Numerical solutions of one-dimensional parabolic convection-diffusion problems arising in biology by the Laguerre collocation method

Burcu Gürbüz,Mehmet Sezer

doi:10.11145/j.biomath.2017.06.047

Abstract

In this work, we present a numerical scheme for the approximate solutions of the one-dimensional parabolic convection-diffusion model problems. Diffusion models form a reasonable basis for studying insect and animal dispersal and invasion, which arise from the question of persistence of endangered species, biodiversity, disease dynamics, multi-species competition so on. Convection diffusion problem is also a form of heat and mass transfer in biological models. The presented method is based on the Laguerre collocation method used for these problems of differential equations. In fact, the approximate solution of the problem in the truncated Laguerre series form is obtained by this method. By substituting truncated Laguerre series solution into the problem and by using the matrix operations and the collocation points, the suggested scheme reduces the problem to a linear algebraic equation system. By solving this equation system, the unknown Laguerre coefficients can be computed. The accuracy and the efficiency of the method is showed by numerical examples and the comparisons by the other methods.

Highlights

Diffusion models form a reasonable basis for studying insect and animal dispersal and Compartment models are general framework
We develop the Laguerre collocation method given in [9,10] and use to obtain the approximate solution of Eq (1) in the truncated Laguerre series form u(x, t) =
0 ≤ t ≤ 1, with = 2.10−4 and the exact solution of the problem is u(x, t) = . ex+t From Table 1, it is seen that the errors from Laguerre Collocation Method (LCM) are in general less than Taylor Collocation Method (TCM)

Summary

Introduction

Diffusion models form a reasonable basis for studying insect and animal dispersal and Compartment models are general frameworkBurcu Gurbuz, Mehmet Sezer, Numerical solutions of one-dimensional parabolic convection-diffusion ...On the other hand, in transport models, we have a physical quantity, such as energy i.e. heat or a quantity of matter, that flows from spatial point to point. We consider the one-dimensional parabolic convection-diffusion problem We develop the Laguerre collocation method given in [9,10] and use to obtain the approximate solution of Eq (1) in the truncated Laguerre series form u(x, t) =

Results

Conclusion