267.