The mechanism of pattern formations has been widely studied and for different types of Reaction-Diffusion equations. This phenomenon has a wide range of applications in the fields of biology, chemistry, engineering, etc. In this paper, we have studied the pattern formation for a Reaction–Diffusion model with nonlinear reaction terms; this equation is different from RDM which has been studied before, and which derived from the interaction between Turing stationery and wave instability. Next, we study the possible traveling wave solution for our RDM and their stability close to the steady states. We discretize the system of Reaction-diffusion equations in one dimension using Semi-Implicit second-order difference method and we investigate the different types of travelling wave solutions (TWS). A finite element package namely COMSOL Multiphysics is used to show some types of pattern formations and for two types of initial conditions. The initial conditions are chosen to investigate the type of spots that can be formulated from the interaction. In parallel, we have proved theoretically the regions where those pattern formations can be found depending on the value of the diffusion coefficients and wave number

In this paper, the optical properties of a pinhole nanorod are studied using finite element package (COMSOL Multiphysics). Both electric and magnetic field distribution for different radius of a hollow nanorod and variable pinhole position has been measured. Furthermore, the scattering cross-section for both fields is calculated. The magnetic field enhancement has been increased by adding a pinhole made of air. The magnetic field enhancement improved 20 percent. In case of scattering cross-section, the response is changed as a function of the pinhole position and diameter. The results provide the ability to tune optical properties through an appropriate geometric feature of the nanorods.

Pattern formations as mathematical models are grows significantly and the development in subject analysis and the type of mathematical tools offer a wide range in the research aspects. The paper shows results and analysis for a novel reaction-diffusion model, which has unstable features, and satisfy the Turing instability conditions when the diffusion coefficient becomes large enough. Two methods are used to analysis this model, namely semi-implicit finite different method and the analysis Finite element method with utilize of COMSOL Multiphysics software. The conditions of diffusion-driven instability are shown and the effect of diffusion coefficient in changing the state of the model to be unstable is explained. Travelling wave solutions for this model in one dimension are founded and compared using the mentioned two methods. Finally, pattern formations for this model are shown in two dimensions and for different values of diffusion coefficients.

The applications of pattern formation in nature attract a huge number of researchers and thus increase the production of researches in this field. In this paper, we introduce a new model of the reaction-diffusion system which satisfies Turing conditions and formulates complicate solutions such as pattern formation. We used for finding the numerical results and forming the patterns software COMSOL Multiphysics finite element package. We have discussed the condition of diffusion-driven instability theoretically and showed the region where these conditions can be satisfied. It was shown that the key fact for instability and the existence of pattern formation is the diffusion coefficient d. When d is large enough we can construct pattern formation with variants rings. The number of rings increases as the domain we use for study increases. Finally, we compared our results to real patterns in nature and we show how they matched together.

In this paper, perturbation and finite difference methods are used to solve a reaction diffusion system. This system is modeled for describing the interaction between species in ecology. The interaction is interpreted as traveling wave solutions for both species under three types of initial conditions which describe some ecological cases. Types of traveling wave solutions are found and studied using numerical and approximate methods when there exists a small parameter appear in one of the equation. The solutions of the two methods are compared and show a good agreement