Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2009, Article ID 319269, 17 pages
doi:10.1155/2009/319269
Research Article
Construction Solutions of PDE in Parametric Form
Alexandra K. Volosova and Konstantin Alexandrovich Volosov
Department of Applied Mathematics, Faculty of Computer Sciences, Moscow State University of Railway
Engineering, 141400 Himki, Moscow region, Russia
Correspondence should be addressed to
Konstantin Alexandrovich Volosov, konstantinvolosov@yandex.ru
Received 22 July 2008; Accepted 31 July 2009
Recommended by Attila Gilanyi
The new important property of wide class PDE was found solely by K. A. Volosov. We make
an arbitrary replacement of variables. In the case of two independent variables x, t, then it
always gives the possibility of expressing all PDE second and more order as AX b. This is a
linear algebraic equations system with regards derivatives to old variables xξ, δ, tξ, δ on new
variables ξ, δ : xξ , xδ , tξ , tδ . This system has the unique solution. In the case of three and more
independent variables x, y, t, . . . , then it gives the possibility of expressing PDE second order as
AX b, if we do same compliment proposes. In the present paper, we suggest a new method
for constructing closed formulas for exact solutions of PDE, then support on this important new
property.
Copyright q 2009 A. K. Volosova and K. A. Volosov. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
A new method of construction of exact solutions for partial differential equations PDE is
proposed in this article. The classical authors in mathematics used change of variables for
classification of linear PDE. However, they did not notice an important property of broad
class of PDE, which was discovered in Volosov’s articles 1–4. Literature reference 3
is available for review on http://eqworld.ipmnet.ru/ This property gives the possibility
of expressing PDE as AX b. This is a linear algebraic equations system with regards
derivatives to old variables on new variables. This equations system has the unique solution.
New identity was obtained which follows from solvability conditions of the system. The
methods of the above calculations and their consequence are described in this article. Let
us consider the:
Z t − KZZ x x FZ 0.
1.1
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Let’s describe the proposed method based on 1.1. The proposed algorithm works providing
that all functions are continuously differentiable functions.
Let’s make an arbitrary replacement of variables:
Zx, t |xxξ,δ, ttξ,δ Uξ, δ.
1.2
The inverse replace of variables defines the function Zx, tof 1.1 from the function Uξ, δ:
Zx, t Uξ, δ |ξξx,t, δδx,t .
1.3
We note that det J x ξ t δ − t ξ x δ /
0 is nonezero and not interminable, where
J
x ξ t ξ
x δ t δ
.
1.4
An inverse transformation exists, at least locally:
ξ ξx, t,
δ δx, t.
1.5
The derivatives of the old independent variables on the new variables are determind
as follows:
∂x
∂δ
det J ,
∂ξ
∂t
∂t
∂δ
− det J ,
∂ξ
∂x
∂x
∂ξ
− det J ,
∂δ
∂t
∂t
∂ξ
det J .
∂δ
∂x
1.6
Let us introduce the following relation:
∂Z KZ
Y ξ, δ,
∂x xxξ,δ, ttξ,δ
∂Z KZ
T ξ, δ.
∂t xxξ,δ, ttξ,δ
1.7
Using 1.6 and 1.7, we obtain the formulas:
∂U ∂t ∂U ∂t
−
Y ξ, δ x ξ t δ − t ξ x δ ,
KUξ, δ
∂ξ ∂δ ∂δ ∂ξ
∂U ∂x ∂U ∂x
KUξ, δ −
T ξ, δ x ξ t δ − t ξ x δ .
∂ξ ∂δ ∂δ ∂ξ
1.8
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Equation 1.1 takes the form
T ξ, δ − KU
∂Y/∂ξ∂t/∂δ − ∂Y/∂δ∂t/∂ξ
KUFU 0.
x ξ t δ − t ξ x δ
1.9
Let us rewrite 1.7 in the form
∂Zx, t
Y ξ, δ
,
∂x
KU|ξξx,t, δδx,t
T ξ, δ
∂Zx, t
.
∂t
KU|ξξx,t, δδx,t
1.10
As Z is continuously differentiable function, with the necessary of ∂/∂tZ x ∂/∂xZ t in the variables ξ, δ.
Taking into consideration on 1.6, 1.7, we can write this equality in the form
Y
∂x ∂
Y
∂t ∂
T
∂t ∂
T
∂x ∂
−
0.
−
∂δ ∂ξ KU
∂ξ ∂δ KU
∂δ ∂ξ KU
∂ξ ∂δ KU
1.11
System 1.8–1.11 will be analyzed in two equationstages. At the first stage,
we consider system 1.8–1.11 as a nonlinear algebraic equationsystem regarding the
derivatives x ξ , x δ , t ξ , t δ .
Theorem 1.1. The implicit linear algebraic equationsystem 1.8–1.11, with regarding the
derivatives x ξ , x δ , t ξ , t δ , has the unique solution:
∂x
Ψ1 ξ, δ,
∂ξ
∂x
Ψ2 ξ, δ,
∂δ
∂t
Ψ3 ξ, δ,
∂ξ
∂t
Ψ4 ξ, δ,
∂δ
1.12
where
def
Ψ1 ξ, δ K − FKU ξ U δ T ξ − T δ U ξ
2
2
− −T U δ Y ξ − T T δ U ξ T U δ T ξ U ξ − Y Y δ T ξ U ξ
T Y δ Y ξ U ξ Y T δ U ξ Y ξ
/P1 ξ, δ,
1.13
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def
2
Ψ2 ξ, δ K −FKU δ U δ T ξ − T δ U ξ − T T ξ U δ Y Y δ T ξ U δ
2
T T δ U ξ U δ − Y T δ Y ξ U δ T Y δ Y ξ U δ − T Y δ U ξ /P1 ξ, δ,
K −Y Y ξ FKU ξ T U ξ U δ Y ξ − Y δ U ξ
,
Ψ3 ξ, δ P1 ξ, δ
def K−Y Y δ FKU δ T U δ U δ Y ξ − Y δ U ξ
Ψ4 ξ, δ ,
P1 ξ, δ
def
1.14
1.15
1.16
where
P1 ξ, δ FK T Y ξ − T ξ Y U δ Y T δ − T Y δ U ξ
T Y −U δ T ξ U ξ T δ Y 2 Y δ T ξ − T δ Y ξ
T 2 U δ Y ξ − Y δ U ξ ,
1.17
2
KU2 Y δ U ξ − U δ Y ξ
det J .
P1 ξ, δ
1.18
and moreover
Proof. Equation 1.11 is linear. Let’s divide the first equation 1.8 by Y , divide second
equation by T deduct second equation from first equation, and obtain linear equation.
Analogously, the linear equation arises from first equation 1.8 and equation 1.9.
We can express any three derivatives x ξ , x δ , t ξ , by means of one derivative. For
example, we can express them by means of tδ .
We can substitute it to 1.9 and obtain a linear algebraic equation for tδ !. This is valid
for any PDE of the second order. This is the substance of the newly discovered property of
PDE.
We obtained AX b.
Matrix A has the form
⎛
Y U δ −Y U ξ T U δ −T U ξ
⎜
⎜ a21
⎜
A⎜
⎜ 0
⎝
0
a22
a23
0
a33
0
0
⎞
⎟
a24 ⎟
⎟
⎟,
a34 ⎟
⎠
a44
1.19
where is a21 −KUY δ Y K UU δ , a22 KUY ξ − Y K UU ξ , a23 −KUT δ T K UU δ , a24 KUT ξ − T K UU ξ , a33 −Y Y δ FUKUU δ T U δ , a34 Y Y ξ −
FUKUU ξ − T U ξ , and a44 P1 ξ, δ. Vectors X, b have the form X x ξ , x δ , t ξ , t δ T ,
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b 0, 0, 0, b4 T , b4 Ψ4 ξ, δP1 ξ, δ. Vector symbol T means conjugation. Eigen values of
matrix A have the form
λ1 −Y Y δ FKU δ T U δ ,
λ2 Y 2 T δ Y ξ − Y δ T ξ − T FK T U δ Y ξ − Y δ U ξ
Y FK T U δ T ξ − T δ U ξ ,
√ 1
M KY ξ Y U δ − K UU ξ ,
λ3,4 M ± D ,
2
2 D 4Y K Y δ U ξ − U δ Y ξ KY ξ Y U δ − K UU ξ .
1.20
The author proposes the alternative classification for PDE solutions on Eigen values.
Eigen pair can be discovered easily. We are not going to discuss here their interesting
properties.
At the second stage, consider the new first-order system 1.12 with the functions x xξ, δ, t tξ, δ.
It is well known that the solvability of a system of this type is verified by calculating
the second mixed derivatives of the functions x xξ, δ and t tξ, δ on the arguments ξ
and δ:
x ξδ x δξ ,
t ξδ t δξ .
1.21
The central result of this article is as follows 1–4.
Theorem 1.2. (1) one has the new identity
∂Ψ1 /∂δ − ∂Ψ2 /∂ξ ∂Ψ3 /∂δ − ∂Ψ4 /∂ξ
≡
,
T
Y
1.22
where the functions Ψi , i 1, . . . , 4 have the form 1.13–1.17.
(2) Two solvability conditions 1.21 of system 1.12 have multiply coefficient (or record
monomial factor) of arbitrary functions U, Y, T
∂
∂
Ψ3 − Ψ4 0,
∂δ
∂ξ
1.23
where Ψ3 , Ψ4 are the right-parts in 1.15, 1.16.
Corollary 1.3. If some free functions U, Y, T satisfy the condition 1.23, then systems 1.12 and
1.8– 1.11 are solvable.
This property Theorems 1.1 and 1.2 of second-order partial differential equations
was not known before.
The specific ways of satisfying the conditions of 1.23 are discussed below.
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Example 1.4. Let’s consider the Zel’dovich equation, which is well known in combustion
theory 5,Kolmogorov, Petrovsky, Piskunov 6–8:
Z t − Z xx − Z2 1 − Z 0.
1.24
Let’s consider 1.1 with KZ 1, and FZ Z2 1 − Z.
Suppose that Gξ, U is a function of two variables and wU, vU are functions of
one variable. We seek the functions Y ξ, δ, T ξ, δ in 1.7 in the form Y ξ, δ Gξ, U hU, T ξ, δ wU vUGξ, U, where U Uξ, δ.
In the articles 3, 4 the case hU 0 was considered.
Theorem 1.5. Let one Gξ, U ∈ C2 R ⊗ R, wU, hU, vU ∈ C2 R. Condition 1.23 takes
the form
v U G3 2 h − v v U vh − 2hv − w U G2
3U2 −1 U 2hv 2w − 4hh v U − 2hvh h2 v 2hw U G
1.25
U−2 3Uhv U2 − 3Uw 1 − UU2 vh 2h −U2 U3 w − hh v
U − 1U2 w − h2 vh h2 w 0.
We can try to satisfy this 1.25 by making equal the coefficients of the powers of G to
zero. We obtain a system of four equations for the two functions w, and v:
v U 0,
2 h − v v U vh − 2hv − w U 0,
3U2 −1 U 2hv 2w − 4hh v U − 2hvh h2 v 2hw U 0,
U−2 3Uhv U2 − 3Uw 1 − UU2 vh 2h −U2 U3 w − hh v
U − 1U2 w − h2 vh h2 w 0.
1.26
1.27
1.28
1.29
It turns out that in a number of interesting cases, all the four equations can be solved.
Moreover, the function Gξ, U, as well as U!, remains arbitrary.
√
√
2
Equation 1.25 can
√ be solved by setting vU 3U/ 2 − 1/ 2, wU 3U 1 −
U/2 −1 3UhU/ 2.
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System 1.12 has the form
w vGG ξ − Fv vw v2 G G2 v Gw − w vGG ξ U ξ
∂x
,
∂ξ
P1
Gv wG δ − Fv v2 G vw G2 v Gw − w vGG U U δ
∂x
,
∂δ
P1
−GG ξ F Gv w − GG U U ξ
∂t
,
∂ξ
P1
∂t
F w Gv − GG U U δ
,
∂δ
P1
1.30
1.31
1.32
where P1 Fw w2 vwG − G2 Gv w . The Jacobian has the form J G ξ U δ /P1 .
The exact solutions of the PDE are already constructed, since relations 1.21, 1.23
were satisfied. However, usually it require more detailed formulas. Let’s extend reviewed
situation.
Let’s choose the function Gξ, U so that the last two equations 1.32 of the system
take the elementary form
tξ, δ ξ,
t ξ 1,
t δ 0.
1.33
Then, we have the following system of two equations for Gξ, U,:
G U 1 − UU2 √
23U − 1G −
√
√
2h 3 2Uh − 2Gh − 2hh
2G h
2 √
√
3 2UU − 1 − 2G − 2h 2G 2h − 2U2
G ξ .
√
8 2G h
,
1.34
Integrating this system, we obtain:
√
C1 3ξ
21 − U
√
2
2UU − 1 − 2G − 2h
√
− ln 2UU − 1 − 2G − 2h
√
ln 2U2 − 2G − 2h .
We still need to analyze the first pair of equations 1.31 for the function xξ, δ.
1.35
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It turns out that, if we can do the variables replacement xξ, δ χU, R, U Uξ, δ, R Gξ, U, then system 1.30–1.32 was integrated. After that, we can return
to the variables ξ, and δ, which have the form
x
√
√
2
C2 2 lnU − 1 − 2 lnU
3
√
4 − lnU − 1 lnU ln −2G 2U − 1U − 2h
√
− ln −2G 2U2 − 2h
√
2 /
× 1 − 3U2 2 21 − 3Uh U 2 h
√
√ 2 3 21 − 3U2 41 − 3Uh U 2 2 h
1.36
√
√
− 2 2U − 1 −1 3U3 − 3 21 − 3U2 h U
√ 3 2
6−1 3U h − 2 2 h
/
√
√ 3 2G − 2U − 1U 2h 1 − 3U 2h
×
√
√ 2 .
21 − 3U2 41 − 3Uh U 2 2 h
We have 1.35, 1.36 exact solution equation 1.24 in parametric form.
The authors believe that the solution 1.36 cannot be constructed based on the classic
technique of the group analysis. However, if we consider hU 0 we have the possibility to
return to the original variables x, t. This equationsimplified solution can be constructed by
other methods, for example, by the method of Satsuma-Hirota.
We can came back to the variables x, t. Let’s introduce the notation
t ξ,
xξ, δ Xt, sx, t,
δ sx, t,
G rt, Ut, sx, t,
1.37
and hU 0.
Let’s consider 1.24 and the change of variables:
Zx, t Ut, sx, t.
1.38
Let a t > 0, and let Ut, s ∈ C2 R ⊗ R, have that the relation for the function rt, U rt, Ut, sx, t takes the form
√
2 21 − U
2 √
2 √
− ln 2UU − 1 − 2r ln 2U2 − 2r ,
C1 t √
3
3 2UU − 1 − 6r 3
1.39
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see 1.35, h 0. Also relation 1.36 takes the form
√
√
√
√ 2
1−U
2
.
2 ln 2r − 2U − 1U − 2 ln 2r − 2U
Xt, s C1 √
3 −2r 2U − 1U
1.40
We have a system of two equations for Xt, s,
∂X
1
3U 3U2 U − 1 U t
√ −√ ,
∂t
2r
r
2
2
∂X U s
.
∂s
r
1.41
This is system 1.30-1.31 for h 0.
Theorem 1.6. Suppose that 1.39, 1.40 are hold, and hU 0. In this case, 1.24 can be solved.
Exact solution equation 1.24 has the form Zx, t Ut, sx, t,
√ 2 expt/2 − 2 exp x/ 2
Zx, t √ √ .
2 expt/2 exp x/ 2 −2 2t x 2
1.42
√
Proof. Let’s express
from 1.40 ln
1 − 2r and substitute in 1.39: rt, U 1 −
√
√ 2UU − √
21 tU UX/ 2C1 − 2t − 2X. We will substitute r in 1.40 and raise it
U−C1 U √
to power 3/2 2 and proceed to exponential functions. Here have put a constant of shift on
X 1.40, C1 0. It always can be restored.
Consider the case hU /
0.
Theorem 1.7. Suppose hU / 0. Exact solution equation 1.24 has the form 1.35, 1.36.
This is a new real family of solution to 1.24.
Example 1.8. Consider the Fitz-Hugh-Nagumo-Semenov semilinear parabolic partial differential equation, which is well known in biophysics theory 6, 9:
Zt − Zxx
− Z 1 − Z2 0.
1.43
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Here we show one more way of integration of system 1.30–1.32. Suppose that Gξ, U ξ
!, wU, vU are functions of one variable. We seek the functions Y ξ, δ, T ξ, δ in 1.7
in the form Y ξ, δ ξ, T ξ, δ wU vUξ, where U Uξ, δ.
√ In this case, equation analogous 1.25 can be solved by setting vU 3U/ 2, wU 3U1 − U2 /2. To avoid any doubts, let’s express 1.30–1.32 in an obvious
form:
x ξ √ √
√
√ 2U 1 − U2 2ξ − 2U2 2U4 − 2ξ − 2 2ξ2 U ξ
P1
x δ t ξ √
√
√ − 2U2 2U4 − 2ξ − 2 2ξ2 U δ
P1
√ −2 2ξ U −1 U2 − 3 2ξ U ξ
t δ 3P1 √
2 U − U3 3 2Uξ U δ
3P1 ,
,
1.44
,
.
Suppose G ξ, and calculate integrals !.
The denominator in the system of ODE 1.31–1.32 can be presented in the form of
six multipliers:
P1 Fw w2 vwG − G2 Gv w
√
√
√
U2 − 2U4 U6 3 2U2 ξ − 3 2U4 ξ − 2ξ2 6U2 ξ2 − 2 2ξ3
⎞⎛
⎞⎛
⎛
√ ⎞
√
√
1
4
2ξ
U
−
−1
−
2 2ξ ⎟⎜ U −
2 2ξ ⎟⎜
⎟
⎜U ⎟
⎜
⎝
⎠⎝
⎠⎝
⎠
2
21/4
21/4
√ ⎞⎛
√ ⎞
U − 1 − 1 4 2ξ
U − −1 1 4 2ξ
⎜
⎟⎜
⎟
⎟⎜
⎟
×⎜
⎝
⎠⎝
⎠
2
2
⎛
√ ⎞
U − 1 1 4 2ξ
⎜
⎟
⎟.
×⎜
⎝
⎠
2
⎛
1.45
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Then the exact solution of 1.43 can be written in parametric form as
⎛
⎡
⎤
√
√
√
2 8ξ 2 8 2ξ
⎜
⎢
⎥
2U
1
⎜
⎥
xξ, δ × arctan⎢
⎣
⎦
√ ⎝
√
√
√
√
2 8 2ξ
−1 − 2 2ξ − 1 4 2ξ
−1 − 2 2ξ − 1 4 2ξ
⎡
⎤⎞
√
√
√
− 2 − 8ξ 2 8 2ξ
⎢
⎥⎟
2U
⎥⎟,
× arctan⎢
⎣
√
√
√
√ ⎦⎠
−1 − 2 2ξ 1 4 2ξ
−1 − 2 2ξ 1 4 2ξ
⎡
√
⎤
⎡
√
⎤
2 2ξ ⎥ 2 ⎢ U 2 2ξ ⎥
2 ⎢U −
tξ, δ − ln⎣
⎦ − ln⎣
⎦
1/4
1/4
3
3
2
2
⎡
√ ⎤
√ ⎤
U −1 1 4 2ξ
U 1 1 4 2ξ
⎥ 1 ⎢
⎥
1 ⎢
⎥ ln⎢
⎥
ln⎢
⎣
⎣
⎦
⎦
3
2
3
2
⎡
⎡
√ ⎤
√ ⎤
U − −1 1 4 2ξ
U − 1 1 4 2ξ
⎥ 1 ⎢
⎥
1 ⎢
⎥ ln⎢
⎥.
ln⎢
⎣
⎣
⎦
⎦
3
2
3
2
⎡
1.46
Theorem 1.9. Suppose that 1.46 is hold. Exact solution equation 1.43 has the form Zx, t Uξ, δ|δδx,t, ξξx,t .
After that, we came back to the variable ξ from 1.46 which takes the form
−U2 exp3t U2 − 1 ± U2 exp3t1 − U2 .
ξ
√ 2 −1 exp3t
1.47
Also δ is arbitrary parameter.
Let’s substitute this expression in the first parity 1.46 x xξ, U and we will receive
the solution in the parametrical form. This is a new real family of solution to 1.43.
Example 1.10. In the three-dimensional case, consider the semilinear parabolic and the change
of variables:
Z t − Z xx − Z yy ft, Z 0,
Z x, y, t |xxξ,δ,τ, yyξ,δ,τ, ttξ,δ,τ Uξ, δ, τ.
1.48
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Suppose that the Jacobian J /
0. Suppose also that there exists at least locally the inverse
transformation ξ ξx, y, t, δ δx, y, t, τ τx, y, t, and the derivatives an related as
∂τ y δ x ξ − x δ y ξ
,
∂t
J
x δ t ξ − t δ x ξ
∂τ
,...
∂y
J
1.49
See 3. We set
∂Z
|xxξ,δ,τ, yyξ,δ,τ, ttξ,δ,τ Y ξ, δ, τ,
∂x
∂Z
|xxξ,δ,τ, yyξ,δ,τ, ttξ,δ,τ Mξ, δ, τ,
∂y
1.50
∂Z
|xxξ,δ,τ, yyξ,δ,τ, ttξ,δ,τ T ξ, δ, τ.
∂t
The functions Y, M, and T are unknown.
Let’s supplement the relations written above by the equalities of mixed derivatives
Zx,y Zy,x ,
Zx,t Zt,x ,
Zy,t Zt,y
1.51
in the variables ξ, δ, and τ.
The nonlinear algebraic system of seven equations in the variables ξ, δ, τ is similar to
system 1.7–1.9, which follows from system 1.48–1.51 with respect to the nine variablederivatives, x ξ , x δ , x τ , y ξ , y δ , y τ , t ξ , t δ , t τ ,which are underdetermined. Hence, there
is much arbitrariness in the choice of functions. In this case we used ”method C” from 3.
Solutions of 1.51 have the form
− T ξ, δ, τt ξ − U ξ Y ξ, δ, τx ξ
,
yξ M
y δ −T ξ, δ, τt δ − U δ Y ξ, δ, τx δ ,
M
y τ −T ξ, δ, τt τ − U τ Y ξ, δ, τx τ .
M
1.52
By integrating the all resulting relations 1.51 and analogous equation 1.9 we have the
theorem.
Theorem 1.11. Suppose that Gξ, δ, τ, U is a twice continuously differentiable function of four
variables and ht, U, wt, U, rt, U are twice continuously differentiable functions of two
variables, where U Uξ, δ, τ.
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Then, the exact solution of 1.48 can by written in parametric form as
T ξ, δ, τ c1 wt, U wt, U
Mξ, δ, τ wt, U,
!U
w t ds
,
2
Uo w t, s
Y ξ, δ, τ c2 wt, U,
xξ, δ, τ rt, U ht, UGξ, δ, τ, U,
yξ, δ, τ −c1 tξ, δ, τ − c2 rt, U − c2 Gξ, δ, τ, Uht, U
!U
ds
c3 , ci / 0, i 1 − 3.
wt,
s
Uo
1.53
The function wt, U is determined from the differential
ft, U
c1 wt, U
!U
wt t, sds
− 2w U t, U 0.
2 t, s
w
Uo
1.54
The twice differentiable functions Gξ, δ, τ, U, rt, U, ht, U, tξ, δ, τ, as well as the
function U!, remain arbitrary.
If f fZ, in 1.48, then hU, wU, rU are functions of one variable. The
function wU is determined from the first-order ordinary differential equation known as
the Abel equation
fU wU c22 1 w U − c1 .
1.55
Example 1.12. Let’s consider the modification equation Fisher-Kolmogorov-PetrovskyPiskunov FKP P 8:
FZ 0,
Zt − Zxx
Z−∞, t a0 ,
Z∞, t− > a1 ,
1.56
where FZ 2Z3 λ2 /9 ± a0 − Za1 − Z. In 5 there were parameters λ 0, a0 0,
a1 1, a1 > a0 .
We can choose FU for 1.1, then vU ±3U/2λ Uλ, wU −3FU/2 and the
relations 1.23, 1.25 hold.
Systems 1.12 has the form:
∂x w vGG ξ − Fv vw v2 G G2 v Gw − w vGG ξ U ξ
,
∂ξ
P1
∂x Gv wG δ − Fv v2 G vw G2 v Gw − w vGG U U δ
,
∂δ
P1
∂t −GG ξ F Gv w − GG U U ξ
,
∂ξ
P1
∂t
F w Gv − GGU Uδ
.
∂δ
P1
1.57
1.58
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International Journal of Mathematics and Mathematical Sciences
The functions P1 , J have the form P1 Fw w2 vwG − G2 Gv w , J G ξ U δ /P1 .
The exact solutions of the PDE are already constructed, since relations 1.23, 1.25
were satisfied. However, usually it require more detailed formulas.
Let’s assume a0 0, a1 1, Gξ, U ξ. The systems 1.57, 1.58 have the form:
3 −6Uλ2 −9 9U 2U2 λ2 18λ 3 2Uλ2 ξ Q1 U ξ
∂x
,
∂ξ
λP2
∂x 3Q1 U δ ,
∂δ
λP2
∂t −6 18λξ Q2 U ξ
,
∂ξ
P2
−3Q2 U δ
∂t
,
∂δ
P2
1.59
1.60
where
Q1 −27Uλ 27U2 λ − 18U2 λ3 24U3 λ3 4U4 λ5 − 81ξ − 54λ2 ξ − 36λ3 ξ2 ,
Q2 −9Uλ 9U2 λ 2U3 λ3 − 27ξ − 18Uλ2 ξ,
P2 −9 9U 2U2 λ2 − 6λξ −9U2 λ 9U3 λ 2U4 λ3 − 27Uξ − 12U2 λ2 ξ 18λξ2
−9 9U 2U2 λ2 − 6λξ ∗ 3Uk2 k 9U 4U2 − 12λξ2
×
k
"
1.61
−3Uk2 k 9U 4U2 − 12λξ2
,
8λk2
9 8λ2 .
Denominator P2 has three square multipliers. We can integration second and fourth
equations 1.59.
Theorem 1.13. The exact solution of 1.56 Zx, t ∈ a0 , a1 can be written in parametric form of
systems 1.57, and 1.58, where vU ±3U/2λ Uλ, wU −3FU/2, and if a0 0, a1 1, and Gξ, U ξ that one has system 1.59.
Exact solution equation of 1.56 in parametric form has the form:
Zx, t Uξ, δ|ξξx,t
1/2 2
−9E1 − 3ηkE2 ± 9E1 3ηkE2 24E1 − 2kE2 λ2 −4kλξE2 E1 3 2λξ
/
4λ2 E1 − 2kE2 |ξξx,t ,
1.62
where k √
9 8λ2 , η k − 3, and E1 exp3kt 4xλ/2η, E2 exp3t/2η.
International Journal of Mathematics and Mathematical Sciences
15
Proof. Exact solution of system 1.59 has the form:
−2 ln−9 9U 2U2 λ2 − 6λξ
3
−k 3 −
ln−3Uk2 k 9U 4U2 λ2 − 12λξ 3k
3 k ln3Uk2 k 9U 4U2 λ2 − 12λξ ,
3k
1 x ln−9 9U 2U2 λ2 − 6λξ
λ
−k 3 4λ2 ln−3Uk2 k 9U 4U2 λ2 − 12λξ 2λk
2λ1 − k
2
2 2
−
ln
k
9U
4U
λ
−
12λξ
3Uk
.
9 8λ2 − 3k
t
1.63
Hence, from 1.63 we have
4xλ 3tk exp
−9 9U 2U2 λ2 − 6λξ
−6 2k
3t
3Uk2 − k 9U 4U2 λ2 − 12λξ 0.
exp
−6 2k
1.64
2. Conclusion
The following new facts complementing PDE theory were discovered in the study of Mr. K.
A. Volosov.
1 The system of four first-order equations 1.8–1.11 which is replacing the secondorder PDE 1.1 is a system of linear algebraic equations regarding derivatives
x ξ , x δ , t ξ , t δ and has unique solution.
2 There is a new identity which allows to record monomial factor 1.23 under the
solvability conditions. It is sometimes called closure condition.
3 Monomial factor 1.23 is the closure condition; the solvability condition is one
between ness relation for three arbitrary functions U, Y, and T . This property was
not noticed earlier. It is useful to consider the respective closure condition together
with each differential equation.
4 Using this approach it is possible to construct new families of exact solutions in
the parametric form, the examples of which were presented in this article. These
solutions cannot be constructed using the classical method of batch properties
analysis.
5 It is useful to study the Eigen pair’s properties for matrix A. The alternative
classification of PDE solutions is possible through matrix A eigenvalues.
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International Journal of Mathematics and Mathematical Sciences
Below you can find the extract from the comments to the article of Mr. M.V.Karasev,
Doctor of Physics and Mathematics, Professor, Head of Chair of Applied Mathematics of
Moscow State Institute of Electronics and Mathematics, Laureate of State award of Russian
Federation.
The article contains several interesting solutions and whole classes of solutions for
nonlinear differential equations with partial secondary derivatives PDE important in
applications. Sometimes in the theory of PDE the approach is used when by making certain
transformation e.g. by replacement of variables which is converted in order to bring it
to another equation whose solution is already known. Thus, the known solution generates
nontrivial solution of initial equation. In Mr. K.A. Volosov’s approach it is suggested a
priori not fixing the type of replacement of variables, leaving that arbitrary on the first
stage. The system derived after replacement of variables contains both required vectorfunction sought after solution and its derivatives and unknown coordinate transformation
Jacobi’s matrix of variables replacements. The number of equations in the system is always
less than the number of variables which opens possibilities for construction of solution by
introduction of different interconnections between the components of sought after vectorfunction. Limitation for the choice of connections requires consistency of the equation for
coordinate replacement function co-ordinate replacement function must be consistent. A
condition of consistency is mandatory for the remaining independent components of sought
after vector-function.
For example, in classic case, for PDE second order with two independent variables,
the sought after vector-function has three components; coordinate replacement contains two
more functions; and all of them together subordinate to four differential equations of firstorder. It was found that all the components of Jacobi 2 × 2 matrix of co-ordinate replacement
cab be explicitly expressed through the components of the sought after vector-function.
On this stage it is not required to introduce any additional connections. Then the received
formula for the Jacobi’s matrix is considered the system of first-order equations relative to two
functions of coordinate replacement, and the condition of this system consistency is noted.
It is reduced only to one equation, but contains three components of sought after vectorfunction. As a result, a big freedom appears to meet the consistency condition. On this stage
it is useful to introduce connections, which allows in many cases to reduce the condition of
consistency to a standard differential equation or even make it explicitly solvable. After that
it is necessary to return to construction of coordinate replacement, that is, to integration of the
system of first order, but this time with right part already known. To the extent this integration
can be executed explicitly in quadrature, to the same extent it is possible to construct the
explicit solution of the initial nonlinear with partial derivatives.
This algorithm is not tied to any group or symmetry attributes. The proposed
transformation of nonlinear differential multi variable equations has common nature with
the mechanism of integration. The coefficients and type of nonlinearity in the equation solved
are not specified, but remain general functions.
This work opens very interesting line in many areas of mathematics and its
applications dealing with nonlinear equations with higher partial derivatives.
Acknowledgments
The author is grateful to V. P. Maslov, M. V. Karasev, S. Yu. Dobrokhotov, V. G. Danilov, A. S.
Bratus for attention and advice.
International Journal of Mathematics and Mathematical Sciences
17
References
1 K. A. Volosov, “Gertsenovskie chteniya,” in Proceedings of Gertsen Readings Conference, St. Petersburg,
Fla, USA, April 2006.
2 K. A. Volosov, “Construction of solutions of quasilinear parabolic equations in parametric form,”
Differential Equations, vol. 43, no. 4, pp. 507–512, 2007.
3 K. A. Volosov, Doctoral dissertation, MIEM, Moscow, Russia, 2007, http://eqworld.ipmnet.ru.
4 K. A. Volosov, “Formulas for the exact solutions of quasilinear partial differential equations in implicit
form,” Rossiı̆skaya Akademiya Nauk. Doklady Akademii Nauk, vol. 418, no. 1, pp. 11–14, 2008.
5 Ya. B. Zeldovich, G. I. Barenblatt, V. B. Librovich, and G. M. Makhviladze, Mathematical Theory of
Combustion and Explosion, Nauka, Moscow, Russia, 1980.
6 V. P. Maslov, V. G. Danilov, and K. A. Volosov, Mathematical Modeling of Heat and Mass Transfer Processes
(Evolution of Dissipative Structures), Nauka, Moscow, Russia, 1987.
7 V. G. Danilov, V. P. Maslov, and K. A. Volosov, Mathematical Modelling of Heat and Mass Transfer Processes,
vol. 348 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands,
1995.
8 A. N. Kolmogorov, I. G. Petrovskii, and I. S. Piskunov, “A study of the diffusion equation with increase
in the quantity of matter, and its application to a biological problem,” Moscow State University Bulletin
A, vol. 1, no. 6, pp. 1–25, 1937.
9 K. A. Volosov, “Construction of solutions of quasilinear parabolic equations in parametric form,”
Sibirskii Journal of Industrial’noi Matematiki, vol. 11, no. 234, pp. 29–39, 2008, English translation in
Journal of Applied and Industrial Mathematics.