On a parabolic PDE [626149]

On a parabolic PDE
Daniel N.Pop
Faculty of Engineering, Lucian Blaga University Sibiu
Abstract
Consider the following parabolic PDE
A@
@x(P@P
@x) =@P
@t
withe 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([2, 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
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
where the constant A is given by the properties of the medium. 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 [2, 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)
with boundary conditions:
u(0) = 1 ; u(1) = 0 (5)
= 0:8 (6)
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 [2, pag 110-114] prove that the ODE (4) is approx-
imately by:
u′′(z) + 2 zu′(z) = 0 (8)
2

2 Main results
2.1 Analytical solution
Using the idea due to L.F. Shampine ( [7, 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 [3] prove that the asymptotic representation of the complementary error function
is :
err u (z)ez2
zp
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 1 The analytical approximations to u(z)andu′(z)are accurate only for large z:
3

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.
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 ;3]
– and the asymptotically area [3, ];
and then we obtained the approximation solutions of the problem like:
-on the interval [0 ;3] with B-splines functions of order k+ 1 method,
-and on [3 ; ) with Runge-Kutta methods kstages by Matlab solver ODE45 ([5]).
The convergence of this embedded method is given in ([1, pp:218-224, 329-355, 386-414]).
The results are depicted in the following gure (1):
Figure 1: BVP for ODE
4

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 ([4]) with
solver MATLAB 2013b PDEPE ([5]). For A= 1:05 we obtain the following results depicted
in gure(2):
Figure 2: Line Method
5

3 Conclusion
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]P.B Bailey, L.F Shampine, P.E. Waltman.- Nonlinear two point boundary value prob-
lems, New-York, Academic Press 1968.
[3]C.M Bender,S.A.Orsag- Advanced Mathematical Methods for Scientists and Engineers
I.-Asymptotic Methods and Perturbation Theory, New-York, Springer Verlag, 1999.
[4]MOL Library and Example Applications: http://www.pdecomp.net/.
[5]Matlab 2013b, Set of Manual, Matworks, Inc-Nattick, MA, 2013.
[6]Nicolae Daniel Pop- Aspects concerning Numerical Methods for Approximate Solution
of BVP-Collocation methods, Editions Universitaires Europeennes, 2017.
[7]L.F Shampine, I.Gladwell, S.Thomson- Solving ODE with Matlab, Cambridge Univer-
sity Press,2003.
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