Variational Formulation For Time Fractional Telegraph Equation With Truncated Lévy Flights

The Cattaneo equation(1948)are a type of partial differential equations describing a diffusion process with a finite velocity of propagation.To describe anomalous trans-port,the time fractional Cattaneo equations have been presented by Compte and Metzler(1997).Following Compte and Metzler,Povstenko(2010)studied the time-and space-nonlocal generalizations of the Fourier law,where the Levy flights have the divergent second order moments.However,in many situations of physical interest,the physical domain is bounded and the involved observables have the finite moments.Then the modified equation can be derived by truncated the Levy measure of the Levy flights and the corresponding tempered space fractional derivative is introduced.Then a theoreti-cal framework for the Galerkin finite element approximation to the modified equation is developed.Moreover,we introduce the fractional ?-norm,and show the existence and uniqueness of variational formulation when ??0.Using the discrete Gronwall's inequality and prior estimate,the stability and global error estimates with O(?2+hr)are rigorously established and numerically verified.