Numerical Solution of Advanced Stochastic Time-delay Differential Equations with Its Effects in E-commerce for Customers Satisfaction


Published: 2022-11-05

Page: 696-715

C. Chibuisi *

Department of Insurance, University of Jos, Jos, Nigeria.

B. O. Osu

Department of Mathematics, Abia State University, Uturu, Nigeria.

C. Olunkwa

Department of Mathematics, Abia State University, Uturu, Nigeria.

*Author to whom correspondence should be addressed.


This paper examined the stochastic time effects on the degree of customers’ satisfaction through the use of e-commerce for business transactions. These days, customers’ satisfaction in the use of e-commerce face serious challenges due to unforeseen, unpredicted and unexpected circumstances between the time orders/purchases of products are made and the delivery time. To improve customers’ satisfaction in the use of e-commerce and drastically reduce these uncertainties which result to advanced stochastic time-delay or random change in network failures during the verification of credit alerts by the sellers when orders are made, road accidents, bad road and long distance, we applied of Block Backward Differentiation Formulae Methods to solve some Advanced Stochastic Time-Delay Differential Equations (ASTDDEs). The convergence and stability analysis of the method were investigated. Based on the findings, it was recommended that companies and suppliers should ensure steady networks, provide alternative means for effective e-commerce between buyers and sellers to improve customer and supplier relationships and orders/purchases should be made from close-by companies/suppliers to avoid advanced stochastic time-delay. The results obtained were presented graphically and compared with other existing methods to prove the computational efficiency and accuracy of our method.

Keywords: Block method, backward differentiation formulae method, lead time, uncertainty, business transaction

How to Cite

Chibuisi, C., Osu, B. O., & Olunkwa, C. (2022). Numerical Solution of Advanced Stochastic Time-delay Differential Equations with Its Effects in E-commerce for Customers Satisfaction. Asian Journal of Pure and Applied Mathematics, 4(1), 696–715. Retrieved from


Download data is not yet available.


Aghaunor L, Fotoh X. Factors affecting e-commerce adoption in Nigeria banks. Sweden. Jönköping International Business School. Jönköping University; 2012.

Al-Alawi AI, Al-Ali FM. Factors affecting e-commerce adoption in SMEs in the GCC: an empirical study of Kuwait. Res J Inf Technol. 2015;7(1):1-21.

Park D, Lee J, Han I. The effect of online consumer reviews on consumer purchasing intension: the moderating role of involvement. Int J Electron Com. 2007;11(4):125-48.

Zhang H, Gan S, Hu L. The split-step backward Euler method for linear stochastic delay deferential equations. Comput Appl Math. 2009;225(2):558-68.

Buckwar E. Introduction to the numerical analysis of stochastic delay differential equations. J Comp Appl Math. 2000;125(1-2):297-307.

Akhtari B. Numerical solution of stochastic state-dependent delay differential equations: convergence and stability [journal]. Adv Differ Equ. 2019;2019(1).

Wang X, Gan S. The improved split-step backward Euler method for stochastic differential delay equations. Int J Comput Math. 2011;88(11):2359-78.

Kazmerchuk YI, Wu JH. Stochastic State-dependent Delay Deferential Equations with Applications in Fnance. Funct. Differ. Equ. 2004;11(1):77-86.

Alesemi M, Iqbal N, Hamoud AA. The analysis of fractional-order proportional delay physical models via a novel transform. Complexity. 2022 Feb 15;2022:1-13.

Hamoud A, Ghadle K. Recent advances on reliable methods for solving Volterra-Fredholm integral and integro-differential equations. Asian J Math Comput Res. 2018 Apr 10;24(3):128-57.

Bhadane P, Ghadle KP, Hamoud AA. Approximate solution of fractional Black-Schole’s European option pricing equation by using ETHPM. Nonlinear Funct Anal Appl. 2020;25(2):331-44.

Hamoud AA, Mohammed NM, Ghadle KP. Some powerful techniques for solving nonlinear Volterra-Fredholm integral equations. J Appl Nonlinear Dyn. 2021;10(3):461-9.

Majid ZA, Radz HM. Solving delay differential equations by the Fve-point one-step block method using Neville’s interpolation. Int J Comput Math. 2013;754015.

Akhtari B, Babolian E, Neuenkirch A. An Euler Scheme for Stochastic Delay Differential Equations on Unbounded Domains: Pathwise Convergence. Discrete Contin. Dyn. Syst. Ser.B. 2015;20(1):23-38.

Onumanyi P, Awoyemi DO, Jator SN, Sirisena UW. New linear multistep methods with continuous coefficients for first order initial value problems. J Niger Math Soc. 1994;13:37-51.

Sirisena UW. A reformulation of the continuous general linear multistep method by matrix inversion for the first order initial value problems [Ph.D. thesis (unpublished)]. Nigeria: University of Ilorin; 1997.

Lambert JD. Computational methods in ordinary differential equations. New York: John Wiley & Sons Inc; 1973.

Dahlquist G. Convergence and stability in the numerical integration of ordinary differential equations. Math Scand. 1956;4:33-53.

Osu BO, Chibuisi C, Egbe GA, Egenkonye VC. The solution of stochastic time-dependent first order delay differential equations using block Simpson’s methods. Int J Math Comput Appl Res (IJMCAR). 2021;11(1):1-20.