Exact Solutions of a Mathematical Model for Fluid Transport in Peritoneal Dialysis
A mathematical model for fluid transport in peritoneal dialysis is constructed. The model is based on a nonlinear system of two-dimensional partial differential equations with corresponding boundary and initial conditions. Using the classical Lie scheme, we establish that the base system of partial...
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irk-123456789-1658202020-02-17T01:27:22Z Exact Solutions of a Mathematical Model for Fluid Transport in Peritoneal Dialysis Cherniga, R.M. Waniewski, J. Статті A mathematical model for fluid transport in peritoneal dialysis is constructed. The model is based on a nonlinear system of two-dimensional partial differential equations with corresponding boundary and initial conditions. Using the classical Lie scheme, we establish that the base system of partial differential equations (under some restrictions on coefficients) is invariant under the infinite-dimensional Lie algebra, which enables us to construct families of exact solutions. Moreover, exact solutions with a more general structure are found using another (non-Lie) technique. Finally, it is shown that some of the solutions obtained describe the hydrostatic pressure and the glucose concentration in peritoneal dialysis. Побудовано математичну модель переносу рідини при очеревинному діалізі, яка базується на нелінійній системі двовимірних диференціальних рівнянь з частинними похідними (ДРЧП) з відповідними крайовими та початковими умовами. Шляхом застосування класичного методу Лі встановлено, що базова система ДРЧП (при певних обмеженнях на коефіцієнти) інваріантна відносно нескінченновимірної алгебри Лі, що дозволило побудувати сім'ї точних розв'язків. Крім того, точні розв'язки більш загальної структури знайдено за допомогою іншого неліївського методу. Також встановлено, що деякі з отриманих розв'язків описують гідростатичний тиск та концентрацію глюкози при очеревинному діалізі. 2005 Article Exact Solutions of a Mathematical Model for Fluid Transport in Peritoneal Dialysis / R.M. Cherniga, J. Waniewski // Український математичний журнал. — 2005. — Т. 57, № 8. — С. 1112–1119. — Бібліогр.: 10 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165820 517.957:58 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Cherniga, R.M. Waniewski, J. Exact Solutions of a Mathematical Model for Fluid Transport in Peritoneal Dialysis Український математичний журнал |
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A mathematical model for fluid transport in peritoneal dialysis is constructed. The model is based on a nonlinear system of two-dimensional partial differential equations with corresponding boundary and initial conditions. Using the classical Lie scheme, we establish that the base system of partial differential equations (under some restrictions on coefficients) is invariant under the infinite-dimensional Lie algebra, which enables us to construct families of exact solutions. Moreover, exact solutions with a more general structure are found using another (non-Lie) technique. Finally, it is shown that some of the solutions obtained describe the hydrostatic pressure and the glucose concentration in peritoneal dialysis. |
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Cherniga, R.M. Waniewski, J. |
| author_facet |
Cherniga, R.M. Waniewski, J. |
| author_sort |
Cherniga, R.M. |
| title |
Exact Solutions of a Mathematical Model for Fluid Transport in Peritoneal Dialysis |
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Exact Solutions of a Mathematical Model for Fluid Transport in Peritoneal Dialysis |
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Exact Solutions of a Mathematical Model for Fluid Transport in Peritoneal Dialysis |
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Exact Solutions of a Mathematical Model for Fluid Transport in Peritoneal Dialysis |
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Exact Solutions of a Mathematical Model for Fluid Transport in Peritoneal Dialysis |
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exact solutions of a mathematical model for fluid transport in peritoneal dialysis |
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Інститут математики НАН України |
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2005 |
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| citation_txt |
Exact Solutions of a Mathematical Model for Fluid Transport in Peritoneal Dialysis / R.M. Cherniga, J. Waniewski // Український математичний журнал. — 2005. — Т. 57, № 8. — С. 1112–1119. — Бібліогр.: 10 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT chernigarm exactsolutionsofamathematicalmodelforfluidtransportinperitonealdialysis AT waniewskij exactsolutionsofamathematicalmodelforfluidtransportinperitonealdialysis |
| first_indexed |
2025-07-14T20:04:32Z |
| last_indexed |
2025-07-14T20:04:32Z |
| _version_ |
1837654052819697664 |
| fulltext |
UDC 517.957:58
R. Cherniha (Inst. Math., Nat. Acad. Sci. Ukraine, Kyiv),
J. Waniewski (Inst. Biocyber. and Biomed. Eng. Pol. Acad. Sci., Warsaw, Poland)
EXACT SOLUTIONS OF A MATHEMATICAL MODEL
FOR FLUID TRANSPORT IN PERITONEAL DIALYSIS∗
TOÇNI ROZV’QZKY ODNI{} MATEMATYÇNO} MODELI
PERENOSU RIDYNY PRY OÇEREVYNNOMU DIALIZI
A mathematical model for fluid transport in peritoneal dialysis is constructed. The model is based on a nonlinear
system of two-dimensional partial differential equations (PDE) with the relevant boundary and initial conditions.
Using the classical Lie scheme, we have established that the based PDE system (under some restrictions on
coefficients) is invariant under the infinite-dimensional Lie algebra, therefore families of exact solutions were
found. Moreover, exact solutions with a more general structure were found using another (non-Lie) technique.
Finally, it was shown that some of the solutions obtained describe the hydrostatic pressure and the glucose
concentration in peritoneal dialysis.
Pobudovano matematyçnu model\ perenosu ridyny pry oçerevynnomu dializi, qka bazu[t\sq na nelinijnij
systemi dvovymirnyx dyferencial\nyx rivnqn\ z çastynnymy poxidnymy (DRÇP) z vidpovidnymy kra-
jovymy ta poçatkovymy umovamy. Ílqxom zastosuvannq klasyçnoho metodu Li vstanovleno, wo bazova
systema DRÇP (pry pevnyx obmeΩennqx na koefici[nty) invariantna vidnosno neskinçennovymirno]
alhebry Li, wo dozvolylo pobuduvaty sim’] toçnyx rozv’qzkiv. Krim toho, toçni rozv’qzky bil\ß za-
hal\no] struktury znajdeno za dopomohog inßoho neli]vs\koho metodu. TakoΩ vstanovleno, wo deqki
z otrymanyx rozv’qzkiv opysugt\ hidrostatyçnyj tysk ta koncentracig hlgkozy pry oçerevynnomu
dializi.
1. Introduction. Peritoneal dialysis is a life saving treatment for chronic patients with
end stage renal disease [1]. Dialysis fluid is infused into the peritoneal cavity, and, during
its dwell there, small metabolites (urea, creatinine) and other uremic toxins diffuse from
blood to the fluid, and after some time (usually a few hours) are removed together with
the drained fluid. The treatment is repeated continuously. The peritoneal transport occurs
between dialysis fluid in the peritoneal cavity and blood passing down capillaries in tis-
sue surrounding the peritoneal cavity. The capillaries are distributed within the tissue at
different distance from the tissue surface that is in contact with dialysis fluid. The solutes,
which are transported between blood and dialysis fluid, have to cross two transport bar-
riers: the capillary wall and a tissue layer. Typically, many solutes are transported from
blood to dialysate, but some solutes that are present in high concentration in dialysis fluid
are transported to blood. This kind of transport system happens also in other medical
treatments, as local delivery of anticancer medications, and some experimental or natu-
ral physiological phenomena. Mathematical description of these systems was obtained
using partial differential equations based on the simplification that capillaries are homo-
geneously distributed within the tissue [2, 3]. Experimental evidence confirmed the good
applicability of such models [2, 3].
Another objective of peritoneal dialysis is to remove excess water from the patient
[1]. This is gained by inducing high osmotic pressure in dialysis fluid by adding a solute
in high concentration. The most often used solute is glucose. This medical application of
high osmotic pressure is rather unique for peritoneal dialysis. Mathematical description of
osmotically induced fluid transport from blood to dialysis fluid has not been formulated
∗ This work was done within the joint project “Application of modern mathematical methods for exact
solving some nonlinear patterns arrising in biology” between Polish Academy of Sciences and National
Academy of Sciences of Ukraine.
c© R. CHERNIHA, J. WANIEWSKI, 2005
1112 ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
EXACT SOLUTIONS OF A MATHEMATICAL MODEL FOR FLUID TRANSPORT . . . 1113
fully yet, in spite of the well known basic physical law for such transport. A previous
attempt did not result in a satisfactory description, and was falsified later on [2].
The paper is organized in the following way. In Section 2, a mathematical model of
fluid transport in peritoneal dialysis is constructed. In Section 3, the classical Lie method
(see, e.g., [4, 5]) is applied for finding exact solutions of a simplification of the model con-
structed. Moreover, exact solutions were found using a non-Lie ansatz. Finally, in Sec-
tion 4, the exact solutions are compared with numerical solutions, which were found using
a numerical technique [6] based on the finite elements method and Galerkin’s method. It
was shown that some of these solutions describe the hydrostatic pressure and the glucose
concentration in peritoneal dialysis.
2. Mathematical model. The mathematical description of transport processes within
the tissue consists in local balance of fluid volume and solute mass. For incompressible
fluid, the change of volume may occur due to elasticity of the tissue. The fractional void
volume, i.e., the volume occupied by the fluid in the interstitium (the rest of the tissue be-
ing cells and macromolecules forming interstitium) expressed per one unit volume of the
whole tissue is denoted ν , and its time evolution is described by the following equation:
∂ν
∂t
= −∂jV
∂x
+ qV (1)
where jV is the volumetric fluid flux across the tissue, qV is the density of volumetric
fluid flux from blood to the tissue, t is time, and x is the distance from the tissue surface
in contact with dialysis fluid (flat geometry of the tissue is here assumed). The solute
(glucose) is distributed only within the interstitial fluid, and its concentration in this fluid
is denoted by CG . The equation that describes the local changes of solute amount, νCG ,
is as follows:
∂(νCG)
∂t
= −∂jG
∂x
+ qG, (2)
where jG is glucose flux through the tissue, and qG is the density of glucose flux from
blood. The flows of fluid and solute are described according to linear nonequilibrium
thermodynamics. Osmotic pressure of glucose is described by van’t Hoff law, i.e., it is
proportional to glucose concentration. The volumetric flux across the tissue is generated
by hydrostatic and osmotic pressure gradients:
jV = −νK
∂P
∂x
+ σTGνKRT
∂CG
∂x
, (3)
whereas for the density of fluid flux from blood to tissue we assume that it is generated
by osmotic pressure difference between blood and tissue:
qV = −LpaσCGRT (CGB − CG). (4)
The solute flux across the tissue is composed of diffusive component (proportional to
glucose concentration gradient) and convective component (proportional to glucose con-
centration and volumetric flux):
jG = −νDG
∂CG
∂x
+ STGCGjV . (5)
Similarly, the density of glucose flux from blood to tissue has diffusive component (pro-
portional to the difference of glucose concentration in blood, CGB , and glucose concen-
tration in tissue, CG ) and convective component (proportional to the density of volumet-
ric flow from blood to tissue, qV ):
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
1114 R. CHERNIHA, J. WANIEWSKI
qG = pGa(CGB − CG) + SCGqV ((1 − FG)CGB + FGCG) . (6)
The coefficients in the above equations are: K — hydraulic permeability of tissue,
σTG — the Staverman reflection coefficient for glucose in tissue, R — gas constant,
T — temperature, Lp — hydraulic permeability of the capillary wall, a — density of
capillary surface area, σCG — the Staverman reflection coefficient for glucose in the
capillary wall, DG — diffusivity of glucose in tissue, STG = 1 − σTG — sieving co-
efficient of glucose in tissue, pG — diffusive permeability of the capillary wall, SCG =
= 1− σCG — sieving coefficient for glucose in the capillary wall, and FG — weighing
factor.
Equations (1), (2) together with equations (3) – (6) for flows form a system of two
nonlinear partial differential equations with three variables: ν, P , and CG . Therefore,
an additional, constitutive, equation is necessary, and this is the equation describing how
fractional fluid volume, ν , depends on interstitial pressure, P :
ν(P ) = νmin +
νmax − νmin
1 +
(
νmax − νmin
ν0 − νmin
− 1
)
exp(−b(P − P0))
, (7)
where νmin, νmax, ν0, and b are empirically measured constants. Boundary conditions
for a tissue layer of width L impermeable at x = L and in contact with dialysis fluid at
x = 0 are as follows:
x = 0 : P = PD, CG = CGD,
x = L :
∂P
∂x
= 0,
∂CG
∂x
= 0.
(8)
Initial conditions describe equilibrium within the tissue without any contact with dialysis
fluid at x = 0:
t = 0 : P = P0, CG = CGB . (9)
It is easily seen that equations (1) – (7) can be united into two equations for finding the
glucose concentration CG ≡ U(t, x) and the hydrostatic pressure P (t, x), namely:
∂ν
∂t
= K
∂
∂x
(
ν
∂P
∂x
)
− σ1
∂
∂x
(
ν
∂U
∂x
)
+ h1U − h0,
(10)
∂(νU)
∂t
= D
∂
∂x
(
ν
∂U
∂x
)
+ SK
∂
∂x
(
νU
∂P
∂x
)
−
−Sσ1
∂
∂x
(
νU
∂U
∂x
)
+ b2U
2 − b1U + b0,
where
σ1 = σTGKRT, D = DG, S = STG,
h0 = CGBh1, h1 = LpaRTσCG,
b0 = PGaCGB − SCG(1 − FG)h0CGB ,
b1 = PGa− (1 − 2FG)SGh0, b2 = SCGFGh1.
(11)
Thus, we obtain the boundary-value problem (7) – (10) to find the functions ν(t, x),
P (t, x) and U(t, x).
Note that possible values of the parameters arising in this problem can be established
from experimental data published in [2].
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
EXACT SOLUTIONS OF A MATHEMATICAL MODEL FOR FLUID TRANSPORT . . . 1115
3. Exact solutions of system (10). In this section we restrict ourselves to the con-
sideration of the nonlinear balance equations (10) together with an approximation of the
switch type relation (7) in the form of the linear piecewise continuous function
ν(P ) =
νmin, P < Pmin,
νmin + b(P − Pmin), Pmin ≤ P ≤ Pmax,
νmax, P > Pmax.
(12)
Substituting (12) into the balance equations (10), we arrive at the following equations:
0 = KPxx − σ1Uxx + h∗
1U − h∗
0,
Ut = DUxx + KS(UPx)x − Sσ1(UUx)x + b∗2U
2 − b∗1U + b∗0
(13)
in the cases ν = νmin or ν = νmax (here h∗
i = hi/νmin or h∗
i = hi/νmax, i = 0, 1 ;
b∗j = bj/νmin or bj/νmax, j = 0, 1, 2) and
bPt = K(νmin − bPmin)Pxx + Kb(PPx)x − h0 + h1U,
(14)
(νmin − bPmin)Ut + b(UP )t = D(νmin − bPmin)Uxx + Db(UxP )x+
+KS(νmin − bPmin)(UPx)x + KSb(UPPx)x−
−Sσ1(νmin − bPmin)(UUx)x − Sσ1b(PUUx)x + b2U
2 − b1U + b0
in the case ν(P ) = νmin + b(P − Pmin).
Let us consider the nonlinear system of equations (13). It can be noted that the nonlo-
cal substitution
V = −KνmPx, (15)
where νm = νmin or νm = νmax (see formula (12)), reduces this system to the form
Vx = h1U − σ1νmUxx − h0,
νmUt = DνmUxx − Sσ1νmU2
x − SV Ux − b20U
2 − b10U + b0,
(16)
where b20 = Sh1 − b2, b10 = b1 − Sh0 .
Theorem 1. The maximal algebra of invariance (MAI) of the nonlinear system (16)
with arbitrary non-zero coefficients is the infinite-dimensional Lie algebra generated by
the operators
Pt =
∂
∂t
, G∞ = f(t)
∂
∂x
+
νm
S
ft
∂
∂V
(17)
where f(t) is an arbitrary smooth function and ft ≡
df
dt
.
The proof of the theorem is based on the classical Lie scheme (see [4, 5] ). Here the
relevant calculations are omitted because of their awkwardness.
It should be noted that the operator G∞ for f(t) = t takes the form
G = t
∂
∂x
+
νm
S
∂
∂V
. (18)
Such a type of invariance operators is known as the Galilei operator because it produces
Galilei transformations of the form
t′ = t, x′ = x + εt,
V ′ = V +
νm
S
ε, U ′ = U,
(19)
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
1116 R. CHERNIHA, J. WANIEWSKI
where ε is a group parameter. Now one can note that the first two formulas in (19)
produce the classical Galilei transformations (see, e.g., [7, 8]).
It seems reasonable to construct exact solutions of system (16) using its Lie symmetry
operators (17). According to the general procedure (see [4, 5]) it is necessary to construct
the general solution of the linear equation
X(Φ(t, x, U, V )) = 0 (20)
where the operator X is a linear combination of the operators of MAI (17) and Φ is an
unknown function. One can easily prove that there are three different types of solutions
that can be found by solving (20) for (i) X = Pt , (ii) X = G∞ and (iii) X = αPt +
+ G∞, 0 �= α ∈ R. Let us consider each of them.
(i) Putting X = Pt , we immediately obtain the ansatz
V = V (x), U = U(x). (21)
One can see that ansatz (21) is the most general form of steady-state solutions. Substitut-
ing (21) into (16), we arrive at the system of ordinary differential equations (ODEs)
Vx = h1U − σ1νmUxx − h0,
0 = DνmUxx − Sσ1νmU2
x − SV Ux − b20U
2 − b10U + b0.
(22)
This system is not integrable if S �= 0 or b20 �= 0 (the case S = b20 = 0 is unrealistic).
However taking into account formulas (11) one can note that
U =
h0
h1
≡ CGB , V = V0, V0 ∈ R, (23)
is a steady-state constant solution of (22). Obviously, the initial conditions (9) can be
obtained from (23) as a particular case at V0 = 0 .
Consider the second case (ii) X = G∞ . Solving the equation (20) for X = G∞,
we obtain the following ansatz for the functions U and V :
U = ϕ1(t), V = ϕ0(t) + ϕ2(t)x, (24)
where ϕ0, ϕ1, ϕ2 ≡ νmft
f
are unknown functions that should be found. Using ansatz (24),
we can reduce the nonlinear system (16) to the ODE equation
νm
dϕ1
dt
= −(Sh1 − b2)ϕ2
1 − (b1 − Sh0)ϕ1 + b0, (25)
and
ϕ2 = h1ϕ1 − h0.
One can see that equation (25) is easily integrated, and, substituting its general solution
into ansatz (24), we obtain the following solution of the nonlinear system (16):
U =
1
2(Sh1 − b2)
[
Sh0 − b1 +
√
H tanh
(√
H
2νm
(t + t0)
)]
,
(26)
V = ϕ0(t)+
+
h1x
2(Sh1 − b2)
[
(Sh0 − b1) +
2(b2 − Sh1)h0
h1
+
√
H tanh
(√
H
2νm
(t + t0)
)]
,
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
EXACT SOLUTIONS OF A MATHEMATICAL MODEL FOR FLUID TRANSPORT . . . 1117
where ϕ0(t) and t0 are an arbitrary function and constant, respectively, and
H = (b1 − Sh0)2 + 4b0(Sh1 − b2).
Finally, consider the case (iii) X = αPt + G∞, α �= 0 . Solving the equation (20)
for this operator, we obtain the ansatz
U = ϕ1(ω), V =
νm
αS
f(t) + ϕ2(ω), (27)
where ω = αx −
∫
f(t)dt . Using this ansatz, the nonlinear system (16) is reduced to
the following ODEs system:
α2σ1νm
d2ϕ1
dω2
+ α
dϕ2
dω
= h1ϕ1 − h0,
α2Dνm
d2ϕ1
dω2
− αSϕ2
dϕ1
dω
= (Sh1 − b2)ϕ2
1 + (b1 − Sh0)ϕ1 − b0.
(28)
Unfortunately, this system is not integrable. Moreover, we have not found any non-
constant particular solution. Of course, one can solve (28) using numerical methods.
It turns out there is another possibility to obtain exact solutions of system (16). One
observes that system (16) contains only quadratic nonlinearities. Several new approaches
were recently suggested to find exact solutions for evolution equations with quadratic
nonlinearities (see, e.g., [9] and references cited therein). Those methods lead to the so
called non-Lie ansätze which cannot be found using the classical Lie method.
Following [10], let us consider the ansatz
U = ψ0(t) + . . . + ψn(t)xn,
V = ϕ0(t) + . . . + ϕm(t)xm,
(29)
where ψi, i = 0, . . . , n, and ϕj , j = 0, . . . ,m, are unknown functions. Obviously,
this ansatz is a generalisation of the Lie ansatz (24). Substituting this ansatz into (16) one
can easily show that the expression obtained is reduced to a ODE system only under the
restriction m = n + 1 . In the particular case n = 2 , the corresponding ODE system
takes the form
νm
dψ0
dt
= 2νmDψ2 − b20ψ
2
0 − b10ψ0 + b0,
νm
dψ2
dt
= (3Sh0 − b1)ψ2 + b20ψ0ψ2.
(30)
Simultaneously, we obtain the expressions
ϕ0 = ϕ2 = 0, ϕ1 = h1ψ0 − 2νmσ1ψ2 − h0, ϕ3 =
h1
3
ψ2 (31)
for the other functions arising in (29) with n = 2, m = 3 and the additional restriction
b2 =
5Sh1
3
. Finally, taking into account formulae (15), (29) and (31), we obtain the exact
solution of (16) with b2 =
5Sh1
3
U = ψ0(t) + ψ2(t)x2,
V = (h1ψ0 − 2νmσ1ψ2 − h0)x +
h1
3
ψ2(t)x3,
(32)
where (ψ0(t), ψ2(t)) is a solution of (30).
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
1118 R. CHERNIHA, J. WANIEWSKI
It turns out that ansatz (29) with n > 2 works only under two additional restrictions
on the coefficients of system (16). Omitting the relevant calculations, we present here
only the result. System (16) with b2 =
2n + 1
n + 1
Sh1
(
i.e., b20 = − n
n + 1
Sh1
)
and
D = 0 has the exact solution
U = ψ0(t) + ψ(t)xn,
V = (h1ψ0 − h0)x− nνmσ1ψ(t)xn−1 +
h1
n + 1
ψ(t)xn+1,
(33)
where (ψ0(t), ψ(t)) is a solution of the ODE system
νm
dψ0
dt
=
n
n + 1
Sh1ψ
2
0 − b10ψ0 + b0,
νm
dψ
dt
=
(
(n + 1)Sh0 − b1 −
n2 − n
n + 1
Sh1ψ0
)
ψ.
(34)
Obviously, the general solution of the nonlinear system (34) can be constructed in explicit
form (see, e.g., [9]).
Remark 1. It can be easily checked by direct calculations that formulae (33), (34)
with arbitrary real value n �= −1 represent the exact solution of the nonlinear system
(16) with b2 =
2n + 1
n + 1
Sh1 and D = 0 .
4. Applications and interpretation. Let us use the particular exact solution (26) for
solving the boundary-value problem (7) – (10) under additional restrictions. The restric-
tions are: 1) we assume t >> 0 , i.e., the initial conditions play no essential role, and
2) we consider the process of dialysis with high P , i.e., with ν(P ) = νmax. Obviously,
we can construct the formula for the pressure using (15), namely:
P = P0(t) −
x
Kνm
ϕ0(t) −
h1x
2
4Kνm(Sh1 − b2)
×
×
[(
2b2
h1
− S
)
h0 − b1 +
√
H tanh
(√
H
2νm
(t + t0)
)]
, (35)
where P0(t) and ϕ0(t) are arbitrary functions. To satisfy the Neunmann and Dirichlet
conditions (8) for the pressure, the functions P0(t) and ϕ0(t) must be specified, and the
expression
P = PD − h1
4Kνm(Sh1 − b2)
×
×
[
b1 +
(
S − 2b2
h1
)
h0 −
√
H tanh
(√
H
2νm
(t + t0)
)]
(2xL− x2) (36)
is obtained. In the similar way, we obtain the formula for the glucose concentration
U =
CGD, x = 0,
1
2(Sh1 − b2)
(
Sh0 − b1 +
√
H tanh
(√
H
2νm
(t + t0)
))
, x > 0, (37)
which also satisfies boundary conditions (8) for U . Thus, formulas (36), (37) present
the exact solution of the balance equations (10) with ν(P ) = νm and the boundary
conditions (8).
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
EXACT SOLUTIONS OF A MATHEMATICAL MODEL FOR FLUID TRANSPORT . . . 1119
Consider the behaviour of (36), (37) at t → ∞ . Taking into account (11) it is easily
calculated that
t → +∞ : P = PD, U =
{
CGD, x = 0,
CGB , x > 0.
In quite similar way, we can construct the exact solution of the balance equations (10)
with ν(P ) = νm, b2 =
2n + 1
n + 1
Sh1, D = 0 and the boundary conditions (8) using
the more general solution (33), (34). The final formulae for the pressure and the glucose
concentration take the form
P = PD − 1
2Kνm
(
(h1ψ0(t) − h0)(x2 − 2xL) − 2σ1νm
(
(x− L)n − (−L)n)
)
ψ(t)+
+
2h1
(n + 1)(n + 2)
(
(x− L)n+2 − (−L)n+2)
)
ψ(t)
)
and
U =
{
CGD, x = 0,
ψ0(t) + (x− L)nψ(t), x > 0,
respectively.
Let us consider a possible interpretation of the solutions obtained. With this in mind,
the numerical solution of the boundary-value problem (7) – (10) was found. Omitting here
the details (this will be done in another paper), we note that a numerical technique based
on the finite elements method and Galerkin’s method was used [6]. We have compared
the numerical solution obtained for parameters given in [2] with the exact solution (36),
(37) and established that it is possible to select the value of the parameter t0 such that
the numerical solution for the pressure P coincides with exact solution (36) if t ≥ t1
(here t1 is a constant that depends on parameters arising in (7) – (10)) and the diffusivity
D = 0 . It was also established that the numerical solution for the concentration U tends
to exact solution (37) if t → ∞ and the diffusivity D = 0 . Thus, we conclude that
the exact solution (36), (37) at sufficiently large values of time describes the hydrostatic
pressure and the glucose concentration in peritoneal dialysis based on the mathematical
model (7) – (10) with D = 0 .
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Received 26.06.2004
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
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