In the present paper we carried out several experiments in oxygen or dry air, at low temperature of some metallic samples. In order to be able to extend or estimate the corrosion phenomenon we made use of the modelling of oxygen diffusion through rust layers (oxides) and of solving the parabolic equations of diffusion, respectively. The diffusion equation is important for modelling the oxygen diffusion within biological systems and for modelling the neutron flux from nuclear reactors.

1. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 7, July (2014), pp. 101-112 © IAEME
101
MODELING OF OXYGEN DIFFUSION THROUGH IRON OXIDES LAYERS
Ion RIZA1
, Marius Constantin POPESCU2
1
University Politehnica of Cluj Napoca, Department of Mathematics, Cluj Napoca, Romania
2
“Vasile Goldis” Westerns University Arad, Department of Computer of Science, Arad, Romania
ABSTRACT
In the present paper we carried out several experiments in oxygen or dry air, at low
temperature of some metallic samples. In order to be able to extend or estimate the corrosion
phenomenon we made use of the modelling of oxygen diffusion through rust layers (oxides) and of
solving the parabolic equations of diffusion, respectively. The diffusion equation is important for
modelling the oxygen diffusion within biological systems and for modelling the neutron flux from
nuclear reactors.
Keywords: Atmospheric Corrosion, Non-Linear Parabolic Equation, Fick Equations, Fokker
Equation, Bessel Function.
1. INTRODUCTION
Although a part of the metal comes back into the circuit by remelting, the losses, in case of
iron, will come to a total of at least 10-15% from the metal got by melting. The corrosion of the
metals and alloys is defined as being the process of their spontaneous destruction, as a result of the
chemical, electrochemical and biochemical interactions with the resistance environment [10]. In
practice, the corrosion phenomena are usually extremely complex and they can appear in several
forms; this is why it is not possible to strictly classify all these phenomena. The chemical corrosion
of metals – or dry corrosion- of alloys takes place by reactions at their surface in contact with dry
gases or non-electrolytes [1], [2], [4]. The products that come out under the action of these
environments generally remain where the metal interacts with the corrosive environments. They
become layers that can have different thicknesses and compositions. Among the most corrosive
factors, O2 has an important contribution. The evolution of the corrosion is related, among other
things, to the evolution of oxygen concentration in oxides and metals. All types of oxidations start
with a law that is proportional or linear with time, followed by another logarithmic or parabolic law.
All equations with partial derivatives that describe and influence diffusion are parabolic.
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ISSN 0976 – 6340 (Print)
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Volume 5, Issue 7, July (2014), pp. 101-112
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2. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
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102
2. EXPERIMENTS IN DRY AIR AT LOW TEMPERATURES
At low temperatures the iron oxides Fe3O4 and Fe2O3, are thermically stable and at normal
temperatures the ordinary rust Fe2O3*nH2O appears. The OL37 iron sample has been vertically
exposed in open atmospheric conditions during different periods of time (about 6, 12, and 24
months), during cold and warm periods. During the cold period the corrosion takes place with values
above the average ones. For studying different parts of the sample, a rectangular part, having the
length of 3.8 [cm], the width of 3.45 [cm], the surface of 13.11 [cm2
], the weight of initial sample
2.5634 [g], the weight without rust 2.2911 [g], the rust weight [g] was taken out.
Fig.1: Explication regarding the thickness of oxide layer at low temperature
The calculation of the thickness of oxide layer makes also possible the calculation of oxygen
diffusion. In order to calculate the thickness of the oxide layer at low temperature we should take
into account some experimental or calculated, such as rust weight gr (0.2723[g]), density ρ (5195
[mg/ܿ݉ଷ
ሿ), number of months of exposure or exposure time, t (1.5552x107
[s], respectively,
3.1x107
[s]), thickness of oxide layer (‫ݕ‬ ൌ ݃௥
ଵ
ఘ
ଵ
ௌ
=0.003997 [cm], for t=1.5552x107
[s]).
3. MATHEMATICAL MODELLING OF DIFFUSION
The equations that describe the diffusion are parabolic partial derivatives, and the
mathematical models are based on three remarkable laws:
- the equation of heat or the Fick second law for diffusion
డ௪
డ ௧
ൌ ‫ܦ‬
డమ௪
డ ௫మ, (1)
- convection-diffusion equation
డ௪
డ௧
ൌ ‫ܦ‬
డమ௪
డ௫మ െ ‫ݒ‬
డ௪
డ௫
, (2)
- and parabolic-diffusion equation
డ௪
డ௧
ൌ ‫ܦ‬
డ
డ௫
ቀ݂ሺ‫ݔ‬ሻ
డ௪
డ௫
ቁ, (3)
where w(x,t) represents the practical value of a concentration, expressed in [mg/cm3
], x is a
distance and t, time.
As a particular case, there is the function f(x)=e-x
, n order to explain the decrease of
concentration in time: this decreases from the air-rust interface (outer air) towards rust-metal
interface (towards the interior). The study in one dimension has been imposed by a diffusion named

3. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 7, July (2014), pp. 101-112 © IAEME
103
D, expressed in [cm2
/s]; due to the fact that we didn’t have any data about D, the time dependency
(t), or (x,t), we considered D=constant (or K). In calculus we considered D=1.12*10-8
[cm2
/s]. The
following abbreviations were used (specific to the calculus program): ODE – normal differential
equation (of variable x or t), PDE – differential equation with partial derivatives (with two variables
x and t) and SOL – solution from an expression or effective solution. In Bessel function, I indicates
the type of function and ‫ܫ‬ ൌ √െ1.
3.1. Parabolic Homogenous Equation of Diffusion
The second law of Fick, (1) for diffusion phenomena that are variable in time and space, in
homogenous and isotropic environments, has been studied with several solving methods:
- the method of separation the variables with a real function represented by a Fourier integral with
Poisson form and solved with erf –Laplace function [7];
- the method of integral transformations, respectively the Fourier transformation [8], [9].
We present five solutions to the heat equation or the second law of Fick about diffusion.
a). After changing the function ‫ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ ൌ ‫ݒ‬ሺ‫ݔ‬ሻ݁ఒ௧
, and, after solving the derivatives
డ
డ௧
,
డ
డ௫
,
డమ
డ௫మ
and their replacement, the following differential equation results
ௗమ
ௗ௫మ
‫ݒ‬ሺ‫ݔ‬ሻ െ
ఒ
௄
‫ݒ‬ሺ‫ݔ‬ሻ ൌ 0, ‫ݒ‬ሺ‫ݔ‬ሻ ൌ ‫ܥ‬ଵ݁
√ഊೣ
√಼ ൅ ‫ܥ‬ଶ݁
ି
√ഊೣ
√಼ ,
(4)
with general solution
‫ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ ൌ ቆ‫ܥ‬ଵ݁
√ഊೣ
√಼ ൅ ‫ܥ‬ଶ݁
ି
√ഊೣ
√಼ ቇ ݁ఒ௧
. (5)
b). A solution having the form w(x,t)= u(y(x,t)) will be determined with y(x,t)=eλx+µt
and, after
derivation and replacements, the following equation will result
ௗమ
ௗ௬మ ‫ݑ‬ሺ‫ݕ‬ሻ ൅ ቀ1 െ
ఓ
௄ఒమቁ
ௗ
ௗ௬
‫ݑ‬ሺ‫ݕ‬ሻ ൌ 0, (6)
The condition is ቀ1 െ
ఓ
௄ఒమ
ቁ ൌ 0 and the result will be a simpler equation
ௗమ
ௗ௬మ ‫ݑ‬ሺ‫ݕ‬ሻ ൌ 0, u(y)= C1 y + C2, (7)
with the general solution
‫ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ ൌ ‫ܥ‬ଵ݁ఒ௫ା௄ఒమ௧
൅ ‫ܥ‬ଶ. (8)
c). Let us determine the solution of Fick’s equation with the form w(x,t)=u(y(x,t)) and ‫ݕ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ ൌ
݁ఒ௫ାఒమ௧
. Calculating the derivatives and replacing the parabolic differential equation the result will
be:
ௗమ
ௗ௬మ
‫ݑ‬ሺ‫ݕ‬ሻ ൅ ቀ1 െ
ଵ
௄
ቁ
ௗ
ௗ௬
‫ݑ‬ሺ‫ݕ‬ሻ ൌ 0, ‫ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ ൌ ‫ܥ‬ଵ ൅ ‫ܥ‬ଶ ݁ି
ሺ಼షభሻ౛λ౮శλమ౪
಼ . (9)

4. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 7, July (2014), pp. 101-112 © IAEME
104
d). Likewise we look for a solution with the form w(x,t) = u(y(x,t)) and ‫ݕ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ ൌ ݁௣௫ି௤௧
ௗమ
ௗ௬మ
‫ݑ‬ሺ‫ݕ‬ሻ ൅ ቀ1 ൅
௤
௄௣మ
ቁ
ௗ
ௗ௬
‫ݑ‬ሺ‫ݕ‬ሻ ൌ 0, ‫ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ ൌ ‫ܥ‬ଵ ൅ ‫ܥ‬ଶ ݁
ି
ሺ಼೛శ೜ሻ౛౦౮ష౧౪
಼೛ . (10)
e). The direct solving of the equation with partial derivatives leads to a solution
w(x,t)=φ1(x) φ2(t), (11)
where
߮ଵሺ‫ݔ‬ሻ ൌ ቀ‫ܥ‬ଵ݁ඥ௖య௫
൅ ‫ܥ‬ଶ݁ିඥ௖య௫
ቁ, ߮ଶሺ‫ݐ‬ሻ ൌ ‫ܥ‬ଵ݁
௄௖య ௧
. (12)
3.2. Convection-diffusion equation
In case of convection-diffusion equation, the phenomenon changes with Fokker–Planck
equation (2) having the general form:
డ௪ሺ௫௧ሻ
డ௧
= -
డ
డ௫
ሾ‫ܦ‬ଵሺ‫,ݔ‬ ‫ݐ‬ሻ‫ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻሿ ൅ ൅
డమ
డ௫మ
ሾ‫ܦ‬ଶሺ‫,ݔ‬ ‫ݐ‬ሻ‫ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻሿ ൅a(x,t)c(x,t)=f(x,t), (13)
where a(x,t) şi f(x,t) represents a disturbing factor and a source, respectively; in the most frequent
case the form is:
డ௖ሺ௫௧ሻ
డ௧
= -
డ
డ௫
ሾ‫ܦ‬ଵሺ‫,ݔ‬ ‫ݐ‬ሻܿሺ‫,ݔ‬ ‫ݐ‬ሻሿ ൅
డమ
డ௫మ
ሾ‫ܦ‬ଶሺ‫,ݔ‬ ‫ݐ‬ሻܿሺ‫,ݔ‬ ‫ݐ‬ሻሿ. (14)
In particular, if D=‫ܦ‬ଶ is considered to be a diffusion coefficient, ‫ܦ‬ଵ ൌ ‫ܦ‬ଵ(x), becoming a
speed, v, by derivation
డ௪
డ௧
ൌ ‫׏‬ሺ ‫ݓ׏ܦ‬ െ ‫ݓݒ‬ሻ, or
డ௪
డ௧
ൌ ‫ݓ∆ܦ‬ െ ‫.ݓ׏ݒ‬ (15)
The term
డ௪
డ௧
is multiplied with a coefficient R named delaying coefficient. This can have a
value higher or lower than one unit and it can delay or accelerate the diffusion process; as a result,
the equation with Fokker partial derivatives becomes
ܴ
డ௪
డ௧
ൌ ‫ܦ‬
డమ௪
డ௫మ െ ‫ݒ‬
డ௪
డ௫
. (16)
A particular case is represented by the introduction of the source (+) or of the consumption
(-), term multiplied with λ coefficient
ܴ
డ௪
డ௧
ൌ ‫ܦ‬
డమ௪
డ௫మ െ ‫ݒ‬
డ௪
డ௫
േ ߣ‫ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ. (17)
For convection-diffusion equation there are two solutions, one with no λ parameter and
another with λ parameter, apart from the transformation into Fick equation [7].

5. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 7, July (2014), pp. 101-112 © IAEME
105
a). The convection-diffusion equation has the solution
߮ଵሺ‫ݔ‬ሻ ൌ ‫ܥ‬ଵ݁
భ
మ
ቆషೇశටೇమశర೎యವቇೣ
ವ ൅ ‫ܥ‬ଶ݁
షభ
మ
ቆశೇశටೇమశర೎యವቇೣ
ವ , ߮ଶሺ‫ݐ‬ሻ ൌ ‫ܥ‬ଵ݁
ష೎య೟
ೃ ; (18)
b). The solution with -λ parameter of the convection-diffusion equation is
߮ଵሺ‫ݔ‬ሻ ൌ ‫ܥ‬ଵ݁
భ
మ
ቆషೇశටೇమశర೎యವቇೣ
ವ ൅ ‫ܥ‬ଶ݁
భ
మ
ቆశೇశටೇమశర೎యವቇೣ
ವ , ߮ଶሺ‫ݐ‬ሻ ൌ ‫ܥ‬ଵ݁
ሺష೎యశഊሻ೟
ೃ , ‫ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ ൌ ߮ଵሺ‫ݔ‬ሻ ߮ଶሺ‫ݐ‬ሻ.
(19)
3.3. Parabolic Diffusion Equation
A). From equation (3) the expression PDE will be obtained, starting from the flux notion (physical
[3]), or from Planck-Nernst equation
‫ܬ‬௜ ൌ ‫ܦ‬௜ ߘ ‫ݓ‬௜ ൅
஽೔௤೔௘
௞ಳ்
‫ݓ‬௜ߘ‫,׎‬ (20)
where, ‫ܬ‬௜ is a species flux i, ‫ݓ‬௜ is the species concentration i, ‫׎‬ is the electrostatic potential,
‫ܦ‬௜ is the diffusion coefϐicient, ‫ݍ‬௜ is the elementary electric load of the electron (1.60217x10ିଵଽ
C),
݇஻ ൌ 1.38065 10ିଶଷ ୎
୏
is Boltzman constant, T is the absolute temperature, expressed in ‫.ܭ‬ The
equation is specialized in modeling the oxygen diffusion through oxide layers (or porous
environments – rust) and it controls the oxygen diffusion through rust layers (oxides). If ܵ௞ is a
source that consumes or give oxygen, then the equation for mass balance is
ܵ௞ ൌ
డ௪ೖ
డ௧
൅ ‫ܬ׏‬௞. (21)
Considering ܵ௞ ൌ 0, the relation (3) becomes
డ௪೔
డ௧
=
ப
ப୶
ቂሺ‫ܦ‬௜ ‫׏‬ ‫ݓ‬௜ሻ െ
஽೔௤೔௘
௞ಳ்
‫ݓ‬௜‫׎׏‬ቃ, (22)
or if the term containing temperature is omitted
డ௪೔
డ௧
=
ப
ப୶
ሾሺ‫ܦ‬௜ ‫׏‬ ‫ݓ‬௜ሻሿ. (23)
If Di is proportional with D through the function x it results
డ௪
డ ௧
ൌ ‫ܦ‬
డ
డ௫
ሺ ݂ሺ‫ݔ‬ሻ
డ௪
డ ௫
ሻ, (24)
with the general form [5]
డ௪
డ௧
െ
డ
௫ೕ
ሺ‫ܣ‬௝௞ሺ௫ሻ
డ௪
డ௫ೖ
ሻ ൅ ‫ܤ‬௞ሺ‫ݔ‬ሻ
డ௪
డ௫ೖ
൅ ‫ܥ‬ሺ‫ݔ‬ሻ‫ݓ‬ ൌ ݂ሺ‫,ݔ‬ ‫ݐ‬ሻ. (25)

6. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
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106
The parabolic equation that describes the diffusion phenomenon transforms into:
- the second law of Fick (1), applied for homogenous environments, for φ(x)=1;
- for φ(x), function of x, given by the nature of the modeled process, the equation is a component part
of Sturm-Liouville operator, in Neumann problem with non-homogenous limit conditions, in
Dirichlet problem for unlimited domains and so on;
- if φ(x)=K, with K=constant, the equation of heat can be obtained, where K(=D) can also be K(w);
- if φ(x) is replaced with w(x,t) or with a function f(w(x,t)), several differential equations with
different forms will be obtained, with f(w) at “m” and/or
డ௪
డ ௫
at ”n” or ሺ߮ሺ‫ݔ‬ሻ
డ௪
డ ௫
ሻ at “p” and some
partial derivatives of w can be added, from (n-1) until one and with a free term w(x,t).
a1). Solving the equation by using the method of variables separation
PDE1:
డ௪
డ௧
ൌ ‫ܭ‬ ቀ݁ି௫ డమ௪
డ௫మ െ ݁ି௫ డ௪
డ௫
ቁ, (26)
the result was the solution ‫1ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ ൌ ߮ଵሺ‫ݔ‬ሻ߮ଶሺ‫ݐ‬ሻ, of components:
߮ଵሺ‫ݔ‬ሻ ൌ ܱܵ‫1ܮ‬ሺ ܱ‫:1ܧܦ‬
݀ଶ
݀‫ݔ‬ଶ
݂1 ൌ െ‫3ܥ‬ ݁௫
݂1ሺ‫ݔ‬ሻ ൅
݀
݀‫ݔ‬
݂1ሺ‫ݔ‬ሻሻ,
߮ଵሺ‫ݔ‬ሻ ൌ ݁
௫
ଶሺ‫1ܥ‬ ‫ܬ݈݁ݏݏ݁ܤ‬ሺ1,2√‫3ܥ‬ ݁௫ ሻ ൅ ‫2ܥ‬ ‫ܻ݈݁ݏݏ݁ܤ‬൫1,2√‫3ܥ‬ ݁௫ ൯ሻ,
߮ଶሺ‫ݐ‬ሻ ൌ ܱܵ‫2ܮ‬ ሺ ܱ‫2ܧܦ‬ ‫׷‬
ௗ
ௗ௧
݂2ሺ‫ݐ‬ሻ ൌ െ‫ܭ‬ ݂2ሺ‫ݐ‬ሻ‫3ܥ‬ , ߮ଶሺ‫ݐ‬ሻ ൌ ‫1ܥ‬ ݁ି ௄ ஼ଷ ௧
. (27)
The constants are determined from a system of initial conditions (x=0, t=0, Ci=1575.745 is
the initial concentration) and of final conditions (x=30x10-4
, t=1.5552x107
, Cf=1279.986 is the final
concentration). There are two solutions for the two cases:
SOL1(C1=57.58669368I; C2=565.9460188 I; C3=1) SOL2(C1=-57.58669368 I; C2= -565.9460188
I; C3=1),
with the graphic representation as shown in Fig.2a.
a) b)
Fig.2: Graphic representation of the solution for: a) t=1.5552x107
[s], b) t=3.1104 x107
[s]

7. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
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According to the development functions of the function ߮ଵሺ‫ݔ‬ሻ it can result a ODE1 variant,
having the form
ODE3: ߮ଵሺ‫ݔ‬ሻ ൌ ܱܵ‫ܮ‬ ቀܱ‫:3ܧܦ‬ ቄ
ௗమ
ௗ௫మ
݂1ሺ‫ݔ‬ሻ െ ൬
ௗ
ௗ௫
݂1ሺ‫ݔ‬ሻ൰ ൅ ൅‫3ܥ‬ ݁௫
݂1ሺ‫ݔ‬ሻቅ , ሼ‫ܨ‬ሺ‫ݔ‬ሻሽቁ.
(28)
The form with F(x), comes from an indefinite derivation. We can find the equivalent solution
of ODE3 equation (normal differential equation by turning ODE3 into ODE4) by bringing it to the
hermitian form
ܱ‫:3ܧܦ‬
ௗమ
ௗ௫మ
‫ݕ‬ሺ‫ݔ‬ሻ െ ൬
ௗ
ௗ௫
‫ݕ‬ሺ‫ݔ‬ሻ൰ ൅ ‫3ܥ‬ ݁௫
‫ݕ‬ሺ‫ݔ‬ሻ ൌ 0. (29)
Any equation having the form
p0(x)y’’
+ p1(x)y’
+ p2(x)y = 0, (30)
can be transformed into
ௗ
ௗ௫
ቀ‫݌‬ሺ‫ݔ‬ሻ
ௗ
ௗ௫
‫ݕ‬ቁ ൅ ‫ݕݍ‬ ൌ 0, (31)
where p(x)=݁
‫׬‬
೛భ
೛బ
ௗ௫
and ‫ݍ‬ሺ‫ݔ‬ሻ ൌ
௣మ
௣బ
݁
‫׬‬
೛భ
೛బ
ௗ௫
.
a2). The hermitian form is
ܱ‫:4ܧܦ‬
ௗమ
ௗ௫మ ‫ݕ‬ሺ‫ݔ‬ሻ ൅ ‫3ܥ‬ ݁ି௫
‫ݕ‬ሺ‫ݔ‬ሻ ൌ 0. (32)
A new solution ‫2ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ ൌ ߮ଷሺ‫ݔ‬ሻ߮ଶሺ‫ݐ‬ሻ, with Bessel functions, results:
߮ଷሺ‫ݔ‬ሻ ൌ ܱܵ‫3ܮ‬ ቆܱ‫:4ܧܦ‬
݀ଶ
݀‫ݔ‬ଶ
‫ݕ‬ሺ‫ݔ‬ሻ ൅ ‫3ܥ‬ ݁ି௫
‫ݕ‬ሺ‫ݔ‬ሻ ൌ 0ቇ,
߮ଷሺ‫ݔ‬ሻ ൌ ቀ‫1ܥ‬ ‫ܬ݈݁ݏݏ݁ܤ‬൫0,2√‫3ܥ‬ ݁ି௫ ൯ ൅ ‫2ܥ‬ ‫ܻ݈݁ݏݏ݁ܤ‬൫0,2√‫3ܥ‬ ݁ି௫ ൯ቁ, φଶ
ሺtሻ ൌ ‫1ܥ‬ ݁ି ௄ ஼ଷ ௧
.
(33)
a) b)
Fig.3. Graphic representation of the solution with hermitian, for: a) t=1.5552x107[s], b) t=
6.2208 x107[s]

8. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
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108
The constants are determined with a system with initial and final conditions, getting two
solutions for the two cases
SOL1(C1=165.4454841 I; C2= -91.23862277I;C3=1) SOL2(C1= -165.4454841I; C2= 91.23862277 I ;
C3=1),
with the cu graphic representation shown in Fig.3a.
Respectively,
SOL1(C1= 296.6675495I; C2= 1548.914183I ; C3=1) SOL2(C1= - 296.6675495 I; C2= 1548.914183I;
C3=1)
with the cu graphic representation shown in Fig.3b.
a3). The variant of the solution ODE4 – hermitian with the special function Eiυ(x), named
exponential integral:
߮ସሺ‫ݔ‬ሻ ൌ ‫1ܥ‬ ൅ ‫݅ܧ‬ ሺ1, െ ݁ି௫ሻ ‫,2ܥ‬ φଶ
ሺtሻ ൌ ‫1ܥ‬ ݁ ௄ ஼ଷ ௧
. (34)
The previous functions are components of the solution
‫3ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ ൌ ߮ସሺ‫ݔ‬ሻ߮ଶሺ‫ݐ‬ሻ, (35)
SOL1(C1=234.2739606–412.0259131I;C2= - 64.08360910 - 112.7061134I),
SOL2(C1=-234.2739606+412.0259131I;C2= + 64.08360910 + 112.7061134I),
with the graphic representation as shown in Fig.4a.
a) b)
Fig.4: Graphic representation of the hermitian solution with the special function Eiυ(x), for:
a) t=1.5552x107[s], b) t=6.2208 x107[s]
a4). In the case of invariant method with solutions of Bessel functions, we transform ODE3 into
an equivalent from with the invariable method. It is known that two ODE have the same solution if
the invariable is common. The equation in question is
ODE3:
ௗమ
ௗ௫మ ‫ݕ‬ሺ‫ݔ‬ሻ െ ൬
ௗ
ௗ௫
‫ݕ‬ሺ‫ݔ‬ሻ൰ ൅ ‫3ܥ‬ ݁௫
‫ݕ‬ሺ‫ݔ‬ሻ ൌ 0, (36)
put under the form

9. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 7, July (2014), pp. 101-112 © IAEME
109
y’’
+p(x)+q(x)=0 (37)
and made the change of variable y=u(x)z, ODE results as
z’’
u+(2u’
+p(x))z’
+(u’’
+p(x)u’
+qu=0. (38)
We cancel the coefficient of zi
,
2u’
+p(x)=0, (39)
we get
‫ݑ‬ሺ‫ݔ‬ሻ ൌ ݁ି
ೣ
మ , (40)
and ܱ‫5ܧܦ‬ ‫׷‬
ௗమ
ௗ௫మ
‫ݖ‬ሺ‫ݔ‬ሻ ൅ ቀെ
ଵ
ସ
‫݌‬ሺ‫ݔ‬ሻଶ
െ
ଵ
ଶ
ௗ
ௗ௫
‫݌‬ሺ‫ݔ‬ሻ ൅ ‫ݍ‬ቁ ‫ݖ‬ ൌ 0, ܱ‫:5ܧܦ‬
ௗమ
ௗ௫మ
‫ݖ‬ሺ‫ݔ‬ሻ ൅ ቀ െ
ଵ
ସ
൅ ‫3ܥ‬ ݁௫
ቁ ‫ݖ‬ ൌ 0,
(41)
߮ହሺ‫ݔ‬ሻ ൌ ܱܵ‫5ܮ‬ሺܱ‫:5ܧܦ‬
ௗమ
ௗ௫మ ‫ݖ‬ሺ‫ݔ‬ሻ ൅ ቀെ
ଵ
ସ
൅ ‫3ܥ‬ ݁௫
ቁ ‫ݖ‬ ൌ 0, (42)
With the same component t, as in the first solution, it results that
φଶ
ሺ‫ݐ‬ሻ ൌ ‫1ܥ‬ ݁ି௄ ஼ଷ ௧
, ߮ହሺ‫ݔ‬ሻ ൌ C1 Bessel J൫1,2 √C3 e୶ ൯ ൅ ‫2ܥ‬ ‫ܻ݈݁ݏݏ݁ܤ‬ ൫1, √C3 e୶ ൯,
‫4ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ ൌ ݁
షೣ
మ ߮ହሺ‫ݔ‬ሻ߮ଶሺ‫ݐ‬ሻ, (43)
The constants are determined with a system with initial and final conditions, getting two
solutions for the two cases
SOL1(C1= 59.95111043 I; C2= 563.3498783 I; C3=1), SOL2(C1=-59.95111043 I; C2=-563.3498783 I;
C3=1).
The graphic representation is identical in both cases and it is shown in Fig.2.
a5). Solutions with Bessel functions, using the Green function [11]. If in parabolic equation (3),
‫݂ܦ‬ሺ‫ݔ‬ሻ
డ௪
డ௫
ൌ ߛሺ‫,ݔ‬ ‫ݐ‬ሻ= φ(w), this becomes:
డ௪
డ௧
ൌ
డఝሺ௪ሻ
డ௫
. (44)
Because
డ௪
డ௧
,
డሺିఝሺ௪ሻሻ
డ௫
are components of the zero divergence of a function θ(x,t,w), the Green
function can be applied [6]
‫׭‬ ቀ
డ௪
డ௧
െ
డఝ
డ௫
ቁ ݀‫ݔ݀ݐ‬ ൌ ‫׬‬ ‫ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ݀‫ݔ‬ ൅ ߮ሺ‫,ݔ‬ ‫ݐ‬ሻ݀‫ݐ‬େ஽
.
(45)
The function lower than the parabolic integral can be proportional with the partial derivatives
of the function U(x,t)
ܷ݀ ൌ
డ௎
డ௫
݀‫ݔ‬ ൅
డ௎
డ௧
݀‫,ݐ‬ ܷ ൌ ‫׬‬
డ௎
డ௫
݀‫ݔ‬ ൅ ‫׬‬
డ௎
డ௧
݀‫.ݐ‬ (46)
Identifying the previous relations, the following system will result
డ௎
డ௫
ൌ ‫ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ,
డ௎
డ௧
ൌ ߮ሺ‫ݓ‬ሻ. (47)

10. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 7, July (2014), pp. 101-112 © IAEME
110
Replacing the second equation of the previous system and coming back to the initial notation
w(x,t), it results
ப୙
ப୲
ൌ ‫݁ܦ‬ି௫ డమ௎
డ௫మ , (48)
Changing the notation, the result is:
డ௪
డ௧
ൌ ‫ܦ‬ ݁ି௫ డమ௪
డ௫మ ,
డ௪
డ௧
ൌ ‫݁ܦ‬ି௫ డమ௪
డ௫మ . (49)
Replacing the integration on [0,∞) with an [a,b] subinterval, it is possible that some solutions
to be lost. The differential equation with partial derivatives of this form can be easily integrated, with
a solution
߮ହሺ‫ݔ‬ሻ ൌ ‫ܥ‬ଵ‫݈݁ݏݏ݁ܤ‬ ‫ܬ‬൫0,2√െ‫3ܥ‬ ݁௫൯ ൅ ‫ܥ‬ଵ ‫݈݁ݏݏ݁ܤ‬ ܻ൫0,2√െ‫3ܥ‬ ݁௫൯,
(50)
or
߮ହሺ‫ݔ‬ሻ ൌ ‫ܥ‬ଵ‫݈݁ݏݏ݁ܤ‬ ‫ܫ‬൫0,2√‫3ܥ‬ ݁௫൯ ൅ ‫ܥ‬ଵ ‫݈݁ݏݏ݁ܤ‬ 2 ‫ܫ‬൫0, √‫3ܥ‬ ݁௫൯, ߮ଶሺ‫ݐ‬ሻ ൌ ‫ܥ‬ଵ݁௄ ஼ଷ ௧
,
(51)
with the solution
‫5ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ ൌ ߮ହሺ‫ݔ‬ሻ߮ଶሺ‫ݐ‬ሻ, (52)
SOL1( C1=147.6003382 I ; C2=-1550.502962 I ; C3= 1),
SOL2( C1= -147.6003382 I; C2=1550.502962 I ; C3= 1).
He graphic representation is identical in both cases, as shown in Fig.5a.
a) b)
Fig.5: Graphic representation of the solution with Green function: a) t=1.5552x107
[s],
b) t= 3.1104 x107
[s]
Respectively,
SOL1(C1=150.0922757 I; C2=-1575.0834191I; C3=1), SOL2(C1=-150.0922757I;
C2=1575.0834191I; C3=1),
with the cu graphic representation shown in Fig.5b.

11. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 7, July (2014), pp. 101-112 © IAEME
111
B). Solutions with exponential component that are solved with Bessel functions:
b1). ‫ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ ൌ ‫ݑ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ݁ିఒ௧
with the solution
߮ଵሺ‫ݔ‬ሻ ൌ ‫ܥ‬ଵ‫݈݁ݏݏ݁ܤ‬ ‫ܬ‬ ൬‫,ܫ‬
ඥି௖భ௞ೣ
୪୬ሺ௞ሻ
൰ ൅ ‫ܥ‬ଵ ‫݈݁ݏݏ݁ܤ‬ ܻ ൬‫,ܫ‬
ඥି௖భ௞ೣ
୪୬ ሺ௞ሻ
൰, ߮ଶሺ‫ݐ‬ሻ ൌ ‫ܥ‬ଵ݁௄௖భ௧
,
‫ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ ൌ ሺ߮ଵሺ‫ݔ‬ሻ ߮ଶሺ‫ݐ‬ሻሻ݁ఒ௫
, (53)
or ‫ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ ൌ ‫ݑ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ݁ఒ௧
with the solution
߮ଵሺ‫ݔ‬ሻ ൌ ‫ܥ‬ଵ‫݈݁ݏݏ݁ܤ‬ ‫ܬ‬ ൬‫,ܫ‬
ඥି௖భ௞ೣ
୪୬ሺ௞ሻ
൰ ൅ ‫ܥ‬ଵ ‫݈݁ݏݏ݁ܤ‬ ܻ ൬‫,ܫ‬
ඥି௖భ௞ೣ
୪୬ ሺ௞ሻ
൰, ߮ଶሺ‫ݐ‬ሻ ൌ ‫ܥ‬ଵ ݁ିఒ௧
݁ି௄௖భ௧
,
‫ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ ൌ ሺ߮ଵሺ‫ݔ‬ሻ ߮ଶሺ‫ݐ‬ሻሻ݁ఒ௧
. (54)
b2). ‫ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ ൌ ‫ݑ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ݁ఒ௫ା ఒమ௧
, with one parameter λ [13], with the solution
߮ଵሺ‫ݔ‬ሻ ൌ ‫ܥ‬ଵ‫݈݁ݏݏ݁ܤ‬ ‫ܬ‬ ൬‫,ܫ‬
ඥି௖భ௞ೣ
୪୬ሺ௞ሻ
൰ ൅ ‫ܥ‬ଵ ‫݈݁ݏݏ݁ܤ‬ ܻ ൬‫,ܫ‬
ඥି௖భ௞ೣ
୪୬ ሺ௞ሻ
൰, ߮ଶሺ‫ݐ‬ሻ ൌ ‫ܥ‬ଵ ݁ିఒమ௧
݁ି௄௖భ௧
,
‫ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ ൌ ሺ߮ଵሺ‫ݔ‬ሻ߮ଶ߬ሻ݁ఒ௫ ା ఒమ௧
, (55)
or ‫ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ ൌ ‫ݑ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ݁ఒ ௫ ି ఒమ௧
with the solution
߮ଵሺ‫ݔ‬ሻ ൌ ‫ܥ‬ଵ‫݈݁ݏݏ݁ܤ‬ ‫ܬ‬ ൬‫,ܫ‬
ඥି௖భ௞ೣ
୪୬ሺ௞ሻ
൰ ൅ ‫ܥ‬ଵ ‫݈݁ݏݏ݁ܤ‬ ܻ ൬‫,ܫ‬
ඥି௖భ௞ೣ
୪୬ ሺ௞ሻ
൰, ߮ଶሺ‫ݐ‬ሻ ൌ ‫ܥ‬ଵ ݁ାఒమ௧
݁ି௄௖భ௧
,
‫ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ ൌ ሺ߮ଵሺ‫ݔ‬ሻ߮ଶ߬ሻ݁ఒ௫ ା ఒమ௧
. (56)
b3). ‫ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ ൌ ‫ݑ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ݁ఒ௫ା ఓ௧
, with two parameters, λ and µ [12], with the solution
߮ଵሺ‫ݔ‬ሻ ൌ ‫ܥ‬ଵ‫݈݁ݏݏ݁ܤ‬ ‫ܬ‬ ൬‫,ܫ‬
ඥି௖భ௞ೣ
୪୬ሺ௞ሻ
൰ ൅ ‫ܥ‬ଵ ‫݈݁ݏݏ݁ܤ‬ ܻ ൬‫,ܫ‬
ඥି௖భ௞ೣ
୪୬ ሺ௞ሻ
൰, ߮ଶሺ‫ݐ‬ሻ ൌ ‫ܥ‬ଵ ݁ିఓ௧
݁ା௄௖భ௧
,
‫ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ ൌ ሺ߮ଵሺ‫ݔ‬ሻ߮ଶ߬ሻ݁ఒ௫ ା ఓ௧
, (57)
or ‫ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ ൌ ‫ݑ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ݁ఒ௫ ି ఓ௧
with the solution
߮ଵሺ‫ݔ‬ሻ ൌ ‫ܥ‬ଵ‫݈݁ݏݏ݁ܤ‬ ‫ܬ‬ ൬‫,ܫ‬
ඥି௖భ௞ೣ
୪୬ሺ௞ሻ
൰ ൅ ‫ܥ‬ଵ ‫݈݁ݏݏ݁ܤ‬ ܻ ൬‫,ܫ‬
ඥି௖భ௞ೣ
୪୬ ሺ௞ሻ
൰, ߮ଶሺ‫ݐ‬ሻ ൌ ‫ܥ‬ଵ ݁ାఓ௧
݁ି௄௖భ௧
,
‫ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ ൌ ሺ߮ଵሺ‫ݔ‬ሻ߮ଶ߬ሻ݁ఒ௫ି ఓ௧
. (58)
4. CONCLUSIONS
The losses of metals and alloys produced by corrosion represent about one third of world’s
production. The change of iron to oxides, more stable – corrosion – is due to the thermodynamic
instability of the iron and to diffusion. The cognition of corrosion development means, among other
things, the cognition of the development of oxygen concentration within oxides and metals. In order
to extend or estimate the development of corrosion phenomenon special mathematics were applied.
Thus, Bessel functions led to precise solutions, using several calculus methods. The solving methods
with Bessel functions of the differential equation with partial derivatives led to identical solutions.

12. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 7, July (2014), pp. 101-112 © IAEME
112
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