We deal with the initial-boundary value problem for a quasilinear degenerate PARABOLIC EQUATION with inhomogeneous density and absorption, which appears in a number of applications to describe the evolution of diffusion processes, in particular non-Newtonian flow in a porous medium. We discuss the extinction of solution and the finite speed of propagation of perturbations.

In this work, the inverse quasi-linear pseudo-PARABOLIC problem was investigated. We demonstrated the solution by the Fourier approximation. The inverse problem was first examined by linearizing and then used implicit finite difference schema for the numerical solution.

In this paper, a variational iteration method (VIM), which is a well-known method for solving nonlinear EQUATIONs, has been employed to solve an inverse PARABOLIC partial differential EQUATION. Inverse problems in partial differential EQUATIONs can be used to model many real problems in engineering and other physical sciences. The VIM is to construct correction functional using general Lagrange multipliers identified optimally via the variational theory. This method provides a sequence of function which converges to the exact solution of the problem. This technique does not require any discretization, linearization or small perturbations and therefore reduces the numerical computations a lot. Numerical examples are examined to show the efficiency of the technique.

In this paper, the PARABOLIC EQUATION method is applied to analyze radio wave propagation through window structures. By this method, a typical window propagation situation is simulated for different window sizes and frame types. The simulation results are represented for both normal and oblique incident cases of uniform and non-uniform plane wave. Results from the implementation of the PARABOLIC EQUATION method show good agreement with FDTD reported simulations. Base on this study, as the PARABOLIC EQUATION method needs less memory size and CPU time against FDTD method, it can be used as an efficient algorithm to analyze this kind of problems.

Identifying  the unknown source terms in diffusion models, including nonlocal ones, is an active research area with significant applications in engineering and scientific fields such as population dynamics, biology, and physics. This study examines an inverse problem focused on recovering a space-dependent source term in a degenerate diffusion model that includes a nonlocal space term, using final-time measured data. As a first step, the inverse problem is reformulated as an optimization one by considering its solution as the minimizer of a well-defined objective function. The existence of a unique solution to the associated direct problem is discussed in a functional framework based on suitable weighted Sobolev spaces. After that, we prove the existence of a minimizer by means of standard arguments, and establish a first-order necessary optimality condition. Using this last one, we obtain some results concerning the stability and local uniqueness property. For the numerical reconstruction of the missing source term, we designed an algorithm based on the Landweber iterative method and showed its effectiveness by providing several numerical tests.

With the rapid growth of indoor wireless communication systems, the need to accurately model radio wave propagation inside the building environments has increased. Many site-specific methods have been proposed for modeling indoor radio channels. Among these methods, the ray tracing algorithm and the finite-difference time domain (FDTD) method are the most popular ones. The ray tracing approach as a high frequency technique is efficient for calculating the received field at a small number of receiver locations. Application of FDTD method as a full wave technique for indoor propagation modeling is time consuming and requires large amounts of memory. The PARABOLIC EQUATION method (PEM) is a fast full-wave technique which allows accurate modeling of the propagation environment and its electrical parameters. This paraxial version of the wave EQUATION can be solved by marching techniques which need far less computation resources than a full elliptic EQUATION. The PEM has been extensively used as an efficient algorithm for outdoor propagation modeling. In this paper we propose an unprecedented application of PEM for indoor propagation problems. Depending on the required speed and accuracy of computations, two and three-dimensional versions of the PEM can be used for indoor problems. Without loss of generality, we restrict ourselves to the two-dimensional problems and use two-dimensional approximation of the PARABOLIC EQUATION for fast and accurate radio wave propagation modeling in indoor environments. The PARABOLIC EQUATION has been derived for lossless media where the refractive index is very close to unity. To the authors' best knowledge the paraxial version of the wave EQUATION has not yet been derived for propagation in general lossy dielectric media. In this paper, we first derive the general form of the PARABOLIC wave EQUATION for lossy dielectric media where it can be used for modeling the radio wave propagation through walls. The special form of the PARABOLIC EQUATION for modeling wave propagation in free space can be derived from this general form. We then apply PEM to model propagation of radio waves through a row of windows, reinforced concrete walls and typical corridors inside buildings. As windows are one of the most prevailing architectural elements in buildings, the phenomenon of plane wave transmission through them is of interest. In this paper PEM is used to model the radio wave propagation through windows. The numerical simulation results are presented for both normal and oblique incidence and compared with some reported results. The transmission and reflection characteristics of inhomogeneous walls have been studied by many numerical and analytical methods such as the finite-element method (FEM) and FDTD. In this paper, we use PEM to characterize reflection and transmission properties of reinforced concrete walls under plane wave incidence. The effect of several parameters namely wall thickness, bar diameter and spacing on the transmission coefficients of reinforced concrete walls will be analyzed. Corridors are also popular elements of buildings, so that the analysis of radio wave propagation in corridors has involved many researchers. The PEM is an effective method for modeling wave propagation in these environments. The effect of obstacles such as cupboards and cabinets inside a corridor can be modeled by PEM. This method is also able to model the effects of variations of the corridor direction on the wave propagation. The numerical simulation results will be presented and compared with the available data in the literature.

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