Theta-type convolution quadrature OSC method for nonlocal evolution equations arising in heat conduction with memory

In this paper, we propose a robust and simple technique with efficient algorithmic implementation for numerically solving the nonlocal evolution problems. A theta-type (\(\theta \)-type) convolution quadrature rule is derived to approximate the nonlocal integral term in the problem under consideration, such that \(\theta \in (\frac{1}{2},1)\), which remains untreated in the literature. The proposed approaches are based on the \(\theta \) method (\(\frac{1}{2}\le \theta \le 1\)) for the time derivative and the constructed \(\theta \)-type convolution quadrature rule for the fractional integral term. A detailed error analysis of the proposed scheme is provided with respect to the usual convolution kernel and the tempered one. In order to fully discretize our problem, we implement the orthogonal spline collocation (OSC) method with piecewise Hermite bicubic for spatial operators. Stability and error estimates of the proposed \(\theta \)-OSC schemes are discussed. Finally, some numerical experiments are introduced to demonstrate the efficiency of our theoretical findings.