Applications of New Time and Spatial Fractional Derivatives with Exponential Kernels

In the paper, we present some applications and features related with the new notions of fractional derivatives with a time exponential kernel and with spatial Gauss kernel for gradient and Laplacian operators. Specifically, for these new models we have proved the coherence with the thermodynamic laws. Hence, we have revised the standard linear solid of Zener within continuum mechanics and the model of Cole and Cole inside electromagnetism by these new fractional operators. Moreover, by the Gaussian fractional gradient and through numerical simulations, we have studied the bell shaped filtering effects comparing the results with exponential and Caputo kernel.