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ORIGINAL ARTICLE

Numerical solutions of one-dimensional
parabolic convection-diffusion problems

arising in biology by the Laguerre collocation
method

Burcu Gürbüz, Mehmet Sezer
Department of Mathematics,

Manisa Celal Bayar University,
Manisa, Turkey

burcugrbz@gmail.com, mehmet.sezer@cbu.edu.tr

Received: 2 October 2016, accepted: 4 June 2017, published: 9 June 2017

Abstract—In this work, we present a numerical
scheme for the approximate solutions of the one-
dimensional parabolic convection-diffusion model
problems which arise in biological models. The pre-
sented method is based on the Laguerre collocation
method used for ordinary differential equations.
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 efficiency of the method is studied
by comparing with other numerical methods when
used to solve some numerical experiments.

Keywords-Convection-diffusion equation models,
Parabolic problem, Laguerre collocation method.

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
[1-3].

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.

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.2017.06.047

Page 1 of 5

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http://dx.doi.org/10.11145/j.biomath.2017.06.047


Burcu Gürbüz, Mehmet Sezer, Numerical solutions of one-dimensional parabolic convection-diffusion ...

Fig. 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 sys-
tems. Furthermore, complex interconnection net-
works 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

∂u

∂t
=
∂2u

∂x2
+ A(x)

∂u

∂x
+ B(x)u + f(x,t),

0 ≤ x ≤ l,0 ≤ t ≤ T,
(1)

with the initial conditions

u(x,0) = g(x), 0 ≤ x ≤ l < ∞, (2)

and the boundary conditions

u(0, t) = h(t), u(l, t) = K(t), 0 ≤ t ≤ T < ∞
(3)

where f(x,t),A(x),B(x),g(x) and h(t) are func-
tions defined in [0, l] × [0,T]; l and T are ap-
propriate 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

u(x,t) =

N∑
r=0

N∑
s=0

ar,sLr,s(x,t); (4)

Lr,s(x,t) = Lr(x)Ls(t)

where ar,s, r,s = 0, ...,N, are the unknown La-
guerre coefficients and Ln(x), n = 0,1,2, ...,N
are the Laguerre polynomials defined by [6-8]

Ln(x) =

n∑
k=0

(−1)k

k!

(
n

k

)
xk,n ∈ N, 0 ≤ x < ∞.

(5)

II. NUMERICAL METHOD

We first consider the series (4) for N = 2, as
follows:

u(x,t) =

2∑
r=0

2∑
s=0

ar,sLr(x)Ls(t)

= a00L0(x)L0(t) + a10L1(x)L0(t)

+a20L2(x)L0(t) + a01L0(x)L1(t) (6)

+a11L1(x)L1(t) + a21L2(x)L1(t)

+a02L0(x)L2(t) + a12L1(x)L2(t)

+a22L2(x)L2(t)

Then we can generalize the approximate solu-
tion (6) for any truncated limit N and can write
the obtained series in the matrix form

[u(x,t)] = L(x)L(t)A (7)

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Burcu Gürbüz, Mehmet Sezer, Numerical solutions of one-dimensional parabolic convection-diffusion ...

where

L(x) =
[
L0(x) L1(x) · · · LN(x)

]
,

L(t) =




L(t) 0 · · · 0
0 L(t) · · · 0
...

...
. . .

...
0 0 · · · L(t)




and

A = [a0,0 a0,1 · · · a0,N · · · aN,0 aN,1 · · · aN,N]
T

Also, we can put the matrix L(x) in the matrix
form

L(x) = X(x)H (8)

where X(x) and H are defined as

X(x) =
[
1 x1 · · · xN

]
and

H =




(−1)0
0!

(
0
0

)
0 · · · 0

(−1)0
0!

(
1
0

)
(−1)1

1!

(
1
1

)
· · · 0

.

.

.
.
.
.

. . .
.
.
.

(−1)0
0!

(
N
0

)
(−1)1

1!

(
N
1

)
· · · (−1)

N

N!

(
N
N

)




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)B2 (9)

where

B =




0 1 0 · · · 0
0 0 2 · · · 0
...

...
...

. . .
...

0 0 0 · · · N
0 0 0 · · · 0


 .

Then, by using the expressions (8) and (9) we
easily find the matrix relations

L′(x) = X(x)BH and L′′(x) = X(x)B2H (10)

L(t) = X(t)H and L
′
(t) = X(t)BH (11)

Now, by means of the relations (7)-(11) we obtain
the following matrix forms:

[u(x,t)] = L(x)L(t)A = X(x)HX(t)HA (12)

[ux(x,t)] = L
′(x)L(t)A

= X(x)BHX(t)HA (13)

[uxx(x,t)] = L
′′(x)L(t)A

= X(x)B2HX(t)HA (14)

[ut(x,t)] = L(x)L(t)A

= X(x)HX(t)BHA (15)

By putting the expressions (8), (12), (13), (14) and
(15) into Eq. (1), we obtain the matrix equation

{X(x)HX(t)B−X(x)B2HX(t)
−A(x)X(x)BHX(t) (16)
−B(x)X(x)HX(t)}HA = f(x,t)

or briefly,

W(x,t)A = f(x,t)

Besides, by substituting the collocation points de-
fined by

xi =
l

N
i, tj =

T

N
j, i,j = 0,1,2, ...,N,

into the Eq.(16), we have the system of the matrix
equations W(xi, tj)A = f(xi, tj) or briefly the
fundamental matrix equation

WA = F =⇒ [W; F]

By using the same procedure for the initial and
boundary conditions we obtain the matrix relations
for
i,j = 0,1, ...,N:

u(xi,0) = X(xi)HX(0)HA = g(xi) = λi

u(0, tj) = X(0)HX(tj)HA = h(tj) = µj

u(y,tj) = X(y)HX(tj)HA = K(tj) = γj

or briefly,

UA = [λ]; [U;λ],VA = [µ]; [V;µ],ZA = [γ]; [Z;γ].

To obtain the approximate solution of Eq. (1) un-
der conditions (2) and (3), we form the augmented
matrix

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Burcu Gürbüz, Mehmet Sezer, Numerical solutions of one-dimensional parabolic convection-diffusion ...

TABLE I
COMPARISON OF THE ABSOLUTE ERRORS WITH TCM AND LCM FOR N = 25, 50 IN EXAMPLE 2.

TCM LCM TCM LCM
x E25 E25 E50 E50

0.0 0.69500E-17 7.000000E-13 0.10000E-18 0.0000000000
0.1 0.15886E-03 3.681538E-06 0.15886E-03 1.040134E-07
0.2 0.63428E-03 2.207214E-05 0.63428E-03 4.157153E-08
0.3 0.14208E-02 4.966673E-05 0.14208E-02 9.332459E-08
0.4 0.25078E-02 5.150761E-05 0.25078E-02 1.652982E-05
0.5 0.38799E-02 2.543790E-11 0.38799E-02 2.569605E-15
0.6 0.55168E-02 1.324508E-04 0.55168E-02 3.676180E-06
0.7 0.73934E-02 3.734395E-04 0.73934E-02 4.964259E-06
0.8 0.94802E-02 7.505606E-04 0.94802E-02 6.423988E-06
0.9 0.11744E-01 1.291408E-03 0.11744E-01 8.044241E-07
1.0 0.14146E-01 2.023575E-03 0.14146E-01 9.812752E-07

[W̃; F̃] =




W; F
U;λ
V;µ
Z;γ




Hence, the unknown Laguerre coefficients are
computed by

A = (
˜̃
W)

−1 ˜̃
F

where [ ˜̃W; ˜̃F] is obtained by using the Gauss
elimination method and then removing zero rows
of augmented matrix [W̃; F̃] [9-11]. By substitut-
ing the determined coefficients into Eq. (4), we
have the Laguerre series solution

uN(x,t) =

N∑
r=0

N∑
s=0

ar,sLr,s(x,t),

Lr,s(x,t) = Lr(x)Ls(t).

III. NUMERICAL RESULTS

Test case[11]

∂u

∂t
=

∂2u

∂x2
+ (2x + 1)

∂u

∂x
+ x2u +

ex+t

�
,

0 ≤ x ≤ 1,0 ≤ t ≤ 1, (17)
with conditions

u(x,0) =
ex

�
, 0 ≤ x ≤ 1

u(0, t) =
et

�
, u(y,t) =

e1+t

�
0 ≤ t ≤ 1,

with � = 2.10−4 and the exact solution of the
problem is u(x,t) = e

x+t

�
. From Table 1, it is seen

that the errors from Laguerre Collocation Method
(LCM) are in general less than Taylor Collocation
Method (TCM).

Table I. shows the comparison between absolute
errors of LCM solutions and TCM solutions for
different N values.

IV. CONCLUSION
We have presented and illustrated the Laguerre

collocation method is based on computing the co-
efficients 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 coeffi-
cients 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 re-
sults 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.

ACKNOWLEDGMENT
This work was financially supported by Society

for Mathematical Biology for during the Bio-
Math 2016, The annual International Conference

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Burcu Gürbüz, Mehmet Sezer, Numerical solutions of one-dimensional parabolic convection-diffusion ...

on Mathematical Methods and Models in Bio-
sciences and it is performed within the ”Numerical
Solutions of Partial Functional Integro Differential
Equations with respect to Laguerre Polynomials
and Its Applications” project, Manisa Celal Ba-
yar University Department of Scientific Research
Projects, with grant ref. 2014-151.

REFERENCES

[1] J. D. Murray, Mathematical Biology 1: An Introduction,
Berlin: Springer, 2002.

[2] B. Zduniak, Numerical analysis of the coupled
modified Van Der Pol equations in a model of
the heart action, Biomath Commun. 3(2014) 1–7.
doi:http://dx.doi.org/10.11145/j.biomath.2013.12.281

[3] A. Prieto-Langarica, H. V. Kojouharov, L.
Tang, Constructing one-dimensional continuous
models from two-dimensional discrete models of
medical implants, Biomath Commun. 1(2012) 1–6.
doi:http://dx.doi.org/10.11145/j.biomath.2012.09.041

[4] J. W. Haefner, Modelling Biological Systems, USA:
Springer, 2005.

[5] B. Hannon, M. Ruth, Modelling Dynamic Biological
Systems, London: Springer, 2014.

[6] G. A. Andrews, R. Askey, R. Roy, Special Functions,
Cambridge: The University Press, 2000.

[7] J. Dieudonne, Orthogonal Polynomials and Applications,
Berlin: Springer, 1985.

[8] J. L. Gracia, E. O’Riordan, Numerical approximation
of solution derivatives of singularly perturbed
parabolic problems of convection-diffusion
type, Math. Comput. 85(2016) 581–599.
doi:http://dx.doi.org/10.1090/mcom/2998

[9] B. Gürbüz, M. Sezer, Laguerre polynomial approach
for solving Lane-Emden type functional differential
equations, Appl. Math. Comput. 242(2014) 255–264.
doi:http://dx.doi.org/10.1016/j.amc.2014.05.058

[10] B. Gürbüz, M. Sezer, Laguerre polynomial solutions of
a class of initial and boundary value problems arising
in science and engineering fields, Acta Phys. Pol. A
130(2016) 194–197. doi: 10.12693/APhysPolA.129.194

[11] Ş. Yüzbaşı, N. Şahin, Numerical solutions of
singularly perturbed one-dimensional parabolic
convection-diffusion problems by the Bessel collocation
method, Appl. Math. Comput. 174(2006) 910–920.
doi:http://dx.doi.org/10.1016/j.amc.2013.06.027

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	Introduction
	Numerical Method
	Numerical Results
	Conclusion
	References