Nonlinear heat-wave propagation in rigid thermal conductors via a distributed-order Caputo fractional model

In this paper, a distributed–order Caputo fractional derivative is used to represent nonlinear heat wave propagation in rigid thermal conductors using a novel numerical approach. The suggested model takes into consideration memory effects that are a part of thermal wave transmission and captures non-Fourier heat conduction behavior. We use a fourth-order compact finite difference approach for spatial discretization to solve the governing nonlinear fractional partial differential equation, guaranteeing good accuracy with a small stencil. In order to handle the fractional–order integration over a continuous range with flexibility and enhanced stability, a weighted average non-standard finite difference method is employed for the temporal discretization of the distributed–order Caputo derivative. The created method’s accuracy, efficiency, and convergence are shown by numerical experiments. The findings demonstrate the model’s capacity to represent intricate thermal behaviors in rigid material by demonstrating the substantial influence of the distributed-order kernel on the propagation properties of thermal waves.