We consider the inverse problem of finding the temperature distribution and the heat source whenever the temperatures at the initial time and the final time are given. The problem considered is one dimensional and the unknown heat source is supposed to be space dependent only. The existence and uniqueness results are proved.

We consider the inverse source problem for a time fractional diffusion equation. The unknown source term is independent of the time variable, and the problem is considered in two dimensions. A biorthogonal system of functions consisting of two Riesz bases of the space L2[(0,1) × (0,1)], obtained from eigenfunctions and associated functions of the spectral problem and its adjoint problem, is used to represent the solution of the inverse problem. Using the properties of the biorthogonal system of functions, we show the existence and uniqueness of the solution of the inverse problem and its continuous dependence on the data. Copyright © 2012 John Wiley & Sons, Ltd.

Bluetooth Low Energy (BLE) represents the low-power, low-cost extension of the Bluetooth communication technology envisioned for the Internet of Things. Mesh protocols on top of BLE are currently emerging and the standard is currently being released. This paper first proposes a detailed measurement based comparison of two mesh approaches that fit within BLE operation: flooding and connection oriented networking. Using metrics such as packet delivery ratio (PDR), end-to-end delay and power consumption we conclude that the optimal mesh approach depends on the application. It is shown that for a comparable performance in terms of PDR and overhead, flooding can trade a lower end-to-end delay for a higher power consumption when compared to the connected mesh. We then propose an architecture, called Bluetooth Now, that is able to automatically switch the network between the two based on message priority. Our measurement results confirm the reliable delivery of important and urgent data sent using the Bluetooth Now paradigm, while saving battery life when transmitting non-time critical messages.

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A generalization of the linear fractional integral equation u(t) = u 0 + ∂ −α Au(t), 1 < α < 2, which is written as a Volterra matrix-valued equation when applied as a pixel-by-pixel technique, has been proposed for image denoising (restoration, smoothing,...). Since the fractional integral equation interpolates a linear parabolic equation and a hyperbolic equation, the solution enjoys intermediate properties. The Volterra equation we propose is well-posed for all t > 0, and allows us to handle the diffusion by means of a viscosity parameter instead of introducing non linearities in the equation as in the Perona-Malik and alike approaches. Several experiments showing the improvements achieved by our approach are provided.

For a space-time fractional diffusion equation, an inverse problem of determination of a space dependent source term along with the solution is considered. The fractional derivatives in time and space are defined in the sense of Caputo. Due to an over-specified data at final time say T, we proved that there exists a unique solution of the inverse source problem. We use the eigenfunction expansion method to prove our main results. Several special cases of space-time fractional diffusion equations are discussed and results are interpolated from generalized results. Some examples are provided.

We investigate the profile of the blowing up solutions to the nonlinear nonlocal system (FDS) u (t) + D α 0 + (u − u 0)(t) = |v(t)| q , t > 0, v (t) + D β 0 + (v − v 0)(t) = |u(t)| p , t > 0, where u(0) = u 0 > 0, v(0) = v 0 > 0, p > 1, q > 1 are given constants and D α 0 + and D β 0 + , 0 < α < 1, 0 < β < 1 stand for the Riemann-Liouville fractional derivatives. Our method of proof relies on comparisons of the solution to the (FDS) with solutions of the subsystems obtained from (FDS) by dropping either the usual derivatives or the fractional derivatives.

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