Forward Parabolic Equations Research Articles

The study for maritime radar coverage based on cylindrical coordinate parabolic equationmethod

In order to predict the omnidirectional propagation characteristics of radar waves over the sea surface, a radio wave propagation model based on cylindrical coordinate parabolic equation is proposed and used to study the coverage range of marine radar. In this paper, the forward parabolic equation in cylindrical coordinate system is derived from Helmholtz Equation. By using the orthogonality of trigonometric functions in the general solution of the parabolic equation, the excitation coefficients of each wave propagation modes are solved, and the stepwise iterative algorithm of the cylindrical coordinate parabolic equation is realized by combining the Split Step Fourier method, to predict the propagation characteristics of the radio waves in space. Numerical examples show that the method proposed in this paper can overcome the azimuth limitation of the traditional three-dimensional cartesian coordinate parabolic equation method, which propagates about 30° along the axis, and realize the omnidirectional calculation. Compared with quasi-three dimensional parabolic equation, it has higher calculation accuracy. Based on the proposed method, the influences of atmospheric waveguide and rough sea surface on radar detection performance are studied, the simulation results have important significance for constructing digital battlefield at sea.

Continuum and lattice models analysis of the aggregation diffusion cell movement

The process by which one may take lattice models of a biophysical process and construct continuous models based on it is of mathematical interest as well as being of practical use. Such a process makes it possible to investigate how processes at the single-cell level affect collective dynamics. In this paper, we first study the singular limit of a class of reinforced random walks on a lattice which simulates cell migration on discrete points and a complete analysis of the existence and stability of solutions is possible. Combined with “the diffusion approximation” assumption, a continuous aggregation diffusion or backward forward parabolic equation is derived. Then the regularity estimate, the nonexistence of the solution with small initial data, and asymptotic behavior of the solution with large initial data are investigated. In contrast to the continuous model, the boundedness, asymptotic behaviors of the lattice model solution in the diffusion region with monotone initial date, and the interface behaviors of the aggregation, diffusion regions are explored. The numerical simulations agree well with theoretical results and biological observations.

Carleman estimates for stochastic parabolic equations with Neumann boundary conditions and applications

By a dual method, two Carleman estimates for forward and backward stochastic parabolic equations with Neumann boundary conditions are established. Then they are used to study a null controllability problem and a state observation problem for some stochastic forward parabolic equations with Neumann boundary conditions.

On a pseudoparabolic regularization of a forward–backward–forward equation

We consider an initial–boundary value problem for a degenerate pseudoparabolic regularization of a nonlinear forward–backward–forward parabolic equation, with a bounded nonlinearity which is increasing at infinity. We prove existence of suitably defined nonnegative solutions of the problem in a space of Radon measures. Solutions satisfy several monotonicity and regularization properties; in particular, their singular part is nonincreasing and may disappear in finite time. The problem is of intrinsic mathematical interest, but also arises naturally when studying, by time reversal, the spontaneous appearance of singularities in a specific application.

Traveling wave in backward and forward parabolic equations from population dynamics

This work is concerned with the properties of the traveling wave of the backward and forward parabolic equation \begin{equation*} u_t= [ D(u)u_x]_x + g(u),\quad t\geq 0, x\in \mathbb{R}, \end{equation*} where $D(u)$ changes its sign once, from negative to positive value, in the interval $u\in [0,1]$ and $g(u)$ is a mono-stable nonlinear reaction term. The existence of infinitely many traveling wave solutions is proven. These traveling waves are parameterized by their wave speed and monotonically connect the stationary states $u\equiv0$ and $u\equiv 1$.

Boundary and initial boundary-value problems for separable backward–forward parabolic problems

A backward–forward parabolic equation is considered in a strip which is infinite in the timelike direction, with boundary conditions on the sides of the strip. The unique bounded solution of the problem is given explicitly by separation of variables. In addition, a similar problem in a semi-infinite strip is treated, with initial data at the end of the strip. It is shown that the solution can be represented arbitrarily closely in the maximum norm by a sum of functions obtain by separation of variables.

Linear regularizing algorithms for positive solutions of linear inverse problems

Linear-regularization methods provide a simple technique for deter­mining stable approximate solutions of linear ill-posed problems such as Fredholm equations of the first kind, Cauchy problems for elliptic equations and backward solution of forward parabolic equations. In most of these problems the solution must be positive to satisfy physical plausibility. In this paper we consider ill-posed first-kind convolution equations and related problems such as numerical differentiation, Radon transform and Laplace-transform inversion. We investigate several linear regularization algorithms which provide positive approximate solutions for these problems at least in the absence of errors on the data. For noisy data the solution is not necessarily positive. Because the appearance of negative values can then only be an effect of the noise, the negative part of the solution should be negligible with a suitable choice of the regularization parameter. A price to pay for ensuring positivity is always, however, a reduction in resolution.