Heat Conduction With Distributed‐Order Fractional Derivative Perturbed by a Stochastic Process

Heat conduction in a rigid body with a stochastic perturbation is studied. The constitutive equation is a generalization of the Cattaneo one so that the first derivative with respect to time is replaced by the distributed order Caputo fractional derivative. In addition, a stochastic term is introduced in the constitutive equation. The Clausius‐Duhem (entropy) inequality is derived by the use of the Bochner‐Schwartz theorem. The solution to the heat conduction fractional equation is obtained in the case of one dimensional space variable. Its evaluation in time is analyzed.