Well-posed coupled fractional hyperbolic problem of thermoelasticity

Abstract

This is an attempt to construct the well-posed hyperbolic heat conduction model based on the Caputo fractional derivative and to study the corresponding coupled thermoelastic problem. The continuous dependence on initial data and energy supply, and the uniqueness of the solutions are mathematically proved. The general closed-form solution of the time fractional conduction model for the initial Dirichlet boundary value problem is obtained analytically by applying the Laplace transform and finite Fourier sine transform in one-dimensional case. The application of theoretical study for heat propagation in the wire is considered. As a special case, two different examples have been discussed to study the analysis of the temperature distributions in the spatial geometry. The influence of the fractional orders on the speed of heat conductivity in the model is discussed. The physical behavior of the temperature distribution has been graphically represented for different fractional orders. Furthermore, the thermal stress analysis is studied using the coupled thermoelasticity theory. In the Laplace domain, the analytical solutions have been obtained. The Gaver–Stehfest technique was employed to numerically perform time domain inversions of the Laplace transforms, which satisfied Kuznetsov’s convergence theorem.