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P.1 (1-19)
by:Rima p. 1
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J. Math. Anal. Appl. ••• (••••) •••–•••
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www.elsevier.com/locate/jmaa
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Nonlocal transformations of Kolmogorov equations
into the backward heat equation
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George Bluman
a,∗
and Vladimir Shtelen
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b
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a Department of Mathematics, University of British Columbia, Vancouver, BC, Canada V6T 1Z2
b Mathematics Department, 110 Frelinghuysen, Rutgers University, Piscataway, NJ 08854, USA
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Received 3 October 2003
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Submitted by K.A. Ames
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Abstract
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Keywords: Kolmogorov equation; Nonlocal transformation; Backward heat equation; Fokker–Planck equation
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We extend and solve the classical Kolmogorov problem of finding general classes of Kolmogorov
equations that can be transformed to the backward heat equation. These new classes include Kolmogorov equations with time-independent and time-dependent coefficients. Our main idea is to
include nonlocal transformations. We describe a step-by-step algorithm for determining such transformations. We also show how all previously known results arise as particular cases in this wider
framework.
 2003 Published by Elsevier Inc.
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1. Introduction
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∂ 2p
∂p
∂p
+ a(t, x) 2 + b(t, x)
= 0, a(t, x) > 0.
(1)
∂t
∂x
∂x
The Kolmogorov equation (1) is the partial differential equation (PDE) satisfied by the
Green’s function for the Fokker–Planck equation
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In this paper, we extend previous work on finding one-dimensional Kolmogorov equations that can be mapped into the backward heat equation. Let p(t, x) satisfy a Kolmogorov
equation
* Corresponding author.
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E-mail addresses: bluman@math.ubc.ca (G. Bluman), shtelen@math.rutgers.edu (V. Shtelen).
0022-247X/$ – see front matter  2003 Published by Elsevier Inc.
doi:10.1016/j.jmaa.2003.11.028
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G. Bluman, V. Shtelen / J. Math. Anal. Appl. ••• (••••) •••–•••
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∂2 ∂q
∂ − 2 a(t, x)q +
b(t, x)q = 0.
∂t
∂x
∂x
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P.2 (1-19)
by:Rima p. 2
1
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We find new sets of coefficients {a(t, x), b(t, x)} for which (1) can be transformed to the
backward heat equation
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∂ p̄
+ 2 = 0.
∂ t¯
∂ x̄
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x
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X = X(t, x) =
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p(t, x),
where
∂ 2X
∂X
∂X
+ a(t, x) 2 + b(t, x)
D(T , X) =
∂t
∂x
∂x
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D(t ,Z) dZ
and, in terms of D(T , X), the coefficient G(T , X) of (3) satisfies
∂G
∂D
1 ∂D ∂ 2 D
+
D
=−
+
.
∂X
2 ∂T
∂X
∂X2
From (6), it follows that if we set
∂
logθ (T , X),
D(T , X) = 2
∂X
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(3)
∂θ
+
+ G(T , X) + Γ (T ) θ = 0
2
∂T
∂X
∂ 2θ
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(4)
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(5)
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(6)
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(7)
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(8)
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for arbitrary Γ (T ).
Let u1 = u1 (T , X) = P (T , X) and let G1 (T , X) = G(T , X). Then Eq. (3) becomes the
canonical PDE
∂u1 ∂ 2 u1
+
+ G1 (T , X)u1 = 0.
∂T
∂X2
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then θ (T , X) satisfies the backward equation
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P (T , X) = e
X
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1/2
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(9)
Consequently, the problem of mapping (1) into (2) reduces to mapping PDE (9) to the
backward heat equation (2).
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dz
,
√
a(t, z)
DP
T = t,
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through the transformation
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∂ 2P
∂P
+
+ G(T , X)P = 0
∂T
∂X2
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(2)
The Kolmogorov equation (1) can be transformed to the backward canonical PDE
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∂ 2 p̄
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P.3 (1-19)
by:Rima p. 3
G. Bluman, V. Shtelen / J. Math. Anal. Appl. ••• (••••) •••–•••
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In his celebrated paper [9], Kolmogorov posed the problem of finding the most general
equation of the form (1) that can be mapped into the backward heat equation (2) by a point
transformation. Cherkasov ([8]; see also [2,11,12]) partially solved this problem by restricting himself to a special class of point transformations considered by Kolmogorov [9].
The complete solution of Kolmogorov’s problem was given in [3].
The main idea of the present paper is as follows. Suppose the Kolmogorov equation (1)
can be embedded in an auxiliary system of PDEs so that the set of all solutions of the
auxiliary system yields all solutions of the Kolmogorov equation but there is not a oneto-one correspondence between solutions of the Kolmogorov equation and those of the
related auxiliary system. Then a point transformation of the variables of the auxiliary
system, which maps a component of the auxiliary system into the backward heat equation, could yield a nonlocal transformation which maps the given Kolmogorov equation
to a backward heat equation. We exhibit such nonlocal transformations, yielding wider
classes of Kolmogorov equations transformable to the backward heat equation than those
previously obtained by point (local) transformations. This is accomplished by embedding
a Kolmogorov equation in an auxiliary “potential” system obtained through replacement
of the Kolmogorov equation by an equivalent conservation law [4].
In Section 2 we present the previously known results on mapping Kolmogorov equations
to the backward heat equation by point transformations.
In Section 3 we present our basic framework for obtaining mappings by nonlocal
transformations. We begin by observing that any solution of the adjoint equation of the
canonical PDE (9) yields a factor for an equivalent conservation law. In turn this conservation law yields a potential system. We then find the most general canonical PDE for
which a point transformation maps a corresponding potential system to a special potential
system related to the backward heat equation. Each component of this special potential
system satisfies the backward heat equation. Such a point transformation of a potential
system is shown to yield a nonlocal transformation of a canonical PDE to the backward
heat equation.
Using the basic framework, in Section 4 we give a step-by-step procedure to obtain new
classes of Kolmogorov equations that are transformable to the backward heat equation.
A theorem is proved which characterizes the richness of our extension of the known (local)
results. We show that our new classes arise from any solution of the backward heat equation
other than its fundamental solution (modulo translations) or its traveling wave solution.
In Section 5 we describe further generalizations emanating from our basic framework
through a recycling procedure which yields chains of Kolmogorov equations transformable
to the backward heat equation. In Section 6 this recycling procedure is shown to be of
particular value when the coefficients of a Kolmogorov equation are time-independent. In
Section 7, as examples we exhibit d-Bessel processes [10] which are only transformable to
the backward heat equation by nonlocal transformations. As a consequence, a spherically
symmetric (2k + 1)-dimensional heat equation can be mapped into the one-dimensional
heat equation only by a nonlocal transformation for k = 2, 3, . . . .
In Section 8 we discuss connections with symmetry analysis.
The approach presented in this paper can be used in other mapping problems. For example, new classes of Schrödinger equations transformable to the free particle equation by
nonlocal transformations were found in [7].
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P.4 (1-19)
by:Rima p. 4
2. Mappings by point transformations
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One can show that a point transformation
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τ = τ (T , X, u1 ),
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y = y(T , X, u1),
ũ1 = ũ1 (T , X, u1 ),
(10)
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maps any PDE (9) into a homologous PDE, namely
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∂ ũ1 ∂ 2 ũ1
+
+ G̃1 (τ, y)ũ1 = 0
∂τ
∂y 2
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for some G̃1 (τ, y) if and only if (10) is of the form
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T
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1
2σ̇ 2 − σ σ̈ 2 2σ̇ ρ̇ − σ ρ̈
G̃1 (τ, y) = 2 G1 (T , X) +
X +
X
σ
4σ 2
2σ 2
2
σ̇
ρ̇
− λ̇ ,
−
+
4σ 2 2σ
(11)
G1 (T , X) = α(T )X + β(T )X + γ (T )
2
The solution of system (15) is given in Appendix A.
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(12)
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(13)
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(14)
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for arbitrary α(T ), β(T ), γ (T ) (see [3]). The corresponding coefficients {α(t, x), b(t, x)}
were given in [3] and will be exhibited in Section 4.
From Eq. (13), we see that if G1 (T , X) is of the form (14), the mapping (12) which
transforms the corresponding PDE (9) to the backward heat equation has σ (T ), ρ(T ),
λ(T ) satisfying the system of ODEs
σ σ̈ − 2σ̇ 2
= α(T ),
4σ 2
σ ρ̈ − 2σ̇ ρ̇
= β(T ),
2σ 2
ρ̇ 2
σ̇
λ̇ = 2 −
+ γ (T ).
2σ
4σ
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where σ (T ), ρ(T ), λ(T ) are arbitrary functions of T and σ̇ = dσ/dT , etc.
Consequently, with respect to a point transformation, PDE (9) can be mapped into the
backward heat equation if and only if G1 (T , X) is of the form
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with
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7
DP
y = σ (T )X + ρ(T ),
τ = σ 2 (µ) dµ,
σ̇ 2
ρ̇
X +
X + λ u1 ,
ũ1 = exp
4σ
2σ
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(15)
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G. Bluman, V. Shtelen / J. Math. Anal. Appl. ••• (••••) •••–•••
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3. The basic framework for nonlocal transformations
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In Rn , for any given linear operator L, its adjoint operator L∗ is defined by
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Di f ,
where x = (x1 , . . . , xn ), the total derivative operator Di = ∂/∂xi , f i is a bilinear expression in u, φ and their derivatives, i = 1, 2, . . . , n. Consequently, if
∗
L φ = 0,
then Lu = 0 if and only if
n
i=1 Di f
Di f = 0
i
i=1
∂2
∂
+
+ G1 (T , X),
L=
∂T
∂X2
its adjoint L∗ is given by
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∂2
∂
+
+ G1 (T , X),
L =−
∂T
∂X2
and the conservation law (19) is given by
∂
∂φ
∂
∂u1
(φu1 ) +
−
u1 = 0.
φ
∂T
∂X
∂X
∂X
∗
∂ 2φ
L∗ φ = −
∂φ
−
+ G1 (T , X)φ = 0.
∂T
∂X2
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(18)
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(19)
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(20)
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(21)
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(22)
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The potential system corresponding to (22), with auxiliary dependent variable v1 (T , X), is
given by
∂φ
∂v1
∂u1
∂v1
= φu1 ,
=
u1 − φ
,
∂X
∂T
∂X
∂X
where φ(T , X) is any solution of the adjoint PDE
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for any φ satisfying its adjoint equation (17). Using (19), one can introduce an auxiliary
potential system whose compatibility conditions yield (18). The set of all solutions of such
a potential system yields the set of all solutions of (18) but the connection between these
solution sets is not one-to-one (see [4]). When n 3, one is confronted with the problem of
a natural gauge arbitrariness. In this case, as shown in [1], the associated potential system
must be augmented by gauge constraints.
Now we specialize to the situation when PDE (18) is the canonical PDE (9). Here the
linear operator L is given by
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(17)
= 0, i.e., a given linear PDE
is equivalent to a conservation law
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i
Lu = 0
n
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i=1
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(16)
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φLu − uL φ =
n
DP
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∗
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(23)
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(24)
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∂w ∂ 2 w
+
+ G2 (T , X)w = 0,
∂T
∂X2
with G2 (T , X) given by
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G2 (T , X) = G1 (T , X) + 2
∂2
∂X2
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log |φ|.
DP
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(27)
k
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(28)
From Eqs. (23) and (26), it immediately follows that if u1 (T , X) solves the canonical PDE
(9) then
X
1
u1 (T , ξ )φ(T , ξ ) dξ + B2 (T ) ,
(29)
w = u2 (T , X) =
φ(T , X)
TE
5
∂v1 ∂ 2 v1
2 ∂φ ∂v1
+
= 0.
(25)
−
∂T
φ ∂X ∂X
∂X2
Moreover, if u1 = U1 (T , X) solves the canonical PDE (9) and φ = Φ(T , X) solves
(24), then from the integrability conditions of (23) it follows that there exist solutions
(u1 (T , X), v1 (T , X)) = (U1 (T , X), V1 (T , X)) of potential system (23) with V1 (T , X)
unique to within an arbitrary constant. Hence for any given φ = Φ(T , X) which solves
(24), the relationship between the solution sets of PDE (9) and system (23) is not one-toone.
For any φ(T , X) that satisfies (24), the point transformation
v1
(26)
w=
φ
maps (25) to the canonical PDE
with B2 (T ) satisfying the condition
∂φ
∂u1
dB2
=
(T , k)u1 (T , k) − φ(T , k)
(T , k)
(30)
dT
∂X
∂X
for any constant k, solves the homologous equation (27) with G2 (T , X) given by (28). Conversely, if w(T , X) solves the canonical PDE (27) and φ(T , X) is any particular solution
of
∂ 2φ
∂2
∂φ
−
+
2φ
log |φ| − G2 (T , X)φ = 0,
∂T
∂X2
∂X2
then solving the forward PDE (24) in terms of G1 (T , X), i.e., setting
∂ 2φ
1 ∂φ
−
,
(31)
G1 (T , X) =
φ ∂T
∂X2
it follows that
∂w w ∂φ
u1 (T , X) =
+
∂X
φ ∂X
solves the homologous canonical equation (9) with G1 (T , X) given by (31).
By direct calculation one can prove the following theorem.
EC
4
1
RR
3
Note that if (u1 (T , X), v1 (T , X), φ(T , X)) solves the potential/adjoint system (23), (24),
then u1 (T , X) solves the canonical PDE (9) and v1 (T , X) solves the backward equation
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G. Bluman, V. Shtelen / J. Math. Anal. Appl. ••• (••••) •••–•••
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G. Bluman, V. Shtelen / J. Math. Anal. Appl. ••• (••••) •••–•••
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Theorem 1. Consider the canonical PDE (9). A point transformation maps the corresponding potential system (23) into the special backward heat equation potential system
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where
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(35)
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1 ∂ψ
σ̇ X + ρ̇
+
= 0,
ψ ∂X
2σ
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then Eq. (33) yields
G1 (T , X) = α(T )X2 + β(T )X + γ (T ) −
13
20
The previously known result [3] about the equivalence of the canonical PDE (9) and the
backward heat equation under a point transformation when the coefficient G1 (T , X) of (9)
is quadratic in X (Eq. (14)) immediately follows as a special case of Theorem 1. Indeed, if
ψ(T , X) satisfies
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19
ρ̇
σ̇ 2
X +
X + λ,
g(T , X) =
4σ
2σ
and (σ (T ), ρ(T ), λ(T )) are related to (α(T ), β(T ), γ (T )) through the system of ODEs
(15) which is solved in Appendix A. The mapping (35) defines a point transformation on
(X, T , u1 , v1 )-space that projects into a nonlocal transformation on (X, T , u1 )-space if the
coefficient of v1 is nonzero in Eq. (35c).
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y = σ (T )X + ρ(T ),
T
τ = σ 2 (µ) dµ,
1 g(T ,X)
σ̇ X + ρ̇ ψX
ψv1 ,
+
u1 +
û = e
σ
2σ
ψ
v̂ = eg(T ,X)ψv1 ,
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11
∂ψ
∂ 2ψ +
+ α(T )X2 + β(T )X + γ (T ) ψ = 0
(34)
2
∂T
∂X
for arbitrary α(T ), β(T ), γ (T ). (Note that ψ(T , X) satisfies (34) if φ(T , X) = 1/ψ(T , X)
satisfies the adjoint equation (24) with G1 (T , X) given by (33).) The corresponding mapping of (23) into (32) is given by
35
36
(33)
RO
13
6
8
DP
12
= 0,
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11
∂2
G1 (T , X) = 2 2 log |ψ| + α(T )X2 + β(T )X + γ (T ),
∂X
where ψ(T , X) is any solution of
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10
= 0, if and on if G1 (T , X) is of the form
OF
∂ û/∂τ
+ ∂ 2 û/∂y 2
4
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for which each component satisfies the backward heat equation, i.e., ∂ v̂/∂τ +∂ 2 v̂/∂y 2
8
9
(32)
TE
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2
3
∂ v̂
∂ û
=− ,
∂τ
∂y
∂ v̂
= û,
∂y
4
6
1
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σ̇
.
σ
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It is easy to check that the mapping (35) yields a nonlocal transformation of (9) to the
backward heat equation if and only if ψ(T , X) is a solution of (34) satisfying the condition
∂3
log |ψ| ≡ 0.
(36)
∂X3
Moreover, the resulting expression (33) for G1 (T , X) is not of the quadratic form (14) if
and only if ψ(T , X) satisfies the condition
OF
2
∂5
log |ψ| ≡ 0.
∂X5
Let ψ̂(τ, y) be any solution of the backward heat equation ∂ ψ̂/∂τ
from (12) it follows that
σ̇ 2
ρ̇
X +
X + λ ψ̂(τ, y)
ψ(T , X) = exp −
4σ
2σ
is a solution of (34), and hence (33) becomes
17
20
21
22
23
24
25
G1 (T , X) = α(T )X2 + β(T )X + γ (T ) − 2σ 2
27
35
36
37
38
39
40
41
42
43
44
45
σ̇
− ,
σ
∂ 3G
1
∂X3
=2
∂5
RR
34
log |ψ|.
for any solution ψ(T , X) of (34).
Now we show how the above result generalizes previous work presented in [3,8].
5
6
7
9
11
12
15
16
17
(38)
∂X5
Consequently, after using (6) with G = G1 , the work of the previous section can be restated
as follows.
Through one of our potential systems, the Kolmogorov equation (1) can be mapped into
the backward heat equation if and only if T , X, D(T , X) defined by (4) and (5) satisfy
∂D ∂ 2 D
∂5
∂ 2 ∂D
+
D
+
=
−4
log |ψ|
(39)
∂X
∂X2 ∂T
∂X2
∂X5
CO
33
ψ̂
2 4
14
Now we apply the results of Section 3 to the Kolmogorov equation (1). Let ψ(T , X) be
any solution of (34) and let G1 (T , X) be given by (33) for arbitrary α(T ), β(T ), γ (T ).
Then
UN
32
ψ̂
+
ψ̂y
3
13
4. New classes of Kolmogorov equations transformable to the backward
heat equation
30
31
= 0. Then
EC
29
ψ̂τ
2
10
T 2
where y = σ X + ρ, τ =
σ (µ) dµ, with σ (T ), ρ(T ) related to α(T ), β(T ) through
(15a,b). Hence through (38) every solution of the backward heat equation yields a coefficient G1 (T , X) for which the corresponding backward equation (9) can be mapped into
the backward heat equation. This relationship will be considered in more detail in the next
section.
26
28
DP
19
(37)
TE
18
+ ∂ 2 ψ̂/∂y 2
1
8
RO
1
P.8 (1-19)
by:Rima p. 8
18
19
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P.9 (1-19)
by:Rima p. 9
G. Bluman, V. Shtelen / J. Math. Anal. Appl. ••• (••••) •••–•••
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
and it was further shown that Cherkasov’s special class of point transformations restricted
D(T , X) to the solutions of (40) that satisfy
6
∂ 2D
= 0.
∂X2
In terms of the original independent variables (x, t), note that
∂X −1 ∂
∂
∂
=
= a(t, x) ,
∂X
∂x
∂x
∂x
−1
∂
∂
∂X ∂X
∂
∂
∂
=
−
=
− a(t, x) Xt .
∂T
∂t
∂t ∂x
∂x ∂t
∂x
8
T
24
y1 = σ1 (T )X + ρ1 (T ),
25
τ1 =
26
∂ 2 ψ̂
30
and let
38
39
40
41
42
43
44
45
4
5
7
9
12
13
14
15
16
17
18
19
20
21
22
23
24
(42)
25
26
27
28
(43)
29
30
31
32
33
34
35
(44)
yields a solution of (34).
(2) Using any ψ(T , X) given by (44), determine G1 (T , X) from (33), i.e.,
CO
37
ψ(T , X) = e−g1 (T ,X)ψ̂(τ1 , y1 )
∂2
G1 (T , X) = 2 2 logψ(T , X) + α(T )X2 + β(T )X + γ (T ).
∂X
(3) For any ψ(T , X) given by (44), and corresponding G1 (T , X) given by (33), use (7)
and (8), with G(T , X) = G1 (T , X), to determine D(T , X), i.e.,
∂
logθ (T , X),
D(T , X) = 2
∂X
UN
36
Then
RR
33
3
11
σ̇1 2
ρ̇1
g1 (T , X) =
X +
X + λ1 .
4σ1
2σ1
32
35
EC
∂ ψ̂
+ 2 =0
∂τ1
∂y1
29
34
σ12 (µ) dµ.
2
10
Let ψ̂(τ1 , y1 ) be any solution of the backward heat equation
28
31
(41)
The previous discussion leads to the following step-by-step procedure to find new classes
of Kolmogorov equations (1) that are transformable to the backward heat equation.
Let α(T ), β(T ), γ (T ) be arbitrary functions of T .
(1) Let ψ(T , X) be a solution of (34), found as follows.
For any given α(T ), β(T ), γ (T ), let σ = σ1 (T ), ρ = ρ1 (T ), λ = λ1 (T ) be any solution
of the system of ODEs (15), solved in Appendix A. Let
23
27
OF
4
1
RO
3
In [3] it was shown that the Kolmogorov equation (1) can be mapped into the backward
heat equation through a point transformation if and only if D(T , X) satisfies
∂D ∂ 2 D
∂ 2 ∂D
+D
+
= 0,
(40)
∂X2 ∂T
∂X
∂X2
DP
2
TE
1
9
36
37
38
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40
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10
where θ (T , X) is any solution of
2
4
8
for arbitrary Γ (T ). The procedure to solve PDE (45) now follows from the results of
Theorem 1 as applied to Eq. (45).
Let σ = σ2 (T ), ρ = ρ2 (T ) be any solution of (15) and let λ = λ2 (T ) be any solution of
9
λ̇ =
10
11
12
ρ̇22 − 2σ2 σ̇2
4σ22
+ γ (T ) + Γ (T ).
Let
T
13
14
σ̇2 2
ρ̇2
g2 (T , X) =
X +
X + λ2 .
4σ2
2σ2
24
25
26
27
28
29
30
31
32
33
34
Then
∂ θ̂
σ̇2 X + ρ̇2 ψX
−
+
θ (T , X) = e−g2 (T ,X) σ2
θ̂
∂y2
2σ2
ψ
TE
23
39
40
41
42
43
44
45
12
13
b(t, x) = 2a(t, x)
∂
1
logθ t, X(t, x) − a(t, x) Xt + ax (t, x)
∂x
2
15
17
18
19
20
21
22
23
(47)
24
25
26
27
28
29
30
31
32
33
In particular, for any solution θ (T , X) of (4.7), and arbitrary a(t, x), the coefficient
34
35
(48)
36
x √
with X(t, x) = (1/ a(t, z) ) dz.
(5) The procedure outlined in Theorem 1 relates the solution of the corresponding
canonical backward equation (3) (or (9)) to any solution of the backward heat equation.
(6) In terms of D(T , X), the corresponding solutions of the Kolmogorov equation (1)
are found through transformation (4).
In the above step-by-step procedure we now show which solutions of the backward
heat equation (43) yield Kolmogorov equations, i.e., coefficients a(t, x), b(t, x), which are
transformable to the backward heat equation only by nonlocal transformations.
37
CO
38
11
∂X
∂X
+ a(t, x) 2 + b(t, x)
= D t, X(t, x) .
∂t
∂x
∂x
∂ 2X
UN
37
8
10
yields a solution of (45) and hence leads to D(T , X) = 2(∂/∂X) log |θ (T , X)|.
(4) For any D(T , X) = 2(∂/∂X) log |θ (T , X)|, the corresponding coefficients {a(t, x),
b(t, x)} of the Kolmogorov equation (1) can be found as follows.
Let D(T , X) = D(t, X(t, x)), where X(t, x) is given by (4b). Then from (5), the coefficients {a(t, x), b(t, x)} are any solutions of
35
36
7
16
20
22
6
Let θ̂(τ2 , y2 ) be any solution of the backward heat equation ∂ θ̂/∂τ2 + ∂ 2 θ̂ /∂y22 = 0, and
let
19
21
5
9
EC
18
4
14
τ2 =
σ22 (µ) dµ.
RR
17
3
(46)
y2 = σ2 (T )X + ρ2 (T ),
15
16
OF
7
(45)
RO
6
2
∂ 2θ
∂θ
+
+ G1 (T , X) + Γ (T ) θ = 0
∂T
∂X2
3
5
1
DP
1
P.10 (1-19)
by:Rima p. 10
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P.11 (1-19)
by:Rima p. 11
G. Bluman, V. Shtelen / J. Math. Anal. Appl. ••• (••••) •••–•••
2
3
4
Theorem 2. Let ψ̂(τ1 , y1 ) be a solution of the backward heat equation (43). By following
the step-by-step procedure (1)–(5), such a solution yields the Kolmogorov equation (1) that
is transformable to the backward heat equation only through a nonlocal transformation if
and only if ψ̂(τ1 , y1 ) is not one of the forms
5
6
7
8
(I)
(II)
9
10
2
where P , y1 , τ̂1 are arbitrary constants.
∂5
16
∂X5
20
∂5
21
∂y15
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
Hence ψ̂(τ1 , y1 ) is of the form
ψ̂(τ1 , y1 ) = exp A(τ1 )y14 + B(τ1 )y13 + C(τ1 )y12 + D(τ1 )y1 + E(τ1 )
TE
22
logψ̂(τ1 , y1 ) ≡ 0.
dC
+ 4C 2 = 0,
dτ1
dE
dD
+ 4CD = 0,
+ 2C + D 2 = 0.
dτ1
dτ1
It is now straightforward to show that the solutions of (51) yield (48). ✷
8
12
13
14
15
16
17
18
19
20
21
22
23
24
(50)
for arbitrary A(τ1 ), B(τ1 ), C(τ1 ), D(τ1 ), E(τ1 ). Substituting (50) into the backward heat
equation (42), we see that
A(τ1 ) = B(τ1 ) ≡ 0,
7
11
DP
19
4
10
Consequently, (∂ 5 /∂X5 ) log |ψ̂(τ1 , y1 )| ≡ 0, which, as it follows from (42) (σ = 0), is
equivalent to
EC
18
RR
17
logψ(T , X) ≡ 0.
3
9
RO
15
25
26
27
28
29
30
(51)
For an arbitrary coefficient a(t, x) in the Kolmogorov equation (1), we next show how
the work presented in this paper generalizes previous works [3,8] on the types of coefficients b(t, x) for which (1) can be transformed to the backward heat equation. Making the
appropriate substitutions in (47), we obtain
√
√
σ̇2
ρ̇2
1
∂
logθ̂ (τ2 , y2 )
− a Xt + ax + 2a
X+
b(t, x) = −2 a
2σ2
2σ2
2
∂x
∂
∂
∂
+ 2a
logσ2
logθ̂ (τ2 , y2 ) − σ1
logψ̂(τ1 , y1 )
∂x
∂y2
∂y1
σ̇1
ρ̇1
σ̇2
ρ̇2 +
X+
−
−
,
(52)
2σ1 2σ2
2σ1 2σ2 CO
14
(49)
Proof. From condition (37), it follows that ψ̂(τ1 , y1 ) yields a Kolmogorov equation that is
transformable to the backward heat equation only by a nonlocal transformation if and only
if ψ(T , X) given by (44) satisfies (∂ 5 /∂X5 ) log |ψ(T , X)| ≡ 0. Now suppose
UN
13
2
6
11
12
1
5
ψ̂(τ1 , y1 ) = e(P y1−P τ1 ) ,
1
(y1 − ŷ1 )2
ψ̂(τ1 , y1 ) = ,
exp
4(τ1 − τ̂1 )
τ1 − τ̂1
OF
1
11
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G. Bluman, V. Shtelen / J. Math. Anal. Appl. ••• (••••) •••–•••
12
3
4
5
6
7
8
9
10
11
12
13
14
with X(t, x), σi (t), ρi (t), τi (t), yi (t, X(t, x)), i = 1, 2, defined by Eqs. (4), (42) and (46);
ψ̂(τ1 , y1 ), θ̂ (τ2 , y2 ) are any solutions of backward heat equations in terms of their respective arguments. Note that if σ1 = σ2 = σ , ρ1 = ρ2 = ρ, then τ1 = τ2 = τ , y1 = y2 = y. One
can show the following:
(I) Cherkasov’s results [8] correspond to σ1 = σ2 = σ , ρ1 = ρ2 = ρ, θ̂ (τ, y) = const,
ψ̂(τ, y) is a solution of the backward heat equation which is one of the special forms (49).
(II) The results presented in [3] also correspond to σ1 = σ2 = σ , ρ1 = ρ2 = ρ, ψ̂(τ, y)
is a solution of the backward heat equation which is one of the special forms (49), but,
unlike in Cherkasov’s results, θ̂ (τ, y) is allowed to be any solution of the backward heat
equation.
(III) The results presented in this paper yield further new classes of b(t, x), a(t, x) if
σ1 = σ2 , ρ1 = ρ2 . If σ1 = σ2 = σ , ρ1 = ρ2 = ρ, then further new classes of b(t, x), a(t, x)
beyond those found in [3], are obtained for any pair of solutions (θ̂(τ, y), ψ̂(τ, y)) of the
backward heat equation except when
OF
2
RO
1
17
(1) ψ̂(τ, y) is one of the special forms (49) or
(2) ψ̂(τ, y) = C θ̂ (τ, y) for some constant C.
18
19
20
21
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
45
8
9
10
11
12
13
14
19
20
21
22
27
(53)
26
28
31
32
33
34
35
36
37
38
39
40
41
42
with B2 (T ) satisfying condition (30), solves
∂u2
2
+
+ Γ2 (T , X)u2 = 0,
∂T
∂X2
25
30
k
∂ 2u
24
29
∂ψ
+
+ α(T )X2 + β(T )X + γ (T ) ψ = 0.
(54)
2
∂T
∂X
This recycling procedure generalizes previous work presented in [6] which dealt with a
T -independent Γ1 (T , X). To establish such chains we proceed as follows.
Suppose u1 (T , X) is any solution of (9) and φ(T , X) = φ1 (T , X) = 1/ψ(T , X) is a
particular solution of (24) with G1 (T , X) = Γ1 (T , X) given by (53) and (54). Then from
Eq. (29) it follows that
X
1
u1 (T , ξ )φ1 (T , ξ ) dξ + B2 (T ) ,
u2 (T , X) =
φ1 (T , X)
∂ 2ψ
UN
44
7
∂2
logψ(T , X) + α(T )X2 + β(T )X + γ (T ),
2
∂X
where ψ(T , X) is any solution of
Γ1 (T , X) = 2
42
43
6
23
TE
26
5
In this section we enhance the results presented in Sections 3 and 4 by describing a recycling procedure which generates sequential chains of Kolmogorov equations transformable
to the backward heat equation. Such chains will emanate from any Kolmogorov equation
with G1 (T , X) = Γ1 (T , X) of the form
EC
25
4
18
RR
24
3
17
CO
23
2
16
5. A recycling procedure further extending classes of Kolmogorov equations
transformable to the backward heat equation
22
1
15
DP
15
16
P.12 (1-19)
by:Rima p. 12
43
44
(55)
45
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P.13 (1-19)
by:Rima p. 13
G. Bluman, V. Shtelen / J. Math. Anal. Appl. ••• (••••) •••–•••
where
1
2
4
5
6
7
8
9
10
11
12
13
14
15
Γ2 (T , X) = Γ1 (T , X) + 2
logφ1 (T , X).
∂X2
Consequently, the canonical equation (55) is transformable to the backward heat equation.
(Since (9), with G1 (T , X) = Γ1 (T , X), can be mapped into the backward heat equation,
in this manner one may obtain all solutions u1 (T , X) by using step (3) of the step-by-step
procedure outlined in Section 4 with Γ1 (T ) ≡ 0 (see formula (47)). However, there is no
general procedure to obtain particular solutions of the adjoint equation (24) when Γ1 (T , X)
is T -dependent. In Section 6 we will show how to obtain the general solution of (24) when
Γ1 (T , X) and φ(T , X) are T -independent.)
In general, suppose un (T , X) is any solution of
∂un ∂ 2 un
+
+ Γn (T , X)un = 0,
∂T
∂X2
and φn (T , X) is a particular solution of
19
20
21
22
23
24
∂un+1 ∂ 2 un+1
+ Γn+1 (T , X)un+1 = 0,
+
∂T
∂X2
25
26
27
where
32
33
34
35
36
37
38
39
40
41
42
43
44
45
6
7
8
9
10
11
12
13
14
18
19
(57)
logφn (T , X).
RR
31
5
17
∂X2
The corresponding nonlocal transformations connecting solutions are given by
X
1
un+1 (T , X) =
un (T , ξ )φn (T , ξ ) dξ + Bn+1 (T ) ,
φn (T , X)
k
with Bn+1 (T ) satisfying the condition
∂un
dBn+1 ∂φn
=
(T , k)un (T , k) − φn (T , k)
(T , k),
dT
∂X
∂X
n = 1, 2, . . . . Consequently, sequential chains of canonical PDEs (57) are transformable to
the backward heat equation.
The coefficients a = an (t, x), b = bn (t, x) of the corresponding chains of Kolmogorov
equations follow from step (4) of the step-by-step procedure presented in Section 4. In
particular, let
∂
logun (T , X)
Dn (T , X) = 2
∂X
CO
30
Γn+1 (T , X) = Γn (T , X) + 2
UN
29
∂2
4
16
EC
28
3
15
∂
∂un
∂
∂φn
(φn un+1 ) = φn un ,
(φn un+1 ) =
un − φn
∂X
∂T
∂X
∂X
yields a nonlocal transformation relating the canonical PDE (56) to the homologous canonical PDE
TE
18
∂φn ∂ 2 φn
+
+ Γn (T , X)φn = 0.
−
∂T
∂X2
Then the system
(56)
DP
16
17
2
OF
3
∂2
RO
1
13
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G. Bluman, V. Shtelen / J. Math. Anal. Appl. ••• (••••) •••–•••
14
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2
3
4
5
P.14 (1-19)
by:Rima p. 14
for any solution un (T , X) of (55). Without loss of generality, a(t, x) is arbitrary. Then
∂
1
logun (t, X) − a(t, x) Xt + ax (t, x),
bn (t, x) = 2 a(t, x)
∂X
2
with X = X(t, x) given by (4b).
6
14
15
16
17
18
19
20
21
22
23
24
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32
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41
42
43
44
45
OF
RO
13
x
dz
,
√
a(z)
1
1 d 2X
dX
=√
b(x) − a (x) ,
D(X) = a(x) 2 + b(x)
dx
dx
2
a(x)
1
1
G(X) = − D (X) + D 2 (X) .
2
2
X(x) =
DP
12
3
4
5
7
8
9
In the important special case when the coefficients a(t, x), b(t, x) of the Kolmogorov
equation (1) are time-independent, i.e, a(t, x) ≡ a(x), b(t, x) ≡ b(x), all of the equations
are completely solvable in the recycling procedure outlined in Section 5.
Here X(t, x) ≡ X(x), D(t, x) ≡ D(x), G(t, x) ≡ G(x). From (4)–(6), one obtains
d 2ψ
+ [αX2 + βX + γ ]ψ = 0
dX2
for some constants α, β, γ , and let
d2
Γ1 (X) = 2 2 log |ψ| + αX2 + βX + γ .
dX
Suppose u1 (T , X) is any solution of
10
11
12
13
14
15
(58)
16
17
(59)
18
19
20
(60)
21
22
In the recycling procedure, the equations are solved as follows.
Let ψ(X) be any solution of the ODE
TE
11
EC
10
23
24
25
(61)
26
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29
(62)
∂u1 ∂ 2 u1
+
+ Γ1 (X)u1 = 0.
∂T
∂X2
All such solutions u1 (T , X) are given by following step (3) of the step-by-step procedure of Section 4 with Γ (T ) ≡ 0 (see formula (47)). Now let φ(X) = φ1 (X) = 1/ψ(X) be
a particular solution of the ODE
RR
9
6. The recycling procedure for Kolmogorov equations
with time-independent coefficients
d 2φ
+ Γ1 (X)φ = 0.
(63)
dX2
Unlike the situation in the time-dependent case, one is able to find the general solution of
(63) since φ(X) = 1/ψ(X) is a particular solution of (63). Specifically,
X
1
2
(64)
K1 + ψ (z) dz
φ(X) = φ1 (X; K1 ) =
ψ(X)
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by:Rima p. 15
G. Bluman, V. Shtelen / J. Math. Anal. Appl. ••• (••••) •••–•••
2
3
4
for arbitrary constant K1 . Then
X
1
u2 (T , X) =
u1 (T , ξ )φ1 (ξ ; K1 ) dξ + B2 (T ) ,
φ1 (X; K1 )
10
6
with B2 (T ) satisfying the condition
dB2 ∂ 2 φ1
∂u1
=
(k; K1 )u1 (T , k) − φ1 (k; K1 )
(T , k)
dT
∂X
∂X
for arbitrary constant k, solves
11
∂u2 ∂ 2 u2
+
+ Γ2 (X; K1 )u2 = 0,
∂T
∂X2
12
13
14
where
15
d2
φ1 (X; K1 ).
log
dX2
In general, suppose un (T , X) is any solution of
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21
22
∂ 2φ
n
∂X2
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25
28
29
30
31
32
+ Γ (X; K1 , . . . , Kn−1 )φn = 0
1
φn (X; K1 , . . . , Kn ) =
φn−1 (X; K1 , . . . , Kn−1 )
X
2
× Kn +
φn−1 (z; K1 , . . . , Kn−1 ) dz .
36
un+1 (T , X) =
37
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40
41
k
with Bn+1 (T ) satisfying the condition
42
dBn+1 ∂φn
∂un
=
(k; K1 , . . . , Kn )un (T , k) − φn (k; K1 , . . . , Kn )
(T , k),
dT
∂X
∂X
43
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45
1
φn (X; K1 , . . . , Kn )
X
×
un (T , ξ )φn (ξ ; K1 , . . . , Kn ) dξ + Bn+1 (T ) ,
solves
UN
35
Then the function
RR
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34
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11
12
13
14
15
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17
18
19
20
21
22
23
24
25
can be obtained in the same manner as that used to obtain expression (64) and it has the
form
EC
27
9
∂un ∂ 2 un
+
+ Γn (X; K1 , . . . , Kn−1 )un = 0.
∂T
∂X2
The general solution of the ODE
23
26
8
DP
17
Γ2 (X; K1 ) = Γ1 (X) + 2
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5
7
8
2
k
5
6
1
OF
1
15
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31
(65)
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P.16 (1-19)
by:Rima p. 16
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16
∂un+1 ∂ 2 un+1
+
+ Γn+1 (X; K1 , . . . , Kn )un+1 = 0,
∂T
∂X2
1
2
(66)
1
2
where
3
4
d2
Γn+1 (X; K1 , . . . , Kn ) = Γn (X; K1 , . . . , Kn−1 ) + 2 2 logφn (X; K1 , . . . , Kn ),
dX
n = 1, 2, . . . , and φ0 (X) = ψ(X) is any solution of (61). Consequently, sequential chains
of canonical equations (66) are transformable to the backward heat equation. For each
member of such a sequential chain,
d
logφn (X; K1 , . . . , Kn ).
Dn (X; K1 , . . . , Kn ) = 2
dX
Hence, with a(x) = an (x) arbitrary, the coefficients b(x) of the corresponding Kolmogorov
equations (1), which are transformable to the backward heat equation, we have
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7
1
d
logφn (X; K1 , . . . , Kn ) + a (x), (67)
b(x) = bn (x; K1 , . . . , Kn ) = 2 a(x)
dX
2
n = 0, 1, 2, . . ., with φ0 (X) = ψ(X). In particular, when a(x) ≡ const = a, formula (67)
becomes
√ d
logφn (X; K1 , . . . , Kn )
(68)
b(x) = bn (x; K1 , . . . , Kn ) = 2 a
dX
√
with X = x/ a.
The term n = 0 of a sequential chain corresponds to the local case which was completely considered in [3], the term n = 1 corresponds to the nonlocal extension which was
completely considered in Section 4 of this paper, and further nonlocal extensions resulting
from recycling correspond to terms n = 2, 3, . . . . In effect there are 4 + n arbitrary fitting
constants in term n.
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ε
1
b(x) = ,
a(x) = ,
2
x
with ε = (d − 1)/2. Using (58)–(60), one finds that
√
X = 2 x,
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For a d-Bessel process (see, for example, [10]) the Kolmogorov equation (1) has coefficients
RR
32
7
29
7. An example: a d-Bessel process
(69)
√ ε
2ε
ε(1 − ε)
D(X) = 2 = , G(X) =
.
(70)
x
X
X2
If ε = 0 (d = 1) or ε = 1 (d = 3), then the coefficient G(X) ≡ 0; if ε = 2 (d = 5), then
the coefficient G(X) from (70) coincides with Γ1 (X) given by (62) with ψ = X, α = β =
γ = 0. Hence the Kolmogorov equation (1), with its coefficients given by (69), for ε = 2
can be mapped by a nonlocal transformation (but not by any local one) into the backward
heat equation.
CO
31
UN
30
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P.17 (1-19)
by:Rima p. 17
G. Bluman, V. Shtelen / J. Math. Anal. Appl. ••• (••••) •••–•••
4
5
6
7
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18
1
1
n+1
a(x) = ,
b(x) = bn (x) =
, n = 0, 1, 2, . . .
(71)
2
x
(which corresponds to d = 3, 5, 7, . . .), can be mapped into the backward heat equation. This follows immediately from (65) and (68) with φn−1 (X) = Xn and Kn = 0,
n = 2, 3, . . . .
A d-Bessel process corresponds to the spherically symmetric heat equation in Rd , i.e.,
3
∂ 2u
(d − 1) ∂u
∂u
=
= 0.
−
∂t
∂R 2
R ∂R
From the above results it follows that (72) can be mapped into the heat equation
10
OF
3
Further application of the recycling procedure of Section 6 leads to the following result.
Any Kolmogorov equation (1) with coefficients
∂ 2u
∂u
= 2 =0
∂t
∂x
(I) by a point transformation if and only if d = 3 (a well-known result);
(II) by a nonlocal transformation if and only if d = 2k + 1, k = 2, 3, . . . .
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8
9
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12
13
14
15
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18
22
2
∂u1 ∂ 2 u1
+
− 2 u1 = 0,
∂T
∂X2
X
and only admits a four-parameter Lie group of point transformations with its infinitesimal
generators given by
35
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28
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21
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20
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26
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23
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4
It is well known that the scalar Kolmogorov equation (1) can be mapped into the backward heat equation by a point transformation if and only if (1) admits a six-parameter Lie
group of point transformations (see [3,5]). One can show that the backward heat equation
potential system (32) admits a six-parameter Lie group of point transformations. Hence it is
necessary that the potential system (23) admit a six-parameter Lie group of point transformations in order that (23) can be mapped into (32) by a point transformation. Consequently,
when such a mapping (35) defines a nonlocal transformation acting on (X, T , u1 )-space, it
follows that (23) must admit a six-parameter Lie group of point transformations whereas
the corresponding canonical equation (9) may not admit a six-parameter Lie group of point
transformations.
As an example, consider the d-Bessel process for d = 5. Here the canonical equation (9)
becomes
∂
∂
∂
,
X2 = 2T
+X
,
∂T
∂T
∂X
∂
1
∂
∂
X3 = T 2
,
+TX
+ (X2 − 2T )u1
∂T
∂X 2
∂u1
X1 =
UN
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19
8. Remarks on connections with symmetry analysis
22
23
(72)
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2
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1
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36
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41
X4 = u 1
∂
.
∂u1
Since φ = X−1 (see Section 7), the corresponding potential system (23) takes the form
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5
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∂
∂
∂
,
X2 = 2T
+X
,
∂T
∂T
∂X
1 2
∂
1 2 3
∂
2 ∂
X3 = T
+TX
+
X − T u1 + X v1
∂T
∂X
4
2
2
∂u1
∂
1 2 3
X − T v1
+
,
4
2
∂v1
1
∂
∂
∂
∂
∂
1
+ v1
+ v1
,
X5 =
− v1
,
X4 = u 1
∂u1
∂v1
∂X X ∂u1 X ∂v1
1
T
T
∂
∂
∂
1
1
v1
v1
+ Xu1 +
X+
X−
+
.
X6 = T
∂X
2
2
X
∂u1
2
X
∂v1
X1 =
OF
3
RO
2
1
1
∂v1
∂v1
1 ∂u1
= u1 ,
= − 2 u1 −
.
(73)
∂X
X
∂T
X
X ∂X
System (73) admits a six-parameter Lie group of point transformations with its infinitesimal generators given by
From the mapping (35), it follows that the point transformation
DP
1
18
19
20
21
X = y,
T = τ,
v̂
v1 = ,
y
v̂
u1 = û −
y
transforms (73) into the backward heat equation potential system (32).
22
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23
24
Acknowledgment
25
26
The authors are grateful to Patrick Doran-Wu for some helpful remarks.
27
EC
28
29
Appendix A. Solution of system (15)
30
33
34
35
36
37
38
39
40
41
42
σ (T ) =
1
.
s(T )
Then Eq. (15a) becomes a linear ODE in terms of s(T ), namely
d 2s
+ 4α(T )s = 0.
(A.1)
dT 2
Let s = S(T ) be any solution of (A.1). Then the general solution of Eqs. (15b,c) is given
by
43
45
T
1
UN
44
RR
32
In system (15), suppose α(T ), β(T ), γ (T ) are given functions of T . Then σ (T ), ρ(T ),
λ(T ) are found as follows.
To find σ (T ), first let
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31
P.18 (1-19)
by:Rima p. 18
ρ(T ) = 2
S 2 (t2 )
dt2
t2
β(t1 )S(t1 ) dt1 ,
1
2
3
4
5
6
7
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P.19 (1-19)
by:Rima p. 19
G. Bluman, V. Shtelen / J. Math. Anal. Appl. ••• (••••) •••–•••
1
2
3
1
λ(T ) = logS(T ) +
2
T 1
2
S (t2 )
2
t2
β(t1 )S(t1 ) dt1
19
+ γ (t2 ) dt2 .
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OF
9
6
[1] S.C. Anco, G. Bluman, Nonlocal symmetries and nonlocal conservation laws of Maxwell’s equations,
J. Math. Phys. 38 (1997) 3508–3532.
[2] A.T. Bharucha-Reid, Elements of the Theory of Markov Processes and Their Applications, McGraw–Hill,
New York, 1960.
[3] G. Bluman, On the transformation of diffusion processes into the Wiener process, SIAM J. Appl. Math. 39
(1980) 238–247.
[4] G. Bluman, P. Doran-Wu, The use of factors to discover potential systems or linearizations, Acta Appl.
Math. 41 (1995) 21–43.
[5] G. Bluman, S. Kumei, Symmetries and Differential Equations, in: Applied Mathematical Sciences, Vol. 81,
Springer, New York, 1989.
[6] G. Bluman, G. Reid, Sequences of related linear PDEs, J. Math. Anal. Appl. 144 (1989) 565–585.
[7] G. Bluman, V. Shtelen, New classes of Schrödinger equations equivalent to the free particle equation through
non-local transformations, J. Phys. A 29 (1996) 4473–4480.
[8] I.D. Cherkasov, On the transformation of the diffusion process to a Wiener process, Theory Probab. Appl. 2
(1957) 373–377.
[9] A. Kolmogoroff, Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung, Math. Ann. 104
(1931) 415–458.
[10] M. Nagasawa, Schrödinger Equations and Diffusion Theory, Birkhäuser, Basel, 1993.
[11] L.M. Ricciardi, On the transformation of diffusion processes into the Wiener process, J. Math. Anal. Appl. 54
(1976) 185–199.
[12] L.M. Ricciardi, Diffusion Processes and Related Topics in Biology, in: Lecture Notes in Biomathematics,
Vol. 14, Springer, Berlin, 1977.
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References
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