Resumen de Differentiation matrices and applications to lotka-volterra competing systems variable coefficients, the biharmonic equation and sherwood number
Frank Richard Prieto Medina
In this thesis we present a pseudospectral method in an interval, in the disk and in the circular annulus. Unlike the methods already known, the disk is not duplicated. Moreover, we solve the Laplace equation under non-homogeneous Dirichlet, Neumann and Robin boundary conditions, by only using the elements of the corresponding differentiation matrices. It is worth mentioningthat we do not use any quadrature, nor need to solve any decoupled system of ordinary differential equations, nor use any pole condition, nor require any lifting. We also solve several numerical examples to show the spectral convergence. The pseudospectral method developed in this thesis is applied to estimate Sherwood numbers integrating the mass flux to the disk and we also implement it to solve Lotka-Volterra systems and nonlinear dffusion problems involving chemical reactions.
Furthermore, we simulate numerically the classical positive solutions, large solutions and metasolutions of the degenerate diffusive logistic equation with spatial heterogeneities in an interval and in the disk through a collocation spectral method presented here. Some metasolutions of the different branches of metasolutions for the logistic equation introduced in the literature are simulated by the first time.
Moreover, we simulate numerically through a collocation spectral method jointly with pathfollowing techniques the classical and non-classical non-negative branches of solutions of Lotka-Volterra competing systems allowing the presence of spatial heterogeneities. Some of these solutions are simulated for the first time.
Also we prove the convergence of an innovative Chebyshev-Gauss-Lobatto( (CGL) pseudospectral method applied to fourth order boundary value problems. The proposed method enjoyed all the advantageous of the pseudospectral methods. Moreover, we can select (N - 3) interior CGL collocation points to enforce the equation, meanwhile we use the remaining four collocation points to assure the boundary conditions.
In addition, the numerical methods introduced in this thesis are extremely innovative because they can be used to solve non-radially symmetric problems. The models are of a huge interest in Spatial Ecology because they enable us to analyse the effects of the spatial heterogeneity on the evolution of the terrestrial ecosystems.
