# On the numerical approximation of Volterra integro-differential equation using Laplace transform

Document Type : Research Paper

Authors

Department of basic sciences and islamiat, University of engineering and technology Peshawar, pakistan

Abstract

In this work we constructed a numerical scheme to approximate the Volterra integro- differential equations of convolution type using Laplace transform. The solution of the problem is recovered using inverse Laplace transform as contour integral in the complex plane. The integral is then approximated along a suitable contour using the trapezoidal rule with equal step size. The solution accuracy depends on optimal contour of integrations to compute accurately the inverse Laplace transform. For better accuracy two types of contour parabolic and hyperbolic are used which are available in the literature. The performance of the numerical scheme is tested for different examples. The actual error well agree with the corresponding error estimates of the proposed numerical scheme for both parabolic as well as hyperbolic contours.

Keywords

### History

• Receive Date: 11 June 2018
• Revise Date: 11 May 2019
• Accept Date: 14 May 2019