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Ibragimov-type invariants for a system of two linear parabolic equations
Commun Nonlinear Sci Numer Simulat 17 (2012) 3140–3147
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Ibragimov-type invariants for a system of two linear parabolic equations F.M. Mahomed a, M. Safdar b,⇑, J. Zama a a Centre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of Witwatersrand, Wits 2050, South Africa b Center for Advanced Mathematics and Physics, National University of Sciences and Technology, Campus H-12, 44000 Islamabad, Pakistan
a r t i c l e
i n f o
Article history: Received 10 September 2011 Received in revised form 14 December 2011 Accepted 15 December 2011 Available online 24 December 2011 Keywords: Complex linear parabolic equation Complex semi-invariants Systems and semi-invariants
a b s t r a c t We obtain new semi-invariants for a system of two linear parabolic type partial differential equations (PDEs) in two independent variables under equivalence transformations of the dependent variables only. This is achieved for a class of systems of two linear parabolic type PDEs that correspond to a scalar complex linear (1 + 1) parabolic equation. The complex transformations of the dependent variables which map the complex scalar linear parabolic PDE to itself provide us with real transformations that map the corresponding system of linear parabolic type PDEs to itself with different coefficients in general. The semi-invariants deduced for this class of systems of two linear parabolic type equations correspond to the complex Ibragimov invariants of the complex scalar linear parabolic equation. We also look at particular cases of the system of parabolic type equations when they are uncoupled or coupled in a special manner. Moreover, we address the inverse problem of when systems of linear parabolic type equations arise from analytic continuation of a scalar linear parabolic PDE. Examples are given to illustrate the method implemented. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction The scalar linear parabolic PDE of one space variable and one time variable
ut ¼ aðt; xÞuxx þ bðt; xÞux þ cðt; xÞu;
ð1Þ
where a – 0, b and c are continuous functions in t and x in some domain, arises in many important applications. The Fokker– Planck PDE [18] belongs to the class (1) and it models many phenomena [4,5]. The Black–Scholes [6] as well as the bondpricing equations [21] are also in the class (1). Algebraic properties of the one and two dimensional Fokker–Planck equations and exact solutions of Eq. (1) have been investigated in [19,20] respectively. The Lie group approach is used to analyze the Black–Scholes and bond pricing equations [8,9], respectively. The symmetry approach was also utilized [17] to reduce onefactor bond pricing parabolic equations to the classical linear heat equation. Fundamental solutions were derived for two zero-coupon bond-pricing PDEs. Lie [12] provided the complete group classification of the parabolic PDE (1). He extracted the four canonical forms of Eq. (1) for which it admits 1, 2, 4 and 6 dimensional nontrivial point symmetry algebras (apart from the infinite dimensional algebra of trivial point symmetries of the superposition operators). The family of parabolic equations (1) with a = 1 and b = 0 reduces to the fourth Lie canonical form which was utilized [16] to study it further. This form was also used to develop a mapping algorithm [7] which was based on the symmetries of the parabolic equation (1). We now focus our discussion on semi-invariants and invariants under equivalence transformations of the linear (1 + 1) parabolic equation (1). Equivalence transformations of the PDE (1) are transformations that map the family (1) into itself. ⇑ Corresponding author. Tel.: +92 344 5288466. E-mail addresses: [email protected] (F.M. Mahomed), [email protected] (M. Safdar), [email protected] (J. Zama). 1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.12.006
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Laplace type semi-invariants of Eq. (1) were derived [10] under linear transformations of the dependent variable only. We refer to these semi-invariants as Ibragimov invariants. Transformations of the independent variables only were also utilized [11] to obtain certain semi-invariants of the linear PDE (1). In the reference cited, the joint invariant equation for (1) was also found by considering transformations of both the dependent and independent variables and this provided necessary and sufficient conditions to reduce (1) to the classical heat PDE via point transformations. The practical criteria for such a reduction were in terms of the coefficients of equation. Furthermore the work on joint invariant equations for the family of Eq. (1) and reduction to the heat equation was addressed [14] along with refinement of the invariant condition for reduction. Moreover, the reducibility to the second Lie canonical form was achieved. A complete invariant characterization was provided for scalar linear parabolic equations under equivalence mappings for all the four Lie canonical forms [13]. Here the invariants and construction of the equivalence transformations of both the dependent and independent variables were studied as well. There exist some special classes of systems of differential equations due to complex symmetry analysis (CSA) [1,2]. These systems are transformable to the same family by utilizing the equivalence mappings of the corresponding complex differential equations if the complex equation is transformable to itself. These systems admit operators via the complex symmetries that are proved to be inequivalent to those obtained by the classical symmetry approach [3]. These classes have fewer arbitrary coefficients than their classical analogues and all these coefficients satisfy Cauchy–Riemann equations. Here we investigate a class of systems of two linear parabolic type PDEs in two independent variables obtainable by a scalar complex linear (1 + 1) parabolic PDE in order to obtain semi-invariants by using CSA as a tool. A complex PDE here involves a complex dependent function of real independent variables; the complex split of this complex function provides a coupled system of two real PDEs. It is not necessary that the algebraic properties of the system corresponds to those of the base complex PDE. But the equivalence transformations for systems are always obtainable by the complex split of the equivalence transformations of the corresponding PDE. The maps and semi-invariants under changes of dependent variables obtained for this class of systems of PDEs are found to correspond to complex maps and Ibragimov invariants of the scalar complex linear (1 + 1) parabolic PDE respectively. The real and imaginary parts of the complex Ibragimov invariants for the complex scalar parabolic PDE are actually the Ibragimov-type invariants for the class of systems obtainable by the complex PDE. Moreover, we answer the inverse problem of when a system of two linear parabolic type PDEs arise from a complex scalar parabolic PDE. In Section 2 we obtain equivalence transformations of the class of systems of two linear parabolic type PDEs that arise from the scalar complex linear (1 + 1) parabolic PDE (1). The next section deals with semi-invariants, Ibragimov-type invariants, under transformations of the dependent variables for such systems. Then we consider some special cases of this system of two parabolic type PDEs, i.e. uncoupled and special coupled parabolic systems. Herein we also mention the inverse problem. The subsequent section is on the applications of our results. Concluding remarks are given in Section 5. 2. Equivalence transformations Equivalence transformations of the dependent variable only for the parabolic PDE (1) are the linear transformations
; u ¼ rðt; xÞu
rðt; xÞ – 0;
ð2Þ
that map the family (1) into itself. That is under the map (2) the parabolic equation (1) remains linear and homogeneous but the transformed PDE
u u xx þ b x þ cu t ¼ a u
ð3Þ
and c which are ; b has new coefficients a
¼ a; a
¼ b þ 2 rx a; b
r
c ¼
LðrÞ
r
ð4Þ
in which the operator L is defined as
L¼a
@2 @ @ þ b þ c: @x @t @x2
ð5Þ
We now focus our attention on the linear parabolic type system of two PDEs in two independent variables t and x
v t ¼ a1 v xx þ a2 wxx þ b1 v x þ b2 wx þ c1 v þ c2 w; wt ¼ a3 v xx þ a4 wxx þ b3 v x þ b4 wx þ c3 v þ c4 w;
ð6Þ
in which the coefficients ai, bi and ci where i = 1, . . . , 4 are continuous functions of t and x in some domain. The equivalence transformations of the dependent variables, as can be verified, of this family (6) of linear parabolic type equations into itself are as follows
v ¼ s1 ðt; xÞv þ s2 ðt; xÞw; w ¼ s3 ðt; xÞv þ s4 ðt; xÞw;
ð7Þ
where the si’s are arbitrary functions of their arguments. The new coefficients are in terms of the si’s and the old coefficients.
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We wish to restrict to a special class of the system (6) which corresponds to a scalar linear parabolic equation (1) considered in the complex domain, i.e. u(t, x) = v(t, x) + iw(t, x), while the coefficients in (1) are also complex functions of the independent variables t and x so as to extract a system of two linear parabolic type PDEs of the form
v t ¼ a1 v xx a2 wxx þ b1 v x b2 wx þ c1 v c2 w; wt ¼ a1 wxx þ a2 v xx þ b1 wx þ b2 v x þ c1 w þ c2 v ;
ð8Þ
which belongs to a special class of systems of linear parabolic PDEs obtainable by CSA. The subset of equivalence transformations of the dependent variables (7) of the form
v ¼ r1 ðt; xÞv r2 ðt; xÞw; w ¼ r2 ðt; xÞv þ r1 ðt; xÞw;
ð9Þ
are arrived at by the complex transformations (2) if r(t, x) = r1(t, x) + ir2(t, x). This subset of the equivalence transformations (9) transforms the system of linear PDEs (8) to
v t ¼ a1 v xx a2 w xx þ b1 v x b2 w x þ c1 v c2 w; 1 w 2 v xx þ b1 w t ¼ a xx þ a x þ b2 v x þ cc1 w þ c2 v ; w
ð10Þ
where
1 ¼ a1 ; a
2 ¼ a2 ; a
1 ¼ b1 þ 2ðr1 ða1 r1x a2 r2x Þ þ r2 ða1 r2x þ a2 r1x ÞÞ ; b r21 þ r22 2ð r ða r a 2 1 1x 2 r2x Þ þ r1 ða1 r2x þ a2 r1x ÞÞ 2 ¼ b2 þ b ; r21 þ r22 r ðL r L2 r2 Þ þ r2 ðL1 r2 þ L2 r1 Þ c1 ¼ 1 1 1 ; r21 þ r22 r ðL r þ L2 r1 Þ r2 ðL1 r1 L2 r2 Þ c2 ¼ 1 1 2 r21 þ r22
ð11Þ
and the operators L1 and L2 are the real and imaginary parts of the complex operator (5), i.e.
L1 ¼ a1
@2 @ @ þ b1 þ c 1 ; @x @t @x2
L2 ¼ a2
@2 @ @ þ b2 þ c 2 : @x @t @x2
ð12Þ
Remark. The linear parabolic type system (8) is a special class of the family of the linear parabolic type system (6). Eq. (8) have six arbitrary coefficients whereas (6) has twelve arbitrary coefficients. We now discuss the case for which system (8) reduces to an uncoupled and a coupled system of two PDEs. This clearly occurs for a2 = b2 = c2 = 0 and a1 = b1 = c1 = 0 respectively, or alternatively the coefficients a, b and c are real or pure imaginary. Thus the uncoupled parabolic system is.
v t ¼ a1 v xx þ b1 v x þ c1 v ; wt ¼ a1 wxx þ b1 wx þ c1 w;
ð13Þ
while the coupled one is
v t ¼ a2 wxx b2 wx c2 w; wt ¼ a2 v xx þ b2 v x þ c2 v :
ð14Þ
We return to these systems (13) and (14) later. To conclude this section we summarize that the system (8) has equivalence transformations which are linear changes in the dependent variables (9) and it is mapped into the same family with in general new coefficients given by (11). 3. Semi-invariants under dependent variables changes In this section we deal with the Ibragimov-type invariants of the linear parabolic type system (8) under the dependent variables transformations (9). We derive Ibragimov-type invariants of the class of linear parabolic type PDEs (8) under the dependent variables (9) by reverting to the base scalar complex linear parabolic equation (1) with complex coefficients. We consider the following two nontrivial semi-invariants a and K [10]
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1 2 b ax þ at þ aaxx a2x b þ ðaax abÞbx abt a2 bxx þ 2a2 cx ; 2
K¼
ð15Þ
referred to as a Laplace-type invariants in [10]. We refer to it as the Ibragimov invariant. The Ibragimov semi-invariant K can be written compactly [13] as
k¼
! 2 @ b b @ b þ ca ; @x 2a x 4a @t 2a
ð16Þ
where K = 2a2k. We have the following results based on the semi-invariants a and k. The scalar linear parabolic PDE (1) is equivalent to the ¼ 0, via the linear change (2) if and only if parabolic equation (3), i.e. Lu
¼ a; a
¼ k; k
ð17Þ
is where k
! ! ! 2 b b @ @ b c a : þ k¼ 2a @x @t 2a 4a
ð18Þ
x
The construction of r in the transformation (2) is achieved by solving the system [13]
! 2 2 b b b þ : c þ a 2 a 4a 4 a x
rx b b rt b ; ¼ ¼ca 2a r 2a 2a r
ð19Þ
x
As an example which we use later, if (1) has constant coefficients, then it can be reduced to the classical heat equation
u xx ; t ¼ a u
¼ constant; a
ð20Þ
under the change of the dependent variable
! 2 b b ; u ¼ exp ct t x u 2a 4a
ð21Þ
¼ k. This transformation is a consequence of the relations (19), viz. ¼ a and k since a 2 rx b rt b ¼ ; ¼c 2a r r 4a
ð22Þ
and its solution is (21) up to a constant. We now derive the Ibragimov-type invariants of the linear parabolic type system (8) under linear changes of the depen i.e. and k dent variables (9) by the complex split of a; k; a
1 ¼ a1 ; a
2 ¼ a2 ; a
1 ¼ k1 ; k
2 ¼ k2 ; k
ð23Þ
where
" 2 #! 2 @ 1 a1 b1 þ a2 b2 a1 b2 a2 b1 1 a1 b1 a1 b2 þ 2a2 b1 b2 1 þ c1 a1 a 2 2 2 2 2 @x 2 4 2 a21 þ a22 a þ a a þ a 1 2 1 2 x x " 2 #! 2 @ 1 a1 b2 a2 b1 a1 b1 þ a2 b2 1 a2 b2 a2 b1 þ 2a1 b1 b2 1 þ k2 ¼ c2 a1 þ a2 2 2 2 2 @x 2 4 2 a21 þ a22 a þ a a þ a 1 2 1 2 x x
k1 ¼
@ a1 b1 þ a2 b2 ; 2 2 @t a1 þ a2 @ a1 b2 a2 b1 : 2 2 @t a1 þ a2
ð24Þ
Note that the transformed k’s are with the bars over the coefficients aj, bj and cj where j = 1, 2. We state the following result. Proposition 1. The linear parabolic system
L1 v L2 w v t þ a1 v xx a2 wxx þ b1 v x b2 wx þ c1 v c2 w ¼ 0; L1 w þ L2 v wt þ a1 wxx þ a2 v xx þ b1 wx þ b2 v x þ c1 w þ c2 v ¼ 0;
ð25Þ
is equivalent to the transformed linear parabolic type system
1 v x b 2 w 1 v xx a 2 w v t þ a xx þ b x þ c1 v c2 w ¼ 0; L1 v L2 w 1 w 2 v x þ cc1 w 1 w 2 v xx þ b t þ a þ L2 v w xx þ a x þ b þ c2 v ¼ 0; L1 w
ð26Þ
by means of the linear transformations (9) if and only if (23) holds, i.e.
1 ¼ a1 ; a
2 ¼ a2 ; a
1 ¼ k1 ; k
2 ¼ k2 ; k
ð27Þ
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where
" ! !# " #! ! þa 2 2 a 1 2 a 2 þ 2a 1 b 2 1 þ a 2 b 1 b 1 b 1 b 2 b 2 b 1 b 2 b 2 b a a 1 a 1 @ a 1 2 1 ¼ @ c1 1 a 1 1 12 k þ ; a 2 1 þ a 21 þ a 21 þ a 21 þ a 22 22 22 22 @x 2 4 2 @t a a a a x x " ! !# " #! ! 2 a 1 1 þ a 2 2 a 2 þ 2a 1 b 2 2 a 1 2 b 2 b 2 b 1 b 2 b 1 b 1 b 2 b 1 b @ 1 1 a 1 @ a a a 1 1 k2 ¼ c2 a1 þ a2 þ 21 þ a 21 þ a 21 þ a 21 þ a 22 22 22 22 @x 2 4 2 @t a a a a x
ð28Þ
x
and r1, r2 in the linear transformations (9) can be deduced from (19) via a complex split. The proof of this proposition follows immediately from the preceding discussion including the construction of the linear transformation encapsulated in r. 3.1. Uncoupled parabolic systems Here we investigate the Ibragimov-type invariants of the system (8) when it maps to the system of the form (13). Notice that these systems are special cases of the system of PDEs (8). Now we want the target parabolic type system to be uncoupled, i.e. of the form
v t ¼ a1 v xx þ b1 v x þ c1 v ;
ð29Þ
1 w 1 w t ¼ a xx þ b x þ c1 w: w 2 ¼ 0 and 1 ¼ a1 ; a 2 ¼ 0; k The transformed semi-invariants thus satisfy a
1 b 1 ¼ @ c1 a 1 k 1 @x 2a
! x
2 b 1 1 4a
!
! 1 @ b þ : 1 @t 2a
ð30Þ
3.2. Special coupled parabolic type systems Suppose that the target system is a coupled system of the special form
v t ¼ a2 w xx b2 w x c2 w; 2 v xx þ b2 v x þ c2 v : t ¼ a w
ð31Þ
1 ¼ 0 and 1 ¼ 0; a 2 ¼ a2 ; k Then the transformed semi-invariants thus satisfy a
2 b 2 ¼ @ c2 a 2 k 2 @x 2a
! x
2 b 2 2 4a
! þ
! 2 @ b : 2 @t 2a
ð32Þ
In summary of these sub-sections, we have the following result. Proposition 2. The linear parabolic type system (8) is reducible to the uncoupled system (29) and the special coupled system (31) ¼ 0 and k as in (30) and a ¼ 0 and k as 1 ¼ a1 ; a 2 ¼ 0; k 2 ¼ a2 ; a 1 ¼ 0; k via equivalence transformations (9) if and only if a 2 1 1 2 in (32), respectively. The proof follows at once from the discussion contained above in Sections 3.1 and 3.2.We can also deduce the following consequence. Corollary. The parabolic type system (8) is reducible to the simpler uncoupled and coupled systems
v t ¼ a1 v xx ; 1 w t ¼ a xx w
ð33Þ
and
v t ¼ a2 w xx ; 2 v xx ; t ¼ a w
ð34Þ
1 ¼ a1 ; a 2 ¼ 0 and a 1 ¼ 0; a 2 ¼ a2 , respectively by means of the linear dependent variables changes of the form (9) if and only if a 1 ¼ k1 ¼ 0 ¼ k2 ¼ k 2 for both the cases, viz. uncoupled and special coupled systems of PDEs. while k Note that if a1 is constant in the Corollary then we get criteria for reduction to the simplest system which is the classical heat system of PDEs. The linear parabolic type uncoupled and coupled systems of PDEs (13) and (14) are not only the special cases of (8) but these also appear as very special and restricted subcases of the general class of systems of two linear parabolic type PDEs (6), i.e. if we extract the uncoupled and coupled systems of parabolic type PDEs from (6) and restrict the remaining non-zero coefficients as they appear in (13) and (14) we obtain subcases of these systems of (6). Thus the Proposition 2 implies that
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the sub-class (9) of the equivalence transformations (7) is found to map special subcases (uncoupled and coupled linear parabolic type systems of two PDEs) of the general class (6) into one of its simplest forms or the same family of equations with different coefficients. We can also address the inverse problem of when a system of two linear parabolic type equations in two independent variables arise from analytic continuation of a scalar (1 + 1) linear parabolic PDE in a simple way. Indeed this is the case if and only if the system (6) has coefficients which satisfy (8). In the next section we pursue some applications of our results. 4. Applications We consider examples of certain parabolic type systems and their reductions. 1. Consider the constant coefficient parabolic type system
v t ¼ a1 v xx a2 wxx þ b1 v x b2 wx þ c1 v c2 w; wt ¼ a1 wxx þ a2 v xx þ b1 wx þ b2 v x þ c1 w þ c2 v :
ð35Þ
This is reducible to the classical heat system
v t ¼ a1 v xx a2 w xx ; 2 v xx þ a 1 w t ¼ a xx ; w
ð36Þ
since
1 ¼ a1 ; a
2 ¼ a2 ; a
1 ¼ k1 ¼ 0; k
2 ¼ k2 ¼ 0: k
ð37Þ
The transformation is given by the complex split of (21) i.e.
v ¼ ea ðv cosða2 Þ w sinða2 ÞÞ; cosða2 Þ þ v sinða2 ÞÞ; w ¼ e a ðw
ð38Þ
1 ! 2 2 a1 b1 b2 þ 2a2 b1 b2 a1 b1 þ a2 b2 @ A t a1 ¼ c1 x; 4 a21 þ a22 2 a21 þ a22 ! 2 2 2a b b a b þ a b a b a b a2 ¼ c2 1 1 2 2 2 12 2 2 t 1 22 22 1 x: 2ða1 þ a2 Þ 4 a1 þ a2
ð39Þ
1
1
where
0
Notice that this transformation is much more complicated than the corresponding transformation (21), but elegantly furnishes the equivalence of (35) and (36). 2. The time-dependent linear parabolic type system
v t ¼ xv xx þ c1 ðtÞv c2 ðtÞw; wt ¼ xwxx þ c1 ðtÞw þ c2 ðtÞv ;
ð40Þ
has a1 = x, k1 = 0 = k2 and by the Corollary is reducible to
v t ¼ xv xx ; xx : t ¼ xw w
ð41Þ
The transformation is
Z sin c2 ðtÞdt w c2 ðtÞdt ; Z Z R cos c2 ðtÞdt : c2 ðtÞdt þ v sin w ¼ e c1 ðtÞdt w R
v ¼e
c1 ðtÞdt
v cos
Z
ð42Þ
This is quite a messy transformation compared to the scalar case of the PDE
ut ¼ xuxx þ cðtÞu;
ð43Þ
1 2 2 A x v xx Bxv x þ C v ; 2 1 wt ¼ A2 x2 wxx Bxwx þ Cw; 2
ð44Þ
R . t ¼ xu xx via u ¼ e cðtÞdt u which reduces to u 3. Consider the variable coefficient linear diffusive system
vt ¼
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which actually corresponds to the complex Black–Scholes equation (set u = v + iw)
1 ut ¼ A2 x2 uxx Bxux þ Cu: 2
ð45Þ
1 ¼ k1 ¼ 0 and k 2 ¼ k2 ¼ 0. Thus the system (44) can be reduced to 1 ¼ a1 ; a 2 ¼ a2 ¼ 0; k Here a
1 2 2 A x v xx ; 2 1 xx ; t ¼ A 2 x2 w w 2
ð46Þ
2 1 1 C B B2 A2 txB=A v ; 2 2
2 1 1 w ¼ C B B2 A2 txB=A w: 2 2
ð47Þ
v t ¼
via
v¼
4. Now we finally consider some special cases, i.e. uncoupled and coupled parabolic type systems of PDEs. If a2 = b2 = c2 = 0 or a1 = b1 = c1 = 0 in (39) then as a result we find the following equivalence transformations
v ¼ ea v ; 3
ð48Þ
a3
w ¼ e w; and
v ¼ ea ðv cosða5 Þ w sinða5 ÞÞ; cosða5 Þ þ v sinða5 ÞÞ; w ¼ ea ðw 4
4
ð49Þ
b2 b2 b1 b2 respectively. Here a3 ¼ c1 4a11 t 2a x; a4 ¼ 2a x and a5 ¼ c2 4a22 t. By the use of (48) and (49) the systems 1 2
v t ¼ a1 v xx þ b1 v x þ c1 v ; wt ¼ a1 wxx þ b1 wx þ c1 w;
ð50Þ
and
v t ¼ a2 wxx b2 wx c2 w; wt ¼ a2 v xx þ b2 v x þ c2 v ;
ð51Þ
are transformable to the following uncoupled and coupled systems of PDEs
v t ¼ a1 v xx ; 1 w xx ; t ¼ a w
ð52Þ
and
v t ¼ a2 w xx ; 2 v xx ; t ¼ a w
ð53Þ
respectively, which satisfy the invariant conditions accordingly. If we have another uncoupled system with variable coefficients
x2 4 x2 wt ¼ wxx þ xwx þ w; 4
v t ¼ v xx þ xv x þ v ;
ð54Þ
it is equivalent to the simpler system of the form
v t ¼ v xx þ v ; xx þ w; t ¼ w w
ð55Þ
via the equivalence transformations 3 2
1 2 4
v ¼ v eð tþ x Þ ; 3
1 2
¼ weð2tþ4x Þ : w 1 ¼ k 2 ¼ 0. ¼ a ¼ 1 and k1 ¼ k2 ¼ k In this case we have a
ð56Þ
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5. Concluding remarks Linear parabolic type equations have been studied by many researchers and several of them have focused mainly on the scalar equation. Here we used complex methods for systems of two linear parabolic type equations of two independent variables of a special class to deduce new semi-invariants called Ibragimov-type semi-invariants. We have shown that the semiinvariants under equivalence transformations of the dependent variables, restricted to a special class, can be achieved for a special class of two linear parabolic type equations in two independent variables that can be obtained from the analytic continuation of the scalar complex linear (1 + 1) parabolic equation. The Ibragimov-type semi-invariants as well as maps for this class of systems of two linear parabolic type equations are found to correspond to the complex maps and Ibragimov invariants of the corresponding complex scalar (1 + 1) parabolic PDE. It is now clear that this special class of systems of two linear parabolic type equations have different algebraic properties when its symmetries are deduced from its complex counterpart compared to those of the system itself (see [15]). However we have proved the consistency of the equivalence criteria, i.e. semi-invariants of the system of two linear parabolic type PDEs always correspond to the complex Ibragimov semiinvariants of the base complex PDE. Moreover we provided an answer to the inverse problem of when a system of two linear parabolic type PDEs arise from a scalar complex linear (1 + 1) parabolic equation. This is the case if the coefficients coincide with those of system (8). Furthermore, we gave various examples to illustrate our approach. Acknowledgements M. Safdar thanks the School of Computational and Applied Mathematics for support and hospitality during the time this work was initiated. FM is grateful to NUST-CAMP and the HEC of Pakistan for a visiting professorship during which time this work was completed. References [1] Ali S, Mahomed FM, Qadir A. Linearizability criteria for systems of two second order differential equations by complex methods. Nonlinear Dyn 2011;66:77. [2] Ali S, Mahomed FM, Qadir A. Complex Lie symmetries for scalar second order ordinary differential equations. Nonlinear Anal: RWA 2009;10:3335. [3] Safdar M, Qadir A, Ali S. Inequivalence of classes of linearizable systems of cubically semi linear ordinary differential equations obtained by real and complex symmetry analysis. Math Comp Appl 2011;16(4):923. [4] Arecchi FT, Rodari GS, Sona A. Statistics of the laser radiation at threshold. Phys Lett 1967;25A(1):59. [5] Barone A, Paterno G. Physics and applications of the Josephson effect. New York: John Wiley; 1982. [6] Black J, Scholes M. The pricing of options and corporate liabilities. J Polit Econ 1973;81:637. [7] Bluman GW. On the transformation of diffusion processes into the Wiener process. SIAM J Appl Math 1980;39(2):238. [8] Gazizov RK, Ibragimov NH. Lie symmetry analysis of differential equations in finance. Nonlinear Dyn 1998;17:387. [9] Goard J. New solutions to the bond-pricing equation via Lie’s classical method. Math Comput Modell 2000;32:299. [10] Ibragimov NH. Laplace type invariants for parabolic equations. Nonlinear Dyn 2002;28:125. [11] Johnpillai IK, Mahomed FM. Singular invariant equation fro the (1 + 1) Fokker–Planck equation. J Phys A: Math Gen 2001;34:11033. [12] Lie S. On integration of a class of linear partial differential equations by means of definite integrals. Archiv for Mathematik og Naturvidenskab 1881;VI(3):328. Reprinted in S. Lie, Gesammelte Abhadlundgen, 3 paper XXXV, [in German]. [13] Mahomed FM. Complete invariant characterization of scalar linear (1 + 1) parabolic equations. JNMP 2008;15:112. [14] Mahomed FM, Pooe CA. Invariant equations for (1 + 1) linear parabolic equations. In: Rudenko OV, Sapozhnikov OA, editor. Proceedings of the international conference modern group analysis 9, 16th ISNA, Moscow State University, Moscow; 2003. p. 525. [15] Mahomed FM, Naz Rehana. A note on the Lie symmetries of complex partial differential equations and their split real systems. Pramana – J Phys 2011;77(3):483. [16] Ovsiannikov LV. Group properties of differential equations. New York: Academic Press; 1982. [17] Pooe CA, Mahomed FM, Wafo Soh C. Fundamental solutions for zero-coupon bond pricing models. Nonlinear Dyn 2004;36:69. [18] Risken H. The Fokker–Planck equation. Methods of solution and applications. Berlin: Springer-Verlag; 1989. [19] Shtelen WM, Stognii VI. Symmetry properties of of one- and two-dimensional Fokker–Planck equations. J Phys A: Math Gen 1989;22:L539. [20] Spichak S, Stognii V. Symmetry classification and exact solutions of the one-dimensional Fokker–Planck equation with arbitrary coefficients of drifts and diffusion. J Phys A: Math Gen 1999;32:8341. [21] Wilmott P. Derivatives: the theory and practice of financial engineering. West Sussex England: John Wiley; 1998.
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Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Ibragimov-type invariants for a system of two linear parabolic equations F.M. Mahomed a, M. Safdar b,⇑, J. Zama a a Centre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of Witwatersrand, Wits 2050, South Africa b Center for Advanced Mathematics and Physics, National University of Sciences and Technology, Campus H-12, 44000 Islamabad, Pakistan
a r t i c l e
i n f o
Article history: Received 10 September 2011 Received in revised form 14 December 2011 Accepted 15 December 2011 Available online 24 December 2011 Keywords: Complex linear parabolic equation Complex semi-invariants Systems and semi-invariants
a b s t r a c t We obtain new semi-invariants for a system of two linear parabolic type partial differential equations (PDEs) in two independent variables under equivalence transformations of the dependent variables only. This is achieved for a class of systems of two linear parabolic type PDEs that correspond to a scalar complex linear (1 + 1) parabolic equation. The complex transformations of the dependent variables which map the complex scalar linear parabolic PDE to itself provide us with real transformations that map the corresponding system of linear parabolic type PDEs to itself with different coefficients in general. The semi-invariants deduced for this class of systems of two linear parabolic type equations correspond to the complex Ibragimov invariants of the complex scalar linear parabolic equation. We also look at particular cases of the system of parabolic type equations when they are uncoupled or coupled in a special manner. Moreover, we address the inverse problem of when systems of linear parabolic type equations arise from analytic continuation of a scalar linear parabolic PDE. Examples are given to illustrate the method implemented. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction The scalar linear parabolic PDE of one space variable and one time variable
ut ¼ aðt; xÞuxx þ bðt; xÞux þ cðt; xÞu;
ð1Þ
where a – 0, b and c are continuous functions in t and x in some domain, arises in many important applications. The Fokker– Planck PDE [18] belongs to the class (1) and it models many phenomena [4,5]. The Black–Scholes [6] as well as the bondpricing equations [21] are also in the class (1). Algebraic properties of the one and two dimensional Fokker–Planck equations and exact solutions of Eq. (1) have been investigated in [19,20] respectively. The Lie group approach is used to analyze the Black–Scholes and bond pricing equations [8,9], respectively. The symmetry approach was also utilized [17] to reduce onefactor bond pricing parabolic equations to the classical linear heat equation. Fundamental solutions were derived for two zero-coupon bond-pricing PDEs. Lie [12] provided the complete group classification of the parabolic PDE (1). He extracted the four canonical forms of Eq. (1) for which it admits 1, 2, 4 and 6 dimensional nontrivial point symmetry algebras (apart from the infinite dimensional algebra of trivial point symmetries of the superposition operators). The family of parabolic equations (1) with a = 1 and b = 0 reduces to the fourth Lie canonical form which was utilized [16] to study it further. This form was also used to develop a mapping algorithm [7] which was based on the symmetries of the parabolic equation (1). We now focus our discussion on semi-invariants and invariants under equivalence transformations of the linear (1 + 1) parabolic equation (1). Equivalence transformations of the PDE (1) are transformations that map the family (1) into itself. ⇑ Corresponding author. Tel.: +92 344 5288466. E-mail addresses: [email protected] (F.M. Mahomed), [email protected] (M. Safdar), [email protected] (J. Zama). 1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.12.006
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Laplace type semi-invariants of Eq. (1) were derived [10] under linear transformations of the dependent variable only. We refer to these semi-invariants as Ibragimov invariants. Transformations of the independent variables only were also utilized [11] to obtain certain semi-invariants of the linear PDE (1). In the reference cited, the joint invariant equation for (1) was also found by considering transformations of both the dependent and independent variables and this provided necessary and sufficient conditions to reduce (1) to the classical heat PDE via point transformations. The practical criteria for such a reduction were in terms of the coefficients of equation. Furthermore the work on joint invariant equations for the family of Eq. (1) and reduction to the heat equation was addressed [14] along with refinement of the invariant condition for reduction. Moreover, the reducibility to the second Lie canonical form was achieved. A complete invariant characterization was provided for scalar linear parabolic equations under equivalence mappings for all the four Lie canonical forms [13]. Here the invariants and construction of the equivalence transformations of both the dependent and independent variables were studied as well. There exist some special classes of systems of differential equations due to complex symmetry analysis (CSA) [1,2]. These systems are transformable to the same family by utilizing the equivalence mappings of the corresponding complex differential equations if the complex equation is transformable to itself. These systems admit operators via the complex symmetries that are proved to be inequivalent to those obtained by the classical symmetry approach [3]. These classes have fewer arbitrary coefficients than their classical analogues and all these coefficients satisfy Cauchy–Riemann equations. Here we investigate a class of systems of two linear parabolic type PDEs in two independent variables obtainable by a scalar complex linear (1 + 1) parabolic PDE in order to obtain semi-invariants by using CSA as a tool. A complex PDE here involves a complex dependent function of real independent variables; the complex split of this complex function provides a coupled system of two real PDEs. It is not necessary that the algebraic properties of the system corresponds to those of the base complex PDE. But the equivalence transformations for systems are always obtainable by the complex split of the equivalence transformations of the corresponding PDE. The maps and semi-invariants under changes of dependent variables obtained for this class of systems of PDEs are found to correspond to complex maps and Ibragimov invariants of the scalar complex linear (1 + 1) parabolic PDE respectively. The real and imaginary parts of the complex Ibragimov invariants for the complex scalar parabolic PDE are actually the Ibragimov-type invariants for the class of systems obtainable by the complex PDE. Moreover, we answer the inverse problem of when a system of two linear parabolic type PDEs arise from a complex scalar parabolic PDE. In Section 2 we obtain equivalence transformations of the class of systems of two linear parabolic type PDEs that arise from the scalar complex linear (1 + 1) parabolic PDE (1). The next section deals with semi-invariants, Ibragimov-type invariants, under transformations of the dependent variables for such systems. Then we consider some special cases of this system of two parabolic type PDEs, i.e. uncoupled and special coupled parabolic systems. Herein we also mention the inverse problem. The subsequent section is on the applications of our results. Concluding remarks are given in Section 5. 2. Equivalence transformations Equivalence transformations of the dependent variable only for the parabolic PDE (1) are the linear transformations
; u ¼ rðt; xÞu
rðt; xÞ – 0;
ð2Þ
that map the family (1) into itself. That is under the map (2) the parabolic equation (1) remains linear and homogeneous but the transformed PDE
u u xx þ b x þ cu t ¼ a u
ð3Þ
and c which are ; b has new coefficients a
¼ a; a
¼ b þ 2 rx a; b
r
c ¼
LðrÞ
r
ð4Þ
in which the operator L is defined as
L¼a
@2 @ @ þ b þ c: @x @t @x2
ð5Þ
We now focus our attention on the linear parabolic type system of two PDEs in two independent variables t and x
v t ¼ a1 v xx þ a2 wxx þ b1 v x þ b2 wx þ c1 v þ c2 w; wt ¼ a3 v xx þ a4 wxx þ b3 v x þ b4 wx þ c3 v þ c4 w;
ð6Þ
in which the coefficients ai, bi and ci where i = 1, . . . , 4 are continuous functions of t and x in some domain. The equivalence transformations of the dependent variables, as can be verified, of this family (6) of linear parabolic type equations into itself are as follows
v ¼ s1 ðt; xÞv þ s2 ðt; xÞw; w ¼ s3 ðt; xÞv þ s4 ðt; xÞw;
ð7Þ
where the si’s are arbitrary functions of their arguments. The new coefficients are in terms of the si’s and the old coefficients.
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We wish to restrict to a special class of the system (6) which corresponds to a scalar linear parabolic equation (1) considered in the complex domain, i.e. u(t, x) = v(t, x) + iw(t, x), while the coefficients in (1) are also complex functions of the independent variables t and x so as to extract a system of two linear parabolic type PDEs of the form
v t ¼ a1 v xx a2 wxx þ b1 v x b2 wx þ c1 v c2 w; wt ¼ a1 wxx þ a2 v xx þ b1 wx þ b2 v x þ c1 w þ c2 v ;
ð8Þ
which belongs to a special class of systems of linear parabolic PDEs obtainable by CSA. The subset of equivalence transformations of the dependent variables (7) of the form
v ¼ r1 ðt; xÞv r2 ðt; xÞw; w ¼ r2 ðt; xÞv þ r1 ðt; xÞw;
ð9Þ
are arrived at by the complex transformations (2) if r(t, x) = r1(t, x) + ir2(t, x). This subset of the equivalence transformations (9) transforms the system of linear PDEs (8) to
v t ¼ a1 v xx a2 w xx þ b1 v x b2 w x þ c1 v c2 w; 1 w 2 v xx þ b1 w t ¼ a xx þ a x þ b2 v x þ cc1 w þ c2 v ; w
ð10Þ
where
1 ¼ a1 ; a
2 ¼ a2 ; a
1 ¼ b1 þ 2ðr1 ða1 r1x a2 r2x Þ þ r2 ða1 r2x þ a2 r1x ÞÞ ; b r21 þ r22 2ð r ða r a 2 1 1x 2 r2x Þ þ r1 ða1 r2x þ a2 r1x ÞÞ 2 ¼ b2 þ b ; r21 þ r22 r ðL r L2 r2 Þ þ r2 ðL1 r2 þ L2 r1 Þ c1 ¼ 1 1 1 ; r21 þ r22 r ðL r þ L2 r1 Þ r2 ðL1 r1 L2 r2 Þ c2 ¼ 1 1 2 r21 þ r22
ð11Þ
and the operators L1 and L2 are the real and imaginary parts of the complex operator (5), i.e.
L1 ¼ a1
@2 @ @ þ b1 þ c 1 ; @x @t @x2
L2 ¼ a2
@2 @ @ þ b2 þ c 2 : @x @t @x2
ð12Þ
Remark. The linear parabolic type system (8) is a special class of the family of the linear parabolic type system (6). Eq. (8) have six arbitrary coefficients whereas (6) has twelve arbitrary coefficients. We now discuss the case for which system (8) reduces to an uncoupled and a coupled system of two PDEs. This clearly occurs for a2 = b2 = c2 = 0 and a1 = b1 = c1 = 0 respectively, or alternatively the coefficients a, b and c are real or pure imaginary. Thus the uncoupled parabolic system is.
v t ¼ a1 v xx þ b1 v x þ c1 v ; wt ¼ a1 wxx þ b1 wx þ c1 w;
ð13Þ
while the coupled one is
v t ¼ a2 wxx b2 wx c2 w; wt ¼ a2 v xx þ b2 v x þ c2 v :
ð14Þ
We return to these systems (13) and (14) later. To conclude this section we summarize that the system (8) has equivalence transformations which are linear changes in the dependent variables (9) and it is mapped into the same family with in general new coefficients given by (11). 3. Semi-invariants under dependent variables changes In this section we deal with the Ibragimov-type invariants of the linear parabolic type system (8) under the dependent variables transformations (9). We derive Ibragimov-type invariants of the class of linear parabolic type PDEs (8) under the dependent variables (9) by reverting to the base scalar complex linear parabolic equation (1) with complex coefficients. We consider the following two nontrivial semi-invariants a and K [10]
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1 2 b ax þ at þ aaxx a2x b þ ðaax abÞbx abt a2 bxx þ 2a2 cx ; 2
K¼
ð15Þ
referred to as a Laplace-type invariants in [10]. We refer to it as the Ibragimov invariant. The Ibragimov semi-invariant K can be written compactly [13] as
k¼
! 2 @ b b @ b þ ca ; @x 2a x 4a @t 2a
ð16Þ
where K = 2a2k. We have the following results based on the semi-invariants a and k. The scalar linear parabolic PDE (1) is equivalent to the ¼ 0, via the linear change (2) if and only if parabolic equation (3), i.e. Lu
¼ a; a
¼ k; k
ð17Þ
is where k
! ! ! 2 b b @ @ b c a : þ k¼ 2a @x @t 2a 4a
ð18Þ
x
The construction of r in the transformation (2) is achieved by solving the system [13]
! 2 2 b b b þ : c þ a 2 a 4a 4 a x
rx b b rt b ; ¼ ¼ca 2a r 2a 2a r
ð19Þ
x
As an example which we use later, if (1) has constant coefficients, then it can be reduced to the classical heat equation
u xx ; t ¼ a u
¼ constant; a
ð20Þ
under the change of the dependent variable
! 2 b b ; u ¼ exp ct t x u 2a 4a
ð21Þ
¼ k. This transformation is a consequence of the relations (19), viz. ¼ a and k since a 2 rx b rt b ¼ ; ¼c 2a r r 4a
ð22Þ
and its solution is (21) up to a constant. We now derive the Ibragimov-type invariants of the linear parabolic type system (8) under linear changes of the depen i.e. and k dent variables (9) by the complex split of a; k; a
1 ¼ a1 ; a
2 ¼ a2 ; a
1 ¼ k1 ; k
2 ¼ k2 ; k
ð23Þ
where
" 2 #! 2 @ 1 a1 b1 þ a2 b2 a1 b2 a2 b1 1 a1 b1 a1 b2 þ 2a2 b1 b2 1 þ c1 a1 a 2 2 2 2 2 @x 2 4 2 a21 þ a22 a þ a a þ a 1 2 1 2 x x " 2 #! 2 @ 1 a1 b2 a2 b1 a1 b1 þ a2 b2 1 a2 b2 a2 b1 þ 2a1 b1 b2 1 þ k2 ¼ c2 a1 þ a2 2 2 2 2 @x 2 4 2 a21 þ a22 a þ a a þ a 1 2 1 2 x x
k1 ¼
@ a1 b1 þ a2 b2 ; 2 2 @t a1 þ a2 @ a1 b2 a2 b1 : 2 2 @t a1 þ a2
ð24Þ
Note that the transformed k’s are with the bars over the coefficients aj, bj and cj where j = 1, 2. We state the following result. Proposition 1. The linear parabolic system
L1 v L2 w v t þ a1 v xx a2 wxx þ b1 v x b2 wx þ c1 v c2 w ¼ 0; L1 w þ L2 v wt þ a1 wxx þ a2 v xx þ b1 wx þ b2 v x þ c1 w þ c2 v ¼ 0;
ð25Þ
is equivalent to the transformed linear parabolic type system
1 v x b 2 w 1 v xx a 2 w v t þ a xx þ b x þ c1 v c2 w ¼ 0; L1 v L2 w 1 w 2 v x þ cc1 w 1 w 2 v xx þ b t þ a þ L2 v w xx þ a x þ b þ c2 v ¼ 0; L1 w
ð26Þ
by means of the linear transformations (9) if and only if (23) holds, i.e.
1 ¼ a1 ; a
2 ¼ a2 ; a
1 ¼ k1 ; k
2 ¼ k2 ; k
ð27Þ
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where
" ! !# " #! ! þa 2 2 a 1 2 a 2 þ 2a 1 b 2 1 þ a 2 b 1 b 1 b 1 b 2 b 2 b 1 b 2 b 2 b a a 1 a 1 @ a 1 2 1 ¼ @ c1 1 a 1 1 12 k þ ; a 2 1 þ a 21 þ a 21 þ a 21 þ a 22 22 22 22 @x 2 4 2 @t a a a a x x " ! !# " #! ! 2 a 1 1 þ a 2 2 a 2 þ 2a 1 b 2 2 a 1 2 b 2 b 2 b 1 b 2 b 1 b 1 b 2 b 1 b @ 1 1 a 1 @ a a a 1 1 k2 ¼ c2 a1 þ a2 þ 21 þ a 21 þ a 21 þ a 21 þ a 22 22 22 22 @x 2 4 2 @t a a a a x
ð28Þ
x
and r1, r2 in the linear transformations (9) can be deduced from (19) via a complex split. The proof of this proposition follows immediately from the preceding discussion including the construction of the linear transformation encapsulated in r. 3.1. Uncoupled parabolic systems Here we investigate the Ibragimov-type invariants of the system (8) when it maps to the system of the form (13). Notice that these systems are special cases of the system of PDEs (8). Now we want the target parabolic type system to be uncoupled, i.e. of the form
v t ¼ a1 v xx þ b1 v x þ c1 v ;
ð29Þ
1 w 1 w t ¼ a xx þ b x þ c1 w: w 2 ¼ 0 and 1 ¼ a1 ; a 2 ¼ 0; k The transformed semi-invariants thus satisfy a
1 b 1 ¼ @ c1 a 1 k 1 @x 2a
! x
2 b 1 1 4a
!
! 1 @ b þ : 1 @t 2a
ð30Þ
3.2. Special coupled parabolic type systems Suppose that the target system is a coupled system of the special form
v t ¼ a2 w xx b2 w x c2 w; 2 v xx þ b2 v x þ c2 v : t ¼ a w
ð31Þ
1 ¼ 0 and 1 ¼ 0; a 2 ¼ a2 ; k Then the transformed semi-invariants thus satisfy a
2 b 2 ¼ @ c2 a 2 k 2 @x 2a
! x
2 b 2 2 4a
! þ
! 2 @ b : 2 @t 2a
ð32Þ
In summary of these sub-sections, we have the following result. Proposition 2. The linear parabolic type system (8) is reducible to the uncoupled system (29) and the special coupled system (31) ¼ 0 and k as in (30) and a ¼ 0 and k as 1 ¼ a1 ; a 2 ¼ 0; k 2 ¼ a2 ; a 1 ¼ 0; k via equivalence transformations (9) if and only if a 2 1 1 2 in (32), respectively. The proof follows at once from the discussion contained above in Sections 3.1 and 3.2.We can also deduce the following consequence. Corollary. The parabolic type system (8) is reducible to the simpler uncoupled and coupled systems
v t ¼ a1 v xx ; 1 w t ¼ a xx w
ð33Þ
and
v t ¼ a2 w xx ; 2 v xx ; t ¼ a w
ð34Þ
1 ¼ a1 ; a 2 ¼ 0 and a 1 ¼ 0; a 2 ¼ a2 , respectively by means of the linear dependent variables changes of the form (9) if and only if a 1 ¼ k1 ¼ 0 ¼ k2 ¼ k 2 for both the cases, viz. uncoupled and special coupled systems of PDEs. while k Note that if a1 is constant in the Corollary then we get criteria for reduction to the simplest system which is the classical heat system of PDEs. The linear parabolic type uncoupled and coupled systems of PDEs (13) and (14) are not only the special cases of (8) but these also appear as very special and restricted subcases of the general class of systems of two linear parabolic type PDEs (6), i.e. if we extract the uncoupled and coupled systems of parabolic type PDEs from (6) and restrict the remaining non-zero coefficients as they appear in (13) and (14) we obtain subcases of these systems of (6). Thus the Proposition 2 implies that
F.M. Mahomed et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 3140–3147
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the sub-class (9) of the equivalence transformations (7) is found to map special subcases (uncoupled and coupled linear parabolic type systems of two PDEs) of the general class (6) into one of its simplest forms or the same family of equations with different coefficients. We can also address the inverse problem of when a system of two linear parabolic type equations in two independent variables arise from analytic continuation of a scalar (1 + 1) linear parabolic PDE in a simple way. Indeed this is the case if and only if the system (6) has coefficients which satisfy (8). In the next section we pursue some applications of our results. 4. Applications We consider examples of certain parabolic type systems and their reductions. 1. Consider the constant coefficient parabolic type system
v t ¼ a1 v xx a2 wxx þ b1 v x b2 wx þ c1 v c2 w; wt ¼ a1 wxx þ a2 v xx þ b1 wx þ b2 v x þ c1 w þ c2 v :
ð35Þ
This is reducible to the classical heat system
v t ¼ a1 v xx a2 w xx ; 2 v xx þ a 1 w t ¼ a xx ; w
ð36Þ
since
1 ¼ a1 ; a
2 ¼ a2 ; a
1 ¼ k1 ¼ 0; k
2 ¼ k2 ¼ 0: k
ð37Þ
The transformation is given by the complex split of (21) i.e.
v ¼ ea ðv cosða2 Þ w sinða2 ÞÞ; cosða2 Þ þ v sinða2 ÞÞ; w ¼ e a ðw
ð38Þ
1 ! 2 2 a1 b1 b2 þ 2a2 b1 b2 a1 b1 þ a2 b2 @ A t a1 ¼ c1 x; 4 a21 þ a22 2 a21 þ a22 ! 2 2 2a b b a b þ a b a b a b a2 ¼ c2 1 1 2 2 2 12 2 2 t 1 22 22 1 x: 2ða1 þ a2 Þ 4 a1 þ a2
ð39Þ
1
1
where
0
Notice that this transformation is much more complicated than the corresponding transformation (21), but elegantly furnishes the equivalence of (35) and (36). 2. The time-dependent linear parabolic type system
v t ¼ xv xx þ c1 ðtÞv c2 ðtÞw; wt ¼ xwxx þ c1 ðtÞw þ c2 ðtÞv ;
ð40Þ
has a1 = x, k1 = 0 = k2 and by the Corollary is reducible to
v t ¼ xv xx ; xx : t ¼ xw w
ð41Þ
The transformation is
Z sin c2 ðtÞdt w c2 ðtÞdt ; Z Z R cos c2 ðtÞdt : c2 ðtÞdt þ v sin w ¼ e c1 ðtÞdt w R
v ¼e
c1 ðtÞdt
v cos
Z
ð42Þ
This is quite a messy transformation compared to the scalar case of the PDE
ut ¼ xuxx þ cðtÞu;
ð43Þ
1 2 2 A x v xx Bxv x þ C v ; 2 1 wt ¼ A2 x2 wxx Bxwx þ Cw; 2
ð44Þ
R . t ¼ xu xx via u ¼ e cðtÞdt u which reduces to u 3. Consider the variable coefficient linear diffusive system
vt ¼
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which actually corresponds to the complex Black–Scholes equation (set u = v + iw)
1 ut ¼ A2 x2 uxx Bxux þ Cu: 2
ð45Þ
1 ¼ k1 ¼ 0 and k 2 ¼ k2 ¼ 0. Thus the system (44) can be reduced to 1 ¼ a1 ; a 2 ¼ a2 ¼ 0; k Here a
1 2 2 A x v xx ; 2 1 xx ; t ¼ A 2 x2 w w 2
ð46Þ
2 1 1 C B B2 A2 txB=A v ; 2 2
2 1 1 w ¼ C B B2 A2 txB=A w: 2 2
ð47Þ
v t ¼
via
v¼
4. Now we finally consider some special cases, i.e. uncoupled and coupled parabolic type systems of PDEs. If a2 = b2 = c2 = 0 or a1 = b1 = c1 = 0 in (39) then as a result we find the following equivalence transformations
v ¼ ea v ; 3
ð48Þ
a3
w ¼ e w; and
v ¼ ea ðv cosða5 Þ w sinða5 ÞÞ; cosða5 Þ þ v sinða5 ÞÞ; w ¼ ea ðw 4
4
ð49Þ
b2 b2 b1 b2 respectively. Here a3 ¼ c1 4a11 t 2a x; a4 ¼ 2a x and a5 ¼ c2 4a22 t. By the use of (48) and (49) the systems 1 2
v t ¼ a1 v xx þ b1 v x þ c1 v ; wt ¼ a1 wxx þ b1 wx þ c1 w;
ð50Þ
and
v t ¼ a2 wxx b2 wx c2 w; wt ¼ a2 v xx þ b2 v x þ c2 v ;
ð51Þ
are transformable to the following uncoupled and coupled systems of PDEs
v t ¼ a1 v xx ; 1 w xx ; t ¼ a w
ð52Þ
and
v t ¼ a2 w xx ; 2 v xx ; t ¼ a w
ð53Þ
respectively, which satisfy the invariant conditions accordingly. If we have another uncoupled system with variable coefficients
x2 4 x2 wt ¼ wxx þ xwx þ w; 4
v t ¼ v xx þ xv x þ v ;
ð54Þ
it is equivalent to the simpler system of the form
v t ¼ v xx þ v ; xx þ w; t ¼ w w
ð55Þ
via the equivalence transformations 3 2
1 2 4
v ¼ v eð tþ x Þ ; 3
1 2
¼ weð2tþ4x Þ : w 1 ¼ k 2 ¼ 0. ¼ a ¼ 1 and k1 ¼ k2 ¼ k In this case we have a
ð56Þ
F.M. Mahomed et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 3140–3147
3147
5. Concluding remarks Linear parabolic type equations have been studied by many researchers and several of them have focused mainly on the scalar equation. Here we used complex methods for systems of two linear parabolic type equations of two independent variables of a special class to deduce new semi-invariants called Ibragimov-type semi-invariants. We have shown that the semiinvariants under equivalence transformations of the dependent variables, restricted to a special class, can be achieved for a special class of two linear parabolic type equations in two independent variables that can be obtained from the analytic continuation of the scalar complex linear (1 + 1) parabolic equation. The Ibragimov-type semi-invariants as well as maps for this class of systems of two linear parabolic type equations are found to correspond to the complex maps and Ibragimov invariants of the corresponding complex scalar (1 + 1) parabolic PDE. It is now clear that this special class of systems of two linear parabolic type equations have different algebraic properties when its symmetries are deduced from its complex counterpart compared to those of the system itself (see [15]). However we have proved the consistency of the equivalence criteria, i.e. semi-invariants of the system of two linear parabolic type PDEs always correspond to the complex Ibragimov semiinvariants of the base complex PDE. Moreover we provided an answer to the inverse problem of when a system of two linear parabolic type PDEs arise from a scalar complex linear (1 + 1) parabolic equation. This is the case if the coefficients coincide with those of system (8). Furthermore, we gave various examples to illustrate our approach. Acknowledgements M. Safdar thanks the School of Computational and Applied Mathematics for support and hospitality during the time this work was initiated. FM is grateful to NUST-CAMP and the HEC of Pakistan for a visiting professorship during which time this work was completed. References [1] Ali S, Mahomed FM, Qadir A. Linearizability criteria for systems of two second order differential equations by complex methods. Nonlinear Dyn 2011;66:77. [2] Ali S, Mahomed FM, Qadir A. Complex Lie symmetries for scalar second order ordinary differential equations. Nonlinear Anal: RWA 2009;10:3335. [3] Safdar M, Qadir A, Ali S. Inequivalence of classes of linearizable systems of cubically semi linear ordinary differential equations obtained by real and complex symmetry analysis. Math Comp Appl 2011;16(4):923. [4] Arecchi FT, Rodari GS, Sona A. Statistics of the laser radiation at threshold. Phys Lett 1967;25A(1):59. [5] Barone A, Paterno G. Physics and applications of the Josephson effect. New York: John Wiley; 1982. [6] Black J, Scholes M. The pricing of options and corporate liabilities. J Polit Econ 1973;81:637. [7] Bluman GW. On the transformation of diffusion processes into the Wiener process. SIAM J Appl Math 1980;39(2):238. [8] Gazizov RK, Ibragimov NH. Lie symmetry analysis of differential equations in finance. Nonlinear Dyn 1998;17:387. [9] Goard J. New solutions to the bond-pricing equation via Lie’s classical method. Math Comput Modell 2000;32:299. [10] Ibragimov NH. Laplace type invariants for parabolic equations. Nonlinear Dyn 2002;28:125. [11] Johnpillai IK, Mahomed FM. Singular invariant equation fro the (1 + 1) Fokker–Planck equation. J Phys A: Math Gen 2001;34:11033. [12] Lie S. On integration of a class of linear partial differential equations by means of definite integrals. Archiv for Mathematik og Naturvidenskab 1881;VI(3):328. Reprinted in S. Lie, Gesammelte Abhadlundgen, 3 paper XXXV, [in German]. [13] Mahomed FM. Complete invariant characterization of scalar linear (1 + 1) parabolic equations. JNMP 2008;15:112. [14] Mahomed FM, Pooe CA. Invariant equations for (1 + 1) linear parabolic equations. In: Rudenko OV, Sapozhnikov OA, editor. Proceedings of the international conference modern group analysis 9, 16th ISNA, Moscow State University, Moscow; 2003. p. 525. [15] Mahomed FM, Naz Rehana. A note on the Lie symmetries of complex partial differential equations and their split real systems. Pramana – J Phys 2011;77(3):483. [16] Ovsiannikov LV. Group properties of differential equations. New York: Academic Press; 1982. [17] Pooe CA, Mahomed FM, Wafo Soh C. Fundamental solutions for zero-coupon bond pricing models. Nonlinear Dyn 2004;36:69. [18] Risken H. The Fokker–Planck equation. Methods of solution and applications. Berlin: Springer-Verlag; 1989. [19] Shtelen WM, Stognii VI. Symmetry properties of of one- and two-dimensional Fokker–Planck equations. J Phys A: Math Gen 1989;22:L539. [20] Spichak S, Stognii V. Symmetry classification and exact solutions of the one-dimensional Fokker–Planck equation with arbitrary coefficients of drifts and diffusion. J Phys A: Math Gen 1999;32:8341. [21] Wilmott P. Derivatives: the theory and practice of financial engineering. West Sussex England: John Wiley; 1998.