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In seismic forward modeling and fluid-flow simulations, sources and initial conditions are two approaches to initiate the perturbation of the medium in order to compute synthetic seismograms and pore-pressure maps of a reservoir, respectively. Assuming delta functions in time and space, source and initial particle velocity are equivalent in the first case (wave equation), while in the second case (diffusion equation) source and diffusion field are equivalent. The differential equation based on fractional derivatives unifies both cases but those equivalences break down when the order of the derivative is not a natural number. A simulation example, based on the fractional wave equation, illustrates the implementation of the source with a band-limited time history on a numerical mesh. In this case, the implementation of the initial condition requires a numerical calculation since the medium is heterogeneous. Body forces and stress sources (e.g., earthquakes) can easily be related in uniform media. A 1D example shows such a relation. [ABSTRACT FROM AUTHOR]

The author of the commented paper applies the Exp-function method for the derivation of generalized solitary solutions to a system of nonlinear partial differential equations. We argue that the derived solution does not exist for all initial conditions and use operator techniques to derive the conditions of existence of the solitary solution. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we argue that the solitary solution of the Liouville equation produced by the Exp-function method does not satisfy the original differential equation for all initial conditions. Moreover, the region where the solution is correct is located entirely on a curve in the parameter plane of initial conditions. We derive the explicit equation for this curve and argue that classical Exp-function type methods cannot identify such constraints related to initial conditions. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the approximation of general multivariable non commensurate fractional systems by integer order state space models. This work contains two main contributions. First, a new state space representation using the fractional integral operator is introduced. Second, the approximate model carries explicitly the initial conditions of the system. Two examples are given to illustrate the accuracy of the approximation. [Copyright &y& Elsevier]