Solution of a partial integro-differential equation arising from viscoelasticity

Abstract

Partial integro-differential equations occur in many fields of science and engineering. This study presents various numerical schemes for solving a partial integro-differential equation with a weakly singular kernel. These schemes are presented for determining the solution of a partial integro-differential equation subject to an initial condition and given boundary conditions. We construct several computational techniques based on finite-difference schemes and the product trapezoidal numerical integration rule. This problem can be found in the modelling of physical phenomena involving viscoelasticity forces. The finite-difference procedures developed are based on the forward Euler explicit scheme, the backward Euler implicit technique, the Crank–Nicolson implicit formula and Crandall’s implicit method. Three of the methods have second-order accuracy with respect to the space variable. The order of accuracy of the Crandall’s scheme is higher than that of the others. The numerical results of a test problem are given to verify the stability and accuracy of the procedures discussed.

Keywords:

• Partial integro-differential equation
• Finite-difference schemes
• Order of accuracy
• Viscoelasticity
• Product numerical integration rule

Additional information

Notes on contributors

Mehdi Dehghan

Email: [email protected]