Numerical solution to an integral equation in geometric process

Abstract

In this article, an integral equation satisfied by the expected number of events occurred by time t in a geometric process is considered; this equation is a generalization of the renewal equation. A numerical solution based on the trapezoidal integration rule to the integral equation is introduced. To explain the method used in the article, four numerical examples with the exponential distribution, gamma distribution, Weibull distribution and lognormal distribution, respectively, are studied.

Keywords:

• Geometric process
• Renewal process
• Bernoulli number
• Trapezoidal integration rule

Acknowledgements

The authors are grateful to Prof. Tao Lü and Miss. Yaping Liu for their helpful discussion. The authors also appreciate the editor and referee for their helpful comments.