Numerical Solutions of One-Dimensional Parabolic Convection-Diﬀusion Problems Arising in Biology by the Laguerre Collocation Method

In this work, we present a numerical scheme for the approximate solutions of the one-dimensional parabolic convection-diffusion model problems. Difusion 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 dierential equations.[1][2]. 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 coecients can be computed. The accuracy and the efficiency of the method is showed by numerical examples and the comparisons by the other methods.[3]. References [1] B. Gurbuz, M. Sezer, Laguerre polynomial approach for solving Lane-Emden type functional dierential equations, Appl. Math. Comput. 242(2014) 255-264. [2] J. D. Murray, Mathematical Biology 1: An Introduction, Springer, Berlin, 2002. [3] Ş. Yuzbasi, N. Sahin, Numerical solutions of singularly perturbed one-dimensional parabolic convection-diffusion problems by the Bessel collocation method, Appl. Math. Comput. 174(2006) 910-920.


I. INTRODUCTION
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 [1][2][3].(a) Flow between imaginary compartments in a continuous one-dimensional system.(b) Discrete grid system used in two-dimensional transport models.(c) A close-up of five grid points showing the similarity to compartment models.

Compartment models are general framework
Citation: Burcu Gürbüz, Mehmet Sezer, Numerical solutions of one-dimensional parabolic convection-diffusion problems arising in biology by the Laguerre collocation method, Biomath 6 (2017), 1706047, http://dx.doi.org/10.11145/j.biomath.2017.06.047Fig. 2. A conceptual rate equation with respect to the convection-diffusion model that has many applications in biology, ecosystems and enzyme kinetics which can be mostly shown by forrester diagrams.The system is decomposed into flows of material as possibly large number of discrete compartments which are very useful.Conversely, it is also useful for the quantities nominally not flow, for instance, blood or water pressure in animal and plant physiological systems.Furthermore, complex interconnection networks can be addressed by these type of models with respect to link many of them together in many different complicated ways (Fig. 1).
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.There are many forces that could influence the flow of the matter, but the following simplified view uses two that will illustrate the qualitative model formulation.Convection moves the substance with a physical flow of water from point to point (i.e.river flow).Diffusion moves a substance in any direction according to the concentration of the substance around each point (Fig. 2) [4][5].
In this study, we consider the one-dimensional parabolic convection-diffusion problem with the initial conditions and the boundary conditions where f (x, t), A(x), B(x), g(x) and h(t) are functions defined in [0, l] × [0, T ]; l and T are appropriate constants.In this study, 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 where a r,s , r, s = 0, ..., N , are the unknown Laguerre coefficients and L n (x), n = 0, 1, 2, ..., N are the Laguerre polynomials defined by [6-8]

II. NUMERICAL METHOD
We first consider the series (4) for N = 2, as follows: Then we can generalize the approximate solution (6) for any truncated limit N and can write the obtained series in the matrix form Biomath 6 (2017), 1706047, http://dx.doi.org/10.11145/j.biomath.2017.06.047 where Also, we can put the matrix L(x) in the matrix form where X(x) and H are defined as Moreover, it is clearly seen that the relations between the matrix X(x) and its derivatives X (x) and X (x) are X (x) = X(x)B and X (x) = X(x)B 2 (9) where Then, by using the expressions ( 8) and ( 9) we easily find the matrix relations Now, by means of the relations ( 7)-( 11) we obtain the following matrix forms: By putting the expressions ( 8), ( 12), ( 13), ( 14) and (15) into Eq.( 1), we obtain the matrix equation Besides, by substituting the collocation points defined by into the Eq.( 16), we have the system of the matrix equations W(x i , t j )A = f (x i , t j ) or briefly the fundamental matrix equation By using the same procedure for the initial and boundary conditions we obtain the matrix relations for i, j = 0, 1, ..., N : To obtain the approximate solution of Eq. ( 1) under conditions (2) and (3), we form the augmented matrix Hence, the unknown Laguerre coefficients are computed by where [ W; F] is obtained by using the Gauss elimination method and then removing zero rows of augmented matrix [ W; F] [9][10][11].By substituting the determined coefficients into Eq.( 4), we have the Laguerre series solution

III. NUMERICAL RESULTS
Test case [11] with conditions

IV. CONCLUSION
We have presented and illustrated the Laguerre collocation method is based on computing the coefficients in the Laguerre expansion of solution of a one dimensional parabolic convection-diffusion model problems.A considerable advantage of the method is that the Laguerre polynomial coefficients of the solution are found very easily by using computer programs; Maple and Matlab.
Illustrative example is included to show the validity and applicability of the technique.Shorter computation time and lower operation count results in reduction of cumulative truncation errors and improvement of overall accuracy.
As a result, the method can also be extended to the system of reaction-diffusion-advection model problems with their residual error analysis, but some modifications are required.

Fig. 1 .
Fig. 1.(a) Flow between imaginary compartments in a continuous one-dimensional system.(b) Discrete grid system used in two-dimensional transport models.(c) A close-up of five grid points showing the similarity to compartment models.

TABLE I COMPARISON
OF THE ABSOLUTE ERRORS WITH TCM AND LCM FOR N = 25, 50 IN EXAMPLE 2.
Biomath 6 (2017), 1706047, http://dx.doi.org/10.11145/j.biomath.2017.06.047 with = 2.10 −4 and the exact solution of the problem is u(x, t) = e x+t .From Table1, it is seen that the errors from Laguerre Collocation Method (LCM) are in general less than Taylor Collocation Method (TCM).TableI.shows the comparison between absolute errors of LCM solutions and TCM solutions for different N values.