Solutions Of Nonlinear Reaction-diffusion Equations Research Articles

Hyperuniform scalar random fields for lensless, multispectral imaging systems.

We propose a novel framework for the systematic design of lensless imaging systems based on the hyperuniform random field solutions of nonlinear reaction-diffusion equations from pattern formation theory. Specifically, we introduce a new class of imaging point-spread functions (PSFs) with enhanced isotropic behavior and controllable sparsity. We investigate PSFs and modulated transfer functions for a number of nonlinear models and demonstrate that two-phase isotropic random fields with hyperuniform disorder are ideally suited to construct imaging PSFs with improved performances compared to PSFs based on Perlin noise. Additionally, we introduce a phase retrieval algorithm based on non-paraxial Rayleigh-Sommerfeld diffraction theory and introduce diffractive phase plates with PSFs designed from hyperuniform random fields, called hyperuniform phase plates (HPPs). Finally, using high-fidelity object reconstruction, we demonstrate improved image quality using engineered HPPs across the visible range. The proposed framework is suitable for high-performance lensless imaging systems for on-chip microscopy and spectroscopy applications.

Nonclassical Symmetries of a System of Nonlinear Reaction-Diffusion Equations

We study the conditional symmetry of a system of nonlinear reaction-diffusion equations and establish the existence of operators of conditional symmetry for systems of nonlinear reaction-diffusion equations with any number of independent variables and find these operators in the explicit form. The exact solutions of nonlinear reaction-diffusion equations with exponential nonlinearity are constructed.

Self-similarity and long-time behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source

This paper deals with the long-time behavior of solutions of nonlinear reaction-diffusion equations describing formation of morphogen gradients, the concentration fields of molecules acting as spatial regulators of cell differentiation in developing tissues. For the considered class of models, we establish existence of a new type of ultra-singular self-similar solutions. These solutions arise as limits of the solutions of the initial value problem with zero initial data and infinitely strong source at the boundary. We prove existence and uniqueness of such solutions in the suitable weighted energy spaces. Moreover, we prove that the obtained self-similar solutions are the long-time limits of the solutions of the initial value problem with zero initial data and a time-independent boundary source.

Blow-up solution of nonlinear reaction-diffusion equations under boundary feedback

In this paper, we study a class of nonlinear reaction-diffusion equations with Neumann boundary control. Some sufficient conditions under which the system admits a unique global solution are given. A time-dependent gain nonlinear proportional boundary feedback control is designed to make the closed-loop system blow-up in a finite time. Finally, some concrete examples are presented to illustrate the applications.

Approximate analytical solutions of fractional reaction-diffusion equations

The homotopy analysis method (HAM) of S.J. Liao has proven useful in obtaining analytical/numerical solutions to various nonlinear differential equations. In this work, the HAM is employed to obtain the analytical/numerical solutions of the nonlinear reaction-diffusion equations with time-fractional derivatives. The fractional derivatives are described in the Caputo sense. This approach transforms the solution of nonlinear reaction-diffusion equations into the solution of a hierarchy of linear equations. The solution is simple yet highly accurate and compare favorably with the solutions obtained early in the literature.

Invasion and adaptive evolution for individual-based spatially structured populations

The interplay between space and evolution is an important issue in population dynamics, that is particularly crucial in the emergence of polymorphism and spatial patterns. Recently, biological studies suggest that invasion and evolution are closely related. Here, we model the interplay between space and evolution starting with an individual-based approach and show the important role of parameter scalings on clustering and invasion. We consider a stochastic discrete model with birth, death, competition, mutation and spatial diffusion, where all the parameters may depend both on the position and on the phenotypic trait of individuals. The spatial motion is driven by a reflected diffusion in a bounded domain. The interaction is modelled as a trait competition between individuals within a given spatial interaction range. First, we give an algorithmic construction of the process. Next, we obtain large population approximations, as weak solutions of nonlinear reaction-diffusion equations. As the spatial interaction range is fixed, the nonlinearity is nonlocal. Then, we make the interaction range decrease to zero and prove the convergence to spatially localized nonlinear reaction-diffusion equations. Finally, a discussion of three concrete examples is proposed, based on simulations of the microscopic individual-based model. These examples illustrate the strong effects of the spatial interaction range on the emergence of spatial and phenotypic diversity (clustering and polymorphism) and on the interplay between invasion and evolution. The simulations focus on the qualitative differences between local and nonlocal interactions.

Local stability of traveling-wave solutions of nonlinear reaction-diffusion equations

We investigate the local stability of traveling-wave solutions of the nonlinear reaction-diffusion equations in various weighted Banach spaces. New methods are used in analyzing the location of the spectrum. The result covers the stability results of the traveling-wave solutions of reaction-diffusion equations including the well known Fisher-KPP-type nonlinearity.

Perturbative linearization of reaction diffusion equations

We develop perturbative expansions to obtain solutions for the initial-value problems of two important reaction-diffusion systems, viz., the Fisher equation and the time-dependent Ginzburg-Landau (TDGL) equation. The starting point of our expansion is the corresponding singular-perturbation solution. This approach transforms the solution of nonlinear reaction-diffusion equations into the solution of a hierarchy of linear equations. Our numerical results demonstrate that this hierarchy rapidly converges to the exact solution.

Existence of coupled quasi-solutions of systems of nonlinear reaction-diffusion equations

Systems of nonlinear parabolic initial boundary value problems arise in many applications, such as epidemies, ecology, biochemistry, biology, chemical and nuclear engineering. Constructive methods of proving existence results for such problems, which can also provide numerical procedures for the computation of solutions, are of greater value than theoretical existence results. The method of upper and lower solutions coupled with monotone iterative technique has been employed successfully to prove existence of multiple solutions of nonlinear reaction-diffusion equations, in special cases, by various authors [335, 10, 11, 15, 181. Recently, in [6, 171 weakly coupled systems of reaction-diffusion equations, when the nonlinear terms are independent of gradient terms, are discussed and some special type of results are obtained. We, in this paper, investigate general systems of nonlinear reaction-diffusion problems when the nonlinear terms possess a mixed quasi-monotone property. We discuss a very general situation and obtain coupled extremal quasi-solutions, which in special cases reduce to minimal and maximal solutions. We shall also indicate how one-step cyclic monotone iterative schemes can be generated which yield accelerated rate of convergence of iterates. This work is in the spirit of our recent paper [12] for elliptic systems.

Integral bounds for solutions of nonlinear reaction-diffusion equations

Differential inequality techniques are used to obtain upper bounds for theL1 norm of solutions of nonlinear reaction-diffusion equations. Essential use is made of Sobolev type integral inequalities. An extension to third order pseudo-parabolic equations is included.