Numerical investigation of generalized tempered-type integrodifferential equations with respect to another function

Abstract

This paper studies two efficient numerical methods for the generalized tempered integrodifferential equation with respect to another function. The proposed methods approximate the unknown solution through two phases. First, the backward Euler (BE) method and first-order interpolation quadrature rule are adopted to approximate the temporal derivative and generalized tempered integral term to construct a semi-discrete BE scheme. Second, the backward differentiation formula (BDF) and second-order interpolation quadrature rule are adopted to establish a semi-discrete second-order BDF (BDF2) scheme. Additionally, the stability and convergence of two semi-discrete methods are deduced in detail. To further demonstrate the effectiveness of proposed techniques, fully discrete BE and BDF2 finite difference schemes are formulated. Subsequently, the theoretical results of two fully discrete difference schemes are presented. Finally, the numerical results demonstrate the accuracy and competitiveness of the theoretical analysis.