Numerical approach to existence and stability of stationary solutions to a SKT cross-diffusion equation

2018 ◽  
Vol 28 (11) ◽  
pp. 2191-2210 ◽  
Author(s):  
Tatsuki Mori ◽  
Takashi Suzuki ◽  
Shoji Yotsutani
Keyword(s):  
Diffusion Equation ◽  
Numerical Investigation ◽  
Numerical Approach ◽  
Stationary Solutions ◽  
Spatial Segregation ◽  
Population Pressure ◽  
Cross Diffusion ◽  
Habitat Area ◽  
Representation Of Solutions ◽  
Existence And Stability

The SKT cross-diffusion equation is proposed by N. Shigesada, K. Kawasaki and E. Teramoto in 1979 to investigate segregation phenomena of two competing species with each other in the same habitat area. The effect of cross-diffusion affects the population pressure between two different species. Lou and Ni derived limiting systems to see whether this effect may give rise to a spatial segregation or not, and to clarify its mechanism. In this paper, we introduce some new representation of solutions to a stationary limiting problem modified from representation by Lou, Ni and Yotsutani. We apply it to the numerical investigation of existence, non-existence, multiplicity and stability.

Nonlinear solution of the reaction–diffusion equation using a two-step third–fourth-derivative block method

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Oluwaseun Adeyeye ◽  
Ali Aldalbahi ◽  
Jawad Raza ◽  
Zurni Omar ◽  
Mostafizur Rahaman ◽  
...  
Keyword(s):  
Diffusion Equation ◽  
Reaction Diffusion ◽  
Numerical Approach ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Block Method ◽  
Wide Range ◽  
Higher Derivatives ◽  
Fourth Derivative ◽  
Improved Accuracy

AbstractThe processes of diffusion and reaction play essential roles in numerous system dynamics. Consequently, the solutions of reaction–diffusion equations have gained much attention because of not only their occurrence in many fields of science but also the existence of important properties and information in the solutions. However, despite the wide range of numerical methods explored for approximating solutions, the adoption of block methods is yet to be investigated. Hence, this article introduces a new two-step third–fourth-derivative block method as a numerical approach to solve the reaction–diffusion equation. In order to ensure improved accuracy, the method introduces the concept of nonlinearity in the solution of the linear model through the presence of higher derivatives. The method obtained accurate solutions for the model at varying values of the dimensionless diffusion parameter and saturation parameter. Furthermore, the solutions are also in good agreement with previous solutions by existing authors.

Existence and stability of static shells for the Vlasov–Poisson system

Analysis ◽  
2006 ◽  
Vol 26 (4) ◽  
Author(s):  
Achim Schulze
Keyword(s):  
Spherical Symmetry ◽  
Variational Approach ◽  
Stationary Solutions ◽  
Poisson System ◽  
Existence And Stability

We prove the existence and stability of stationary solutions to the Vlasov–Poisson System with spherical symmetry, which describe static shells, i.e., the support of their densities is bounded away from the origin. We use a variational approach which was established by Y. Guo and G. Rein.

scholarly journals Experimental and Numerical Investigations on the Behaviour of Masonry Walls Reinforced with an Innovative Sisal FRCM System

2017 ◽  
Vol 747 ◽  
pp. 190-195 ◽  
Author(s):  
Claudia Brito de Carvalho Bello ◽  
Antonella Cecchi ◽  
Emilio Meroi ◽  
Daniel V. Oliveira
Keyword(s):  
Numerical Investigation ◽  
Compression Test ◽  
Composite System ◽  
Numerical Approach ◽  
Masonry Walls ◽  
Test Results ◽  
Numerical Investigations ◽  
Cementitious Matrix ◽  
Diagonal Compression ◽  
Sisal Fibers

An experimental and numerical investigation on an innovative composite reinforced with sisal fibers for masonry strengthening is presented in this paper. A FEM numerical approach is also developed, based on diagonal compression test results, to simulate the shear in-plane response of unreinforced masonry panels (URM) and masonry strengthened with a Fibre Reinforced Cementitious Matrix (FRCM) composite system made with sisal fibers (RM-SISAL).

scholarly journals Existence and stability of static shells for the Vlasov–Poisson system with a fixed central point mass

2009 ◽  
Vol 146 (2) ◽  
pp. 489-511
Author(s):  
ACHIM SCHULZE
Keyword(s):  
Black Hole ◽  
Simple Model ◽  
Variational Approach ◽  
Existence Result ◽  
Stationary Solutions ◽  
Point Mass ◽  
Classical Solutions ◽  
Poisson System ◽  
Existence And Stability ◽  
Global Existence Result

AbstractWe consider the Vlasov–Poisson system with spherical symmetry and an exterior potential which is induced by a point mass in the center. This system can be used as a simple model for a newtonian galaxy surrounding a black hole. For this system, we establish a global existence result for classical solutions with shell-like initial data, i.e. the support of the density is bounded away from the point mass singularity. We also prove existence and stability of stationary solutions which describe static shells, where we use a variational approach which was established by Y. Guo and G. Rein.

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