On a parabolic PDE [626150]

On a parabolic PDE
Daniel N.Pop, Narcisa I.Vr^ nceanu
Faculty of Engineering, Lucian Blaga University Sibiu
Abstract
Consider the following parabolic PDE
A@
@x(P@P
@x) =@P
@t
with the initial condition
P(x;0) = P0
and the boundary condition
P(0; t) =P1;where A= 1P2
0
P2
1
The origin of this problem is in the study of a gas
ow through a semi-innite porous
medium([3, pag 122]).
We present two solution for this problem: by reduction to a BVP for ODE and by
using line method.
2010 Mathematics subject classications
35K20,34B05,65M12
Keywords and Phrases
Parabolic equation and systems, Boundary value problems, Numerical analysis
1

1 Preliminaries
The application of these problems involves chemical kinetics, astrophysics, experimental and
mathematical physics, nuclear charge in heavy atoms, thermal behavior of a spherical cloud of
gas, thermodynamics, population models,
uid mechanics and many other topics. Transport
phenomena and diffusion in micro-nano porous materials have attracted the researchers
attention for a long time. The modeling of gas
ow through a porous media is quite valuable
because of its importance in investigating gas-solid processes. This example originates in
study of the unsteady
ow of a gas through a semi-innite porous medium initially lled
with gas at a uniform pressure P 0:
At the time
t= 0
the pressure at the out
ow face is suddenly reduced from P 0to P 1and is thereafter main-
tained at this lower pressure. The unsteady isothermal
ow of gas has been described by
the nonlinear partial differential equation
∇2(P2) = 2 A@P
@t;
A=ϕ
k
where ϕis the porosity, is the viscosity and kis the permeability.
In the one-dimensional medium extending from x= 0 to x=1, this reduces to
@
@x(P@P
@x) =A@P
@t(1)
with the boundary conditions
P(x;0) = P0;0< x <1 (2)
P(0; t) =P1(< P 0);0t <1
To obtained a similarly solution P.E. Waltman [3, pag122-124] introduce the new inde-
pendent variable
z=xp
t(A
4P0)1
2
and the dimension-free dependent variable udened by
u(z) =1(1P2(z)
P2
0)
where
= 1P2
1
P2
0(3)
In the term of these new variables, the problems take the form:
u′′(z) +2z√
1u(z)u′(z) = 0 (4)
The typical boundary conditions imposed by the physical properties are:
u(0) = 1 ; u(1) = 0 (5)
= 0:8 (6)
2

is a real parameter, 0 <  < 1 give by (3).
Obviously that we should expect u(z) to belong to the interval J= [0;1] in which case
we would have
L1(z) = 2 z2z
[1u(z)]1
22(1)1=2z=L2(z)
The solution u(z) should respect the physical requirements:
0u(z)1 (7)
Using Theorem 7.1 and 7.5 Bailey [3, pag 110-114] prove that the ODE (4) is approx-
imately by:
u′′(z) + 2 zu′(z) = 0 (8)
Remark 1 This problem was also study by S.Abbasbandy [2]. He use shooting method for
= 0:5.
2 Main results
2.1 Analytical solution
Using the idea due to L.F. Shampine ( [8, pag 153-154]) the ODE (8) becomes approx:
u′′(z)
u′(z)=2z
lnu′(z) =z2+
u′(z)ez2
Integrating and imposing the boundary conditions at innite we obtained:
u(z) =∫z
0et2dt+

u(1) = 0
)
∫1
0et2dt+
= 0
But: ∫1
0et2dt=p
2
implies

=p
2
So
u(z)∫1
zet2dtp
2err u (z)
Bender in [4] prove that the asymptotic representation of the complementary error function
is :
err u (z)ez2
zp
3

So:
u(z) 1
2ez2
z
To solve the ODE (8) with conditions (5 )we could do that by treating as a unknown
parameter:
u′(z) =ez2
u(z) =
2zez2
instead of
u(Z) = 0
Remark 2 The analytical approximations to u(z)andu′(z)are accurate only for large z:
2.2 Numerical solution
2.2.1 Reduction to a BVP for ODE
In the following we propose to solve the problem (8)+(5) with an embedded method: B-
splines functions andRunge Kutta methods ([7, pag 106-108]).
If in ODE (8) we make the substitution:
u(z) =y(z)f(z); f(z) =ez2
2
we obtained:
u′(z) =y′(z)ez2
2zez2
2y(z) (9)
u′′(z) =y′′(z)ez2
2zy′(z)ez2
2ez2
2y(z) +z2ez2
2y(z)zy′(z)ez2
2 (10)
Multiply (9) by 2 zand adding (10), the ODE (8) begin:
y′′(z)(z2+ 1)y(z) = 0 (11)
with boundary conditions
y(0) = 1 ; y(1) = 0 (12)
We divide the interval [0, 1) in two subintervals :
– A transient area [0 ;4]
– and the asymptotically area [4, ];
and then we obtained the approximation solutions of the problem (11+12) like:
-on the interval [0 ;4] with B-splines functions of order ( k+ 1),
-and on [4 ; ) with Runge-Kutta kstages by Matlab solver ODE45 ([6]).
The convergence of this embedded method is given in ([1, pp:218-224, 329-355, 386-414]).
Also we use the solver Matlab BVP4C. The results are depicted in gure (1).
4

(a) bvp with Bspline
(b) bvp4c
Figure 1: BVP for ODE
2.2.2 Line Method (MOL)
Recall given the partial differential equation (1+2) that describes this process, we postulate
a solution of the form:
P(x; t) =u(x)f(t); f(t) =et2
2
where u(x) is the solution of BVP (11+12).
We propose to solve the problem (1) with the conditions (2) using MOL method [5] and
MATLAB 2013b solver PDEPE [6].
ForA= 0:5 and A= 0:8 we obtain the following results depicted in gure(2).
(a)
(b)
Figure 2: Line Method
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3 Conclusion
Using the functions tic-toc of Malab we obtained the following results:
1.for BVP method the times is 0.733778 seconds.
2.for MOL method the times is 2.001758 seconds.
3.for BVP4C method of Matlab time is 1.352202 seconds.
The problem (11) with boundary conditions (12) is set on an innite interval, so some
experimentation is necessary to verify that a sufficiently large has been specied.
4 Acknowledgements
The writing of this work beneted enormously from a lot of discussion with conf.univ. dr.
Radu T.Tr^ mbit a s from Babe s-Bolyai University of Cluj-Napoca.
References
[1]U.Ascher, B.Mateij, R.D.Russsel- Numerical solution of boundary problems for ordinary
differential equations, S.I.A.M, Pretince Hall, 1997.
[2]S.Abbasbandy- Numerical Study of Gas Flow through a Micro-Nano Porous Media, Acta
Physica Polonia A, vol.121, no.3, 2012.
[3]P.B Bailey, L.F Shampine, P.E. Waltman.- Nonlinear two point boundary value prob-
lems, New-York, Academic Press 1968.
[4]C.M Bender,S.A.Orsag- Advanced Mathematical Methods for Scientists and Engineers
I.-Asymptotic Methods and Perturbation Theory, New-York, Springer Verlag, 1999.
[5]MOL Library and Example Applications: http://www.pdecomp.net/.
[6]Matlab 2013b, Set of Manual, Matworks, Inc-Nattick, MA, 2013.
[7]Nicolae Daniel Pop- Aspects concerning Numerical Methods for Approximate Solution
of BVP-Collocation methods, Editions Universitaires Europeennes, 2017.
[8]L.F Shampine, I.Gladwell, S.Thomson- Solving ODE with Matlab, Cambridge Univer-
sity Press,2003.
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