A geometric numerical integration method for solving the Volterra integro-differential equations

ABSTRACT

In this paper, a geometric approach is proposed to the integration of the system of Volterra integro-differential equations (IDEs). To do this, the equivalent system of ordinary differential equations of IDEs are obtained and converted into a Lie type augmented dynamical system. Then we construct the group preserving scheme (GPS) on the system of Volterra IDEs which is formulated by an exponential mapping to keep the group properties of $S{O}_o(n,1)$. Some linear and nonlinear examples are solved as a description for the power and efficiency of GPS. Finally, in order to demonstrate the validity and efficiency of the present method, the convergency is numerically discussed and accuracy of results are compared with the reported results in the literature.

KEYWORDS:

• Group preserving scheme
• Minkowski space
• Volterra integro-differential equation

2010 AMS SUBJECT CLASSIFICATIONS:

• 45J05
• 34K28
• 34A26

Acknowledgements

The authors are grateful to the anonymous referees for their comments which substantially improved the quality of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author highly appreciate the support from University of Bonab, under contact number: 93/I/ER/291.