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Hybrid ·Peer-reviewed·ISSN (Online): 2169-012X·ISSN (Print): 0972-0502

Monthly Journal: Publishes the methodological and theoretical role of mathematics and mathematical applications underpinning scientific research.

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Open Access Research Article

Solving stochastic delay differential equation by using Runge-Kutta method and method of lines

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pp. 1871–1879Vol. 28Issue 5August 2025DOI: 10.47974/JIM-2143XML
Received:
10 Dec 2024
Published Online:
28 Aug 2025
Article type:
Research Article
Language:
EN
Article no.:
JIM-2143
Pages:
1871–1879

Abstract

In this study, we have examined a stochastic delay differential equation in one dimension. We employed the Runge-Kutta method along with the method of lines to address the issue, establishing stability analysis and studying the spread of discontinuities. Additionally, we derived a semi-discretization in the temporal domain. Subsequently, the Runge-Kutta method of order four and cubic Hermite Interpolation is derived and numerical examples are illustrated for the theoretical result.

Keywords

Subject Classifications

37H1060G2035R60

References

[1] A. N. Al-Mutib, “Stability properties of numerical methods for solving delay differential equations,” Journal of Computational and Applied Mathematics, vol. 10, no. 1, pp. 71-79 (1984).     
[2] X. D. Liu, “Maximum principle satisfying modification of triangle based adapative stencils for the solution of scalar hyperbolic conservation laws,” SIAM Journal on Numerical Analysis, vol. 30, no. 3, pp. 701-716 (1993). 
[3] Zdzisław Kamont and Drumi D. Bainov, “Comparison principles for impulsive hyperbolic equations of first order,” Journal of Computational and Applied Mathematics, vol. 60, pp. 379-388 (1995). 
[4] Ernst Hairer and Gerhard Wanner, “Solving ordinary differential equations II,” Springer, Berlin (1996). 
[5] Alfredo Bellen and Marino Zennaro, “Numerical methods for delay differential equations,” Oxford, New York (2003). 
[6] Pardeep Singh and K. K. Sharma, “Numerical solution of first-order hyperbolic partial differential-difference equation with shift,” Numerical Methods for Partial Differential Equations, vol. 26, no. 1, pp. 107-116 (2010). 
[7] H. Brunner and E. Hairer, “A general-purpose implicit-explicit Runge-Kutta integrator for delay differential equations,” BIT Numerical Mathematics, vol. 52, no. 2, pp. 293-314 (2012). 
[8] H Yuan, C Song, and P Wang, “Nonlinear stability and convergence of two step Runge-Kutta methods for neutral delay differential equations,” Mathematical Problems in Engineering, Vols. 1-14, p. 2013 (2013). 
[9] JB Niu, Y Ding, LM Zhu, and H Ding, “Eigenvalue assignment for control of time-delay systems via the generalized Runge-Kutta method,” Journal of Dynamic Systems, Measurement, and Control, vol. 137, no. 9, p. 091003 (2015). 
[10] S Singh, V. K Patel, and V. K Singh, “Application of wavelet collocation method for hyperbolic partial differential equations,” Applied Mathematics and Computation, vol. 320, pp. 407-424 (2018). 
[11] V. Chauhan, and P. K. Srivastava, “Computational techniques based on Runge-Kutta Method of various,” Engineering and Management Sciences, vol. 4, no. 2, pp. 375-386 (2019). 
[12] T. Bochacik, M. Goćwin, P. M. Morkisz, and P. Przybyłowicz, “Randomized Runge-Kutta method–stability and convergence under inexact information,” Journal of Complexity, vol. 65, p. 101554 (2021). 
[13] M. Masud Rana, V. E. Howle, K. Long, A. Meek, and W. Milestone, “A new block preconditioner for implicit Runge–Kutta Methods for parabolic PDE problems,” SIAM Journal on Scientific Computing, vol. 43, pp. 475-495 (2021). 
[14] G. L. Zhang and C. Liu, “Two schemes of impulsive Runge–Kutta methods for linear differential equations with delayed impulses,” Mathematics, vol. 12, no. 13, p. 2075 (2024). 
[15] N. Shahid, A. Raza, S. Iqbal, N. Ahmed, E. Fadhal, and B. Ceesay, “Stochastic delayed analysis of coronavirus model through efficient computational method,” Scientific Reports, vol. 14, no. 1, p. 21170 (2024). 
[16] D. Stosovic and E. Čajić, “Optimization of numerical solutions of stochastic differential equations With time delay,” Current Research in Statistics & Mathematics, vol. 3, no. 1, pp. 1-9 (2024). 
[17] H. Wen, “Delay-dependent stability of predictor–corrector methods of Runge–Kutta type for stochastic delay differential equations,” Calcolo, vol. 61, no. 2 (2024). 
[18] F. V. Difonzo, P. Przybyłowicz, Y. Wu, and X. Xie, “A randomized Runge-Kutta method for time-irregular delay differential equations,” arXiv preprint arXiv (2024). 
[19] H. Zhang, L. Liu, X. Qian, and S. Song, “Quantifying and eliminating the time delay in stabilization exponential time differencing Runge-Kutta schemes for the Allen-Cahn equation,” ESAIM: Mathematical Modelling and Numerical Analysis, vol. 58, pp. 191-221 (2024). 

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