Linearization criteria for systems of two second-order stochastic ordinary differential equations

Applied Mathematics and Computation 301 (2017) 25–35

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Linearization criteria for systems of two second-order stochastic ordinary differential equations T.G. Mkhize a,b, K. Govinder b, S. Moyo c, S.V. Meleshko d,∗ a

Durban University of Technology, Department of Mathematics, PO Box 1334, Durban 4000, South Africa University of KwaZulu-Natal, School of Mathematics, Statistics and Computer Science, Private Bag X5400, Durban 4000, South Africa c Durban University of Technology, Department of Mathematics & Institute for Systems Science, PO Box 1334, Durban 4000, South Africa d Suranaree University of Technology, School of Mathematics, Nakhon Ratchasima 30000, Thailand b

a r t i c l e

i n f o

MSC: 34F05 60H10 Keywords: Stochastic ordinary differential equations Linearization Itô formula Equivalence transformation

a b s t r a c t We provide the necessary and sufficient conditions for the linearization of systems of two second-order stochastic ordinary differential equations. The linearization criteria are given in terms of coefficients of the system followed by some illustrations. This paper gives a new treatment for the linearization of two second-order stochastic ordinary differential equations and with some examples. © 2016 Elsevier Inc. All rights reserved.

1. Introduction Stochastic ordinary differential equations (SODEs) include a stochastic component which describes the randomness within the differential equations. SODEs are in general nonlinear and their solutions are difficult to obtain. Various methods of solving differential equations involve applying a change of variables to transform a given differential equation in to another equation with known properties. The class of linear equations is known to be the simplest class of equations for which it is easier to find a solution, hence, the existence of the problem of transforming a given differential equation into a linear equation. This problem, called a linearization problem, is a particular case of an equivalence problem [1,3,7]. Linear SODEs play a role similar to that of linear equations in the deterministic theory of ordinary differential equations (ODEs). However, the change of variables in SODEs differs from that in ODEs due to the Itô formula. The transformation of nonlinear SODEs into linear ones via an invertible stochastic mapping prove to be useful in obtaining the closed form solutions [2,7,9,13]. In this paper, we present a general linearizability criteria for the systems of two second-order SODEs. We consider the system of two second-order SODEs,

˙ Y˙ ) dt + g1 (t, X, Y, X, ˙ Y˙ ) dW dX˙ = f1 (t, X, Y, X, ˙ ˙ ˙ ˙ dY = f2 (t, X, Y, X,Y ) dt + g2 (t, X, Y, X,Y˙ ) dW,

(1)

where fi and gi , (i = 1, 2 ) are deterministic functions and dW is the infinitesimal increment of the Wiener process [8]. System (1) is said to be linear if the functions fi and gi are linear functions with respect to variables X and Y and their respective derivatives. For the linearization problem one considers the class of equations equivalent to linear equations. ∗

Corresponding author. E-mail addresses: [email protected] (T.G. Mkhize), [email protected] (K. Govinder), [email protected] (S. Moyo), [email protected], [email protected] (S.V. Meleshko). http://dx.doi.org/10.1016/j.amc.2016.12.019 0 096-30 03/© 2016 Elsevier Inc. All rights reserved.

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Thus a linear system of two second-order SODEs has the form,

 α11 (t )X + α12 (t )Y + α13 (t )X˙ + α14 (t )Y˙ + α10 (t ) dt   + β11 (t )X + β12 (t )Y + β13 (t )X˙ + β14 (t )Y˙ + β10 (t ) dW,   dY˙ = α21 (t )X + α22 (t )Y + α23 (t )X˙ + α24 (t )Y˙ + α20 (t ) dt   + β21 (t )X + β22 (t )Y + β23 (t )X˙ + β24 (t )Y˙ + β20 (t ) dW.

dX˙ =



(2)

We can rewrite (2) in the form of first-order SODEs:

dX = X˙ dt,     dX˙ = AX + BX˙ + a dt + F1 X + F2 X˙ + b dW,

(3)

where A(t), B(t), Fi (i = 1, 2 ) are 2 × 2 matrices; a(t), b(t) are vectors and

 

X=

X . Y

Similar to the treatment of ordinary differential equations (ODEs), the linearization problem involves finding a change of the dependent variables,

x¯ = ϕ (t, x, y ), y¯ = ψ (t, x, y ),  = ϕx ψy − ϕy ψx = 0, which can transform the system of equations given in (1) into linear SODEs (2). Lie [6] laid a foundation for the linearization criteria of the second-order ODEs via an invertible point transformation. He showed that the second-order ODE

x¨ = f (t, x, x˙ ),

(4)

is linearizable by a change of both the independent and dependent variables provided f is a polynomial of the third degree with respect to the first-order derivative,

x¨ + F x˙ 3 + Gx˙ 2 + H x˙ + L = 0, where the coefficients F(t, x), G(t, x), H(t, x) and L(t, x) satisfy the conditions

K1 = 3Ftt − 2Gxt + Hxx − 3Ft H + 3Fx L + 2Gt G − 3Ht F − Hx G + 6Lx F = 0

(5)

K2 = Gtt − 2Hxt + 3Lxx − 6Ft L + Gt H + 3Gx L − 2Hx H − 3Lt F + 3Lx G = 0. Eq. (4) is also linearizable by a change of the dependent variable x provided F = 0 in which conditions (5) become

K1 = (−2Gt + Hx )x − G(−2Gt + Hx ) = 0

(6)

K2 = (Gt − 2Hx )t + H (Gt − 2Hx ) + 3(Lx + GL )x = 0. Lie’s linearizability criteria for second-order ODEs was extended to the system of second-order ODEs by the authors in [1,10,14] and the references therein. In Bagderina [1], a study of the linearization problem of the system of two second-order ODEs was completed. Modifiying Lie’s work for ODEs to SODEs has been done by [11], extended by [7] to the second-order SODEs and in [9,12,13], the conditions for the invertible transformations which linearize the jump-diffusion are obtained. The reducibility approach was used in [15] to study the linearization problem of stochastic differential equations (SDEs) with fractional Brownian motion. This work, however, misused the fractional Itô formula to derive the reducibility conditions of nonlinear fractional SDEs to linear fractional SDEs. This was reviewed and corrected in [5]. The rest of the paper is organized as follows: Section 2 discusses an equivalence transformation used to reduce the number of coefficients α ij in system (2). In Section 3, the determining equations are derived in an Itô calculus context. These determining equations are non-stochastic. The linearization criteria for a system of two second-order SODEs are given in terms of coefficients of the system. The later part of the paper deals with the β ij s also from Eq. (2) and the analysis of relations for them is given. The main result and Theorem are given in Section 3. Section 4 gives some examples and the conclusion is given in Section 5. To the best of our knowledge this is a new contribution on the linearization problem of systems of two second-order SODEs. 2. Equivalence transformation We consider the transformation

X1 = C (t )X + h(t ), where C = C (t ) is a nonsingular matrix and h(t) a vector.

(7)

T.G. Mkhize et al. / Applied Mathematics and Computation 301 (2017) 25–35

27

Using transformation (7), system (3) becomes

dX1 = X˙ 1 dt,     dX˙ 1 = A¯ X1 + B¯ X˙ 1 + a¯ dt + F¯1 X1 + F¯2 X˙ 1 + b¯ dW,

(8)

where

B¯ = C −1 (BC − 2C˙ ), F¯1 = CF1 ,

F¯2 = CF2 ,

A¯ = C −1 (AC + BC˙ − C¨ ),

a¯ = C −1 (Ah + Bh˙ − h¨ ),

b¯ = C b.

Choosing C and h such that

C˙ =

1 BC, 2

h¨ = Bh˙ + Ah,





we have

1 1 A¯ = C −1 A + B2 − B˙ C, 4 2 where we assume that for solving the linearization problem,

B = 0,

a=0

or

α10 = α13 = α14 = α20 = α23 = α24 = 0. Hence the equivalence transformation here is used to reduce the number of coefficients α ij in system (2). The rest of the α ij ’s will be given in the next section. 3. Linearization criteria for system of two second-order SODEs Given a system of two second-order SODEs,

dX˙ = f1 (t, X, Y, X˙ , Y˙ ) dt + g1 (t, X, Y, X˙ , Y˙ ) dW dY˙ = f2 (t, X, Y, X˙ , Y˙ ) dt + g2 (t, X, Y, X˙ , Y˙ ) dW,

(9)

system (9) is reduced to a system of first-order stochastic ordinary differential equations

d X = P dt dX˙ = f1 (t, X, Y, P, Q ) dt + g1 (t, X, Y, P, Q ) dW dY = Q dt dY˙ = f2 (t, X, Y, P, Q ) dt + g2 (t, X, Y, P, Q ) dW.

(10)

We then apply the change of variables

x1 = ϕ (t, x, y ), p1 = ϕ2 (t, x, y, p, q );

y1 = ψ (t, x, y ), q1 = ψ2 (t, x, y, p, q )

(11)

with

 = ϕx ψy − ϕy ψx = 0, and using the Itô formula [8] for (10), we obtain

dX1 = ϕ2 (t, X, Y, P, Q )dt, dX˙1 = f˜1 (t, X, Y, P, Q )dt + g˜1 (t, X, Y, P, Q )dW ;

(12)

dY1 = ψ2 (t, X, Y, P, Q )dt, dY˙1 = f˜2 (t, X, Y, P, Q )dt + g˜2 (t, X, Y, P, Q )dW ; where

ϕ2 = ϕt + pϕx + qϕy , ψ2 = ψt + pψx + qψy ; f˜1 (t, x, y, p, q ) =

 1 1 ϕ2t + pϕ2x + qϕ2y + f1 ϕ2 p + f2 ϕ2q + g21 ϕ2 pp + g1 g2 ϕ2 pq + g22 ϕ2qq (t, x, y, p, q );



g˜1 (t, x, y, p, q ) = g1 ϕ2 p + g2 ϕ2q (t, x, y, p, q );

2

2

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and

1 1 ψ2t + pψ2x + qψ2y + f1 ψ2 p + f2 ψ2q + g21 ψ2 pp + g1 g2 ψ2 pq + g22 ψ2qq (t, x, y, p, q ); 2 2

 g˜2 (t, x, y, p, q ) = g1 ψ2 p + g2 ψ2q (t, x, y, p, q ). f˜2 (t, x, y, p, q ) =



Equating f˜1 , g˜1 , f˜2 and g˜2 with the linear form (2), we obtain four equations,

1 2

1 2

1 2

ϕ2t + pϕ2x + qϕ2y + f1 ϕ2 p + f2 ϕ2q + g21 ϕ2 pp + g1 g2 ϕ2 pq + g22 ϕ2qq = α11 ϕ + α12 ψ + α13 ϕ2 + α14 ψ2 + α10 ; g1 ϕ2 p + g2 ϕ2q = β11 ϕ + β12 ψ + β13 ϕ2 + β14 ψ2 + β10 ; and

1 2

1 2

1 2

ψ2t + pψ2x + qψ2y + f1 ψ2 p + f2 ψ2q + g21 ψ2 pp + g1 g2 ψ2 pq + g22 ψ2qq = α21 ϕ + α22 ψ + α23 ϕ2 + α24 ψ2 + α20 ; g1 ψ2 p + g2 ψ2q = β21 ϕ + β22 ψ + β23 ϕ2 + β24 ψ2 + β20 . Substituting the functions ϕ 2 and ψ 2 from (11) into the above equations yields the following conditions:

ϕxx p2 + 2ϕxy pq + ϕyy q2 + (2ϕxt − α13 ϕx − α14 ψx ) p + (2ϕyt − α13 ϕy − α14 ψy )q + f1 ϕx + f2 ϕy + ϕtt − α13 ϕt − α14 ψt − α11 ϕ − α12 ψ − α10 = 0, ψxx p2 + 2ψxy pq + ψyy q2 + (2ψxt − α23 ϕx − α24 ψx ) p + (2ψyt − α23 ϕy − α24 ψy )q + f1 ψx + f2 ψy + ψtt − α23 ϕt − α24 ψt − α21 ϕ − α22 ψ − α20 = 0,

(13)

g1 ϕx + g2 ϕy = β11 ϕ + β12 ψ + β13 (ϕt + pϕx + qϕy ) + β14 (ψt + pψx + qψy ) + β10 , g1 ψx + g2 ψy = β21 ϕ + β22 ψ + β23 (ϕt + pϕx + qϕy ) + β24 (ψt + pψx + qψy ) + β20 .

(14)

Thus the two pairs of conditions (13) and (14) are necessary and sufficient for the SODEs to be linearizable. The necessary representation of the functions f1 , f2 , g1 and g2 are

f1 = a11 p2 + 2a12 pq + a13 q2 + a14 p + a15 q + a10 , f2 = a21 p2 + 2a22 pq + a23 q2 + a24 p + a25 q + a20 ,

(15)

g1 = b11 p + b12 q + b10 , g2 = b21 p + b22 q + b20 .

(16)

Here the coefficients aij and bik , (i = 1, 2 ), ( j = 0, .., 5 ), (k = 0, 1, 2 ) are functions of x, y and t. Substituting the representations of the function fi , (i = 1, 2 ) in Eq. (15) into (13) and splitting them with respect to p and q, we obtain the overdetermined system of equations for the functions ϕ and ψ , that is,

ϕxx = −a11 ϕx − a21 ϕy ; ϕyy = −a13 ϕx − a23 ϕy ; ϕxy = −a12 ϕx − a22 ϕy ; 1 1 2 2 = −a10 ϕx − a20 ϕy + α11 ϕ + α12 ψ ;

ϕxt = − [a14 ϕx + a24 ϕy ]; ϕyt = − [a15 ϕx + a25 ϕy ]; ϕtt

(17)

and

ψxx = −a11 ψx − a21 ψy ; ψyy = −a13 ψx − a23 ψy ; ψxy = −a12 ψx − a22 ψy ; 1 1 2 2 = −a10 ψx − a20 ψy + α21 ϕ + α22 ψ .

ψxt = − [a14 ψx + a24 ψy ]; ψyt = − [a15 ψx + a25 ψy ]; ψtt

(18)

We assume that the coefficients aij and bik are given. Compatibility analysis of system (17) and system (18) gives conditions for these coefficients which are sufficient for linearization. Comparing all mixed derivatives of the third order leads to the following equations:

A11,1 ϕx + A11,2 ϕy = 0, A11,1 ψx + A11,2 ψy = 0,

(19)

A12,1 ϕx + A12,2 ϕy = 0, A12,1 ψx + A12,2 ψy = 0,

(20)

A13,1 ϕx + A13,2 ϕy = 0, A13,1 ψx + A13,2 ψy = 0,

(21)

A14,1 ϕx + A14,2 ϕy = 0, A14,1 ψx + A14,2 ψy = 0,

(22)

T.G. Mkhize et al. / Applied Mathematics and Computation 301 (2017) 25–35

29

A15,1 ϕx + A15,2 ϕy = 0, A15,1 ψx + A15,2 ψy = 0,

(23)

A16,1 ϕx + A16,2 ϕy = 0, A16,1 ψx + A16,2 ψy = 0,

(24)

(A17,1 − α11 )ϕx + A17,2 ϕy − α12 ψx = 0, −α21 ϕx + (A17,1 − α22 )ψx + A17,2 ψy = 0,

(25)

A18,1 ϕx + (A18,2 − α11 )ϕy − α12 ψy = 0, −α21 ϕy + A18,1 ψx + (A18,2 − α22 )ψy = 0,

(26)

where

A11,1 = −a11y + a12x − a12 a22 + a13 a21 ,

A11,2 = −a21y + a22x + a11 a22 − a12 a21 + a21 a23 − a222 ,

A12,1 =

1 (a14x − 2a11t − a12 a24 + a15 a21 ), 2

A12,2 =

1 (a24x − 2a21t + a11 a24 − a14 a21 + a21 a25 − a22 a24 ), 2

A13,1 =

1 (a14y − 2a12t − a13 a24 + a15 a22 ), 2

A13,2 =

1 (a24y − 2a22t + a12 a24 − a14 a22 + a22 a25 − a23 a24 ), 2

A14,1 = −a12y + a13x − a11 a13 + a212 − a12 a23 + a13 a22 ,

A14,2 = −a22y + a23x + a12 a22 − a13 a21 ,

A15,1 =

1 (−a14y + a15x − a11 a15 + a12 a14 − a12 a25 + a13 a24 ), 2

A15,2 =

1 (−a24y + a25x + a14 a22 − a15 a21 − a22 a25 + a23 a24 ), 2

A16,1 =

1 (a15y − 2a13t − a12 a15 + a13 a14 − a13 a25 + a15 a23 ), 2

A16,2 =

1 (a25y − 2a23t + a13 a24 − a15 a22 ), 2

A17,1 =

1 (4a10x − 2a14t − 4a10 a11 − 4a12 a20 + a214 + a15 a24 ), 4

A17,2 =

1 (4a20x − 2a24t − 4a10 a21 + a14 a24 − 4a20 a22 + a24 a25 ), 4

A18,1 =

1 (4a10y − 2a15t − 4a10 a12 − 4a13 a20 + a14 a15 + a15 a25 ), 4

A18,2 =

1 (4a20y − 2a25t − 4a10 a22 + a15 a22 − 4a20 a23 + a225 ). 4

Since  = 0, we can rewrite Eq. (19) in matrix form



ϕx ϕy ψx ψy



A11,1 A11,2



 

=

0 . 0

(27)

Then from Eqs. (19)–(24), it is necessary and sufficient that

A1i, j = 0,

(28)

for (i = 1, . . . , 6 ) and ( j = 1, 2 ). Solving Eqs. (25) and (26) one finds α ij , i = 1, 2; j = 1, 2 as follows:

α11 = −1 (−A18,1 ϕx ψx + A17,1 ϕx ψy − A18,2 ϕy ψx + A17,2 ϕy ψy ), α12 = −1 (A18,1 ϕx2 + (−A17,1 + A18,2 )ϕx ϕy − A17,2 ϕy2 ), α21 = −1 (−A18,1 ψx2 + (A17,1 − A18,2 )ψx ψy + A17,2 ψy2 ), α22 = −1 (A18,1 ϕx ψx + A18,2 ϕx ψy − A17,1 ϕy ψx − A17,2 ϕy ψy ), where

 = ϕx ψy − ϕy ψx . Differentiating α ij in (29) with respect to x and y, we have the conditions

A17,1x = a12 A17,2 − a21 A18,1 ;

A17,2x = −a11 A17,2 + a21 (A17,1 − A18,2 ) + a22 A17,2 ,

A18,1x = a11 A18,1 − a12 (A17,1 − A18,2 ) − a22 A18,1 ;

A18,2x = −a12 A17,2 + a21 A18,1 ,

(29)

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T.G. Mkhize et al. / Applied Mathematics and Computation 301 (2017) 25–35

A17,1y = a13 A17,2 − a22 A18,1 ;

A17,2y = −a12 A17,2 + a22 (A17,1 − A18,2 ) + a23 A17,2 ,

A18,1y = a12 A18,1 − a13 (A17,1 − A18,2 ) − a23 A18,1 ;

A18,2y = −a13 A17,2 + a22 A18,1 .

(30)

The next step involves finding the β ij and relations for them. To do this we split Eqs. (14) and (16) with respect to p and q, to obtain

−b10 ϕx − b20 ϕy + β10 + β11 ϕ + β12 ψ + β13 ϕt + β14 ψt = 0,

(31)

−b11 ϕx − b21 ϕy + β13 ϕx + β14 ψx = 0,

(32)

−b12 ϕx − b22 ϕy + β13 ϕy + β14 ψy = 0,

(33)

−b10 ψx − b20 ψy + β20 + β21 ϕ + β22 ψ + β23 ϕt + β24 ψt = 0

(34)

−b11 ψx − b21 ψy + β23 ϕx + β24 ψx = 0,

(35)

−b12 ψx − b22 ψy + β23 ϕy + β24 ψy = 0.

(36)

and

From Eqs. (31)–(36) we find

β13 = −1 (−b12 ϕx ψx + b11 ϕx ψy − b22 ϕy ψx + b21 ϕy ψy ), β14 = −1 (b12 ϕx2 + (−b11 + b22 )ϕx ϕy − b21 ϕy2 ), β23 = −1 (−b12 ψx2 + (b11 − b22 )ψx ψy + b21 ψy2 ), β24 = −1 (b12 ϕx ψx + b22 ϕx ψy − b11 ϕy ψx − b21 ϕy ψy ), β10 = −1 (B10,1 ϕt + B10,2 ψt + b10 ϕx + b20 ϕy − β11 ϕ − β12 ψ ), β20 = −1 (B20,1 ϕt + B20,2 ψt + b10 ψx + b20 ψy − β21 ϕ − β22 ψ ),

(37)

where

B10,1 = b12 ϕx ψx − b11 ϕx ψy + b22 ϕy ψx − b21 ϕy ψy , B20,1 = b12 ψx2 + (−b11 + b22 )ψx ψy − b21 ψy2 ,

B10,2 = −b12 ϕx2 + (b11 − b22 )ϕx ϕy + b21 ϕy2 ,

B20,2 = −b12 ϕx ψx − b22 ϕx ψy + b11 ϕy ψx + b21 ϕy ψy .

The equations (βi0 )x = 0 and (βi0 )y = 0, i = 1, 2 compose an algebraic system of linear equations with respect to β ij , (i = 1, 2; j = 1, 2 ) with a none vanishing determinant. Hence one can find β 11 , β 12 , β 21 , β 22 from this system. For this representation we introduce the notations

ξ1 = 2b10y − 2a12 b10 − 2a13 b20 + a15 b11 + a25 b12 , ξ2 = 2b10x − 2a11 b10 − 2a12 b20 + a14 b11 + a24 b12 , ξ3 = 2b20y + a15 b21 − 2a22 b10 − 2a23 b20 + a25 b22 , ξ4 = 2b20x + a14 b21 − 2a21 b10 − 2a22 b20 + a24 b22 , ξ5 = b12y − a12 b12 + a13 b11 − a13 b22 + a23 b12 , ξ6 = b11x − a12 b21 + a21 b12 , ξ7 = b22y + a13 b21 − a22 b12 , ξ8 = b21x + a11 b21 − a21 b11 + a21 b22 − a22 b21 , ξ9 = b12x − a11 b12 + a12 b11 − a12 b22 + a22 b12 , ξ10 = b21y + a12 b21 − a22 b11 + a22 b22 − a23 b21 , ξ11 = b11y − a13 b21 + a22 b12 , ξ12 = b22x + a12 b21 − a21 b12 .

(38)

The final representation of all β ij and relations are presented in the appendix. Notice that from the definitions of ξ i and found relations we obtain the following corollaries:

ξ2y = ξ1x − a11 ξ1 + a12 (ξ2 − ξ3 ) + a13 ξ4 + a14 ξ11 − a15 ξ6 + a24 ξ5 − a25 ξ9 , ξ4y = ξ3x + a14 ξ10 − a15 ξ8 − a21 ξ1 + a22 (ξ2 − ξ3 ) + a23 ξ4 + a24 ξ7 − a25 ξ12 , ξ6y = ξ11x − a12 ξ10 + a13 ξ8 + a21 ξ5 − a22 ξ9 ξ8y = ξ10x + a11 ξ10 − a12 ξ8 + a21 (ξ7 − ξ11 ) + a22 (ξ6 − ξ10 − ξ12 ) + a23 ξ8 ξ9y = ξ5x − a11 ξ5 − a12 (ξ7 − ξ9 − ξ11 ) − a13 (ξ6 − ξ12 ) + a22 ξ5 − a23 ξ9 ξ12y = ξ7x + a12 ξ10 − a13 ξ8 − a21 ξ5 + a22 ξ9 .

(39)

The equations (βi j )x = 0, (i = 1, 2; j = 1, 2 ) compose an algebraic system of linear equations with respect to ξ 1x , ξ 2x , ξ 3x , ξ 4x with a none vanishing determinant. Hence one can find ξ 1x , ξ 2x , ξ 3x , ξ 4x from this system. We omit their expressions here. The equations (βi j )x = 0, (i = 1, 2; j = 3, 4 ) compose an algebraic system of linear equations with respect to ξ 6 , ξ 8 , ξ 9 , ξ 12 with a none vanishing determinant. Hence one can find ξ 6 , ξ 8 , ξ 9 , ξ 12 from this system:

ξ6 = 0, ξ8 = 0, ξ9 = 0, ξ12 = 0.

(40)

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31

The equations (βi j )y = 0, (i = 1, 2; j = 3, 4 ) compose an algebraic system of linear equations with respect to ξ 5 , ξ 7 , ξ 10 , ξ 11 with none vanishing determinant. Hence one can find ξ 5 , ξ 7 , ξ 10 , ξ 11 from this system:

ξ5 = 0, ξ7 = 0, ξ10 = 0, ξ11 = 0.

(41)

From the equations (β11 )y = 0 and (β21 )y = 0; one finds

ξ1y = a12 ξ1 − a13 (ξ2 − ξ3 ) − a23 ξ1 , ξ3y = −a13 ξ4 + a22 ξ1 .

(42)

In order to obtain sufficiency conditions, assume that all the identities listed above are satisfied. Hence we have proven the following theorem: Theorem. A system of two second-order stochastic ordinary differential equations,

˙ Y˙ ) dt + g1 (t, X, Y, X, ˙ Y˙ ) dW dX˙ = f1 (t, X, Y, X, ˙ Y˙ ) dt + g2 (t, X, Y, X, ˙ Y˙ ) dW, dY˙ = f2 (t, X, Y, X, is linearizable by an invertible point transformation if and only if,

f1 = a11 x˙ 2 + 2a12 x˙ y˙ + a13 y˙ 2 + a14 x˙ + a15 y˙ + a10 ; f2 = a21 x˙ 2 + 2a22 x˙ y˙ + a23 y˙ 2 + a24 x˙ + a25 y˙ + a20 ; g1 = b11 x˙ + b12 y˙ + b10 , g2 = b21 x˙ + b22 y˙ + b20 ; where fi , gi , ai j , (i = 1, 2; j = 0, . . . , 5 ) and bi j , (i = 1, 2; j = 0, 1, 2 ) satisfy the conditions

A1i, j = 0, (i = 1, .., 6 ); ( j = 1, 2 ). Here the A1i, j are listed as follows:

A17,1x = a12 A17,2 − a21 A18,1 ;

A17,2x = −a11 A17,2 + a21 (A17,1 − A18,2 ) + a22 A17,2 ,

A18,1x = a11 A18,1 − a12 (A17,1 − A18,2 ) − a22 A18,1 ; A17,1y = a13 A17,2 − a22 A18,1 ;

A18,2x = −a12 A17,2 + a21 A18,1 ,

A17,2y = −a12 A17,2 + a22 (A17,1 − A18,2 ) + a23 A17,2 ,

A18,1y = a12 A18,1 − a13 (A17,1 − A18,2 ) − a23 A18,1 ;

A18,2y = −a13 A17,2 + a22 A18,1 .

In addition,

ξ1y = (a12 − a23 )ξ1 − a13 (ξ2 − ξ3 ); ξ2y = ξ1x − a11 ξ1 + a12 (ξ2 − ξ3 ) + a13 ξ4 , ξ3y = −a13 ξ4 + a22 ξ1 ; ξ4y = ξ3x − a21 ξ1 + a22 (ξ2 − ξ3 ) + a23 ξ4 , b12y − a12 b12 + a13 b11 − a13 b22 + a23 b12 = 0; b11x − a12 b21 + a21 b12 = 0, b22y + a13 b21 − a22 b12 = 0; b21x + a11 b21 − a21 b11 + a21 b22 − a22 b21 = 0, b12x − a11 b12 + a12 b11 − a12 b22 + a22 b12 = 0; b21y + a12 b21 − a22 b11 + a22 b22 − a23 b21 = 0, b11y − a13 b21 + a22 b12 = 0, b22x + a12 b21 − a21 b12 = 0, where

ξ1 = 2b10y − 2a12 b10 − 2a13 b20 + a15 b11 + a25 b12 ; ξ2 = 2b10x − 2a11 b10 − 2a12 b20 + a14 b11 + a24 b12 , ξ3 = 2b20y + a15 b21 − 2a22 b10 − 2a23 b20 + a25 b22 ; ξ4 = 2b20x + a14 b21 − 2a21 b10 − 2a22 b20 + a24 b22 . 4. Examples In this section, we apply the obtained Theorem to some selected examples to illustrate how to linearize a system of two second-order SODEs. For checking whether a system of two second-order SODEs is linearizable we develop a code using REDUCE [4]. We demonstrate the use of this code using three illustrative examples.

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4.1. Example 1 Consider a nonlinear system of two second-order SODEs

dx˙ = x dW, dy˙ = −2x−1 pqdt,

(43)

where the first equation is linear, and the second equation is a nonlinear one without Itô’s integral. First we check that all sufficient conditions for linearization (19) – (42) are satisfied. To find a linearization transformation we obtain the overdetermined system of equations:

ϕxx = 0, ϕyy = 0, ϕxt = 0, ϕyt = 0, ϕtt = 0, ϕxy = x−1 ϕy ,

(44)

ψxx = 0, ψyy = 0, ψxt = 0, ψyt = 0, ψtt = 0, ψxy = x−1 ψy .

(45)

and

Notice that this overdetermined system of equations is compatible. Hence we can find a solution of this system of equations. The general solution of Eqs. (44) and (45) is

ϕ = c1 xy + c2 x + c3t + c4 ; ψ = c5 xy + c6 x + c7t + c8 ,

(46)

where ci , (i = 1, 2, . . . , 8 ) are constant. We choose the constants ci such that  = ϕx ψy − ϕy ψx = 0, for example,

c1 = 0, c2 = 1 , c3 = 0 , c4 = 0 , c5 = 1 , c6 = 0 , c7 = 0, c8 = 0, which give the transformation

ϕ = x, ψ = xy. The latter change maps the nonlinear system (43) into the linear system

dx˙ = x dW, dy˙ = y dW.

(47)

4.2. Example 2 Consider a nonlinear system of two second-order SODEs



1



2ϑ  ϑ p − x(2(ϑ  )2 − ϑ  ϑ ) dt +

ϑ3

(yp − xq )dW, x2    1 1  4 dy˙ = 2 2 −2yϑ 2 p2 + 2xϑ 2 pq + 4xyϑ  ϑ p − 2x2 ϑ  ϑ q − 2x2 y(ϑ  )2 dt + 3 (x + y2 ϑ 4 ) p − xyϑ 4 q dW, x ϑ x ϑ dx˙ =

ϑ2

(48)

where ϑ = ϑ (t ) = 0. First we check that all sufficient conditions for linearization (19)–(42) are satisfied. To find a linearization transformation we obtain the overdetermined system of equations:

ϕxx = ϕtt =

1 2y ϑ ϕ , ϕ = − ϕ , ϕ = 0 , ϕ = − ( xϕ x + 2 yϕ y ) , y xy y yy xt x xϑ x2



1

ϑ2

ϕyt =



ϑ ϕ, ϑ y

x(2(ϑ  )2 − ϑ  ϑ )ϕx + 2y(ϑ  )2 ϕy ,

(49) (50)

and

ψxx = ψtt =

2y ϑ 1 ψy , ψxy = − ψy , ψyy = 0, ψxt = − ( xψ x + 2 yψ y ) , 2 x 2 xϑ x



1

ϑ

2



x(2(ϑ  )2 − ϑ  ϑ )ψx + 2y(ϑ  )2 ψy .

ψyt =

ϑ ψ, ϑ y

(51) (52)

Notice that this overdetermined system of equations is compatible. Hence we can find a solution of this system of equations. Solving the first five sets of equations in (49) and (51) we obtain

y x

y x

ϕ = ϑλ1 + λ3 x + λ10 , ψ = ϑλ2 + λ4 x + λ20 ,

(53)

where λi , (i = 1, . . . , 4 ) are constant and λ10 (t) and λ20 (t) are arbitrary functions. Substituting (53) into (50) and (52) we obtain λ10 = 0 and λ20 = 0. We can choose the trivial solution of these equations λ10 = 0, λ20 = 0. Since

 = ϕx ψy − ϕy ψx = 0, we also can choose a particular set of constants such that λ1 = λ4 = 1 and λ2 = λ3 = 0. This gives us the transformation

x y

ϕ=ϑ ,ψ =

x

ϑ

,

which maps the nonlinear system (48) into the linear system

dx˙ = (ϑ  y + ϑ q )dW, dy˙ = (ϑ  x + ϑ p)dW.

(54)

T.G. Mkhize et al. / Applied Mathematics and Computation 301 (2017) 25–35

33

4.3. Example 3 For the third example we have,

dx˙ = − dy˙ =

2(

2(

x2

x2

 1 (2x( p2 + q2 ) − 4ypq − x3 + xy2 )dt + y(x2 − y2 )dW , 2 −y )

 1 (2y( p2 + q2 ) − 4xpq + x2 y − y3 )dt + x(x2 − y2 )dW . 2 −y )

(55)

Eq. (55) satisfies all the sufficient conditions (19)–(42) and hence the compatible overdetermined system of equations is given by

ϕxx = ϕyy =

1 1 (xϕx − yϕy ), ϕxy = − 2 (yϕx − xϕy ), ϕxt = ϕyt = 0, x2 − y2 x − y2

1 2

ϕtt = − (xϕx + yϕy − 2ϕ ),

(56) (57)

and

ψxx = ψyy =

1 1 (xψx − yψy ), ψxy = − 2 (yψx − xψy ), ψxt = ψyt = 0. x2 − y2 x − y2

1 2

ψtt = − (xψx + yψy − 2ψ ).

(58) (59)

Solving the system of Eqs. (56) and (58) we get

ϕ = x2 (C1 + C2 ) + 2xy(C1 − C2 ) + y2 (C1 + C2 ) + C10 , ψ = x2 (C3 + C4 ) + 2xy(C3 − C4 ) + y2 (C3 + C4 ) + C20 ,

(60)

where Ci , (i = 1, . . . , 4 ) are constant and C10 (t) and C20 (t) are arbitrary functions. After substituting (60) into (57) and  = 0 and C  = 0. For simplicity we choose the trivial solution of these equations C = 0, C = 0. Since (59) we obtain C10 10 20 20

 = ϕx ψy − ϕy ψx = 0, we also choose a particular set of constants such that C1 = C2 = C3 = Hence the transformation

1 2

and C4 = − 21 .

ϕ = x2 + y2 , ψ = 2xy

(61)

linearizes the Eq. (55) into

dx˙ = xdt + ydW,

dy˙ = ydt + xdW.

(62)

5. Conclusion In this paper, we have completely solved the linearization problem of systems of two second-order stochastic ordinary differential equations. Necessary and sufficient conditions for linearization by an invertible transformation are given in terms of coefficients of the system. The result is given in terms of a Theorem with three examples. In addition, we have also shown that the system of two nonlinear second-order stochastic ordinary differential equations is linearizable via an invertible transformation when certain conditions are satisfied. Moreover, we have developed a code using REDUCE for checking whether a system of two second-order stochastic ordinary differential equations is linearizable. Certain nonlinear second-order stochastic ordinary differential equations appeared to be linearizable via invertible transformations. Acknowledgments TGM thanks Suranaree University of Technology, School of Mathematics for the support and hospitality during the period of the visit in 2015 and the Durban University of Technology for supporting the visit. Appendix Final conditions are

A17,1x = a12 A17,2 − a21 A18,1 ;

A17,2x = −a11 A17,2 + a21 (A17,1 − A18,2 ) + a22 A17,2 ,

A18,1x = a11 A18,1 − a12 (A17,1 − A18,2 ) − a22 A18,1 ; A17,1y = a13 A17,2 − a22 A18,1 ;

A18,2x = −a12 A17,2 + a21 A18,1 ,

A17,2y = −a12 A17,2 + a22 (A17,1 − A18,2 ) + a23 A17,2 ,

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T.G. Mkhize et al. / Applied Mathematics and Computation 301 (2017) 25–35

A18,1y = a12 A18,1 − a13 (A17,1 − A18,2 ) − a23 A18,1 ;

A18,2y = −a13 A17,2 + a22 A18,1 .

β11 = −1 (−ξ1 ϕx ψx + ξ2 ϕx ψy − ξ3 ϕy ψx + ξ4 ϕy ψy ), β12 = −1 (ξ1 ϕx2 − (ξ2 − ξ3 )ϕx ϕy − ξ4 ϕy2 ), β13 = −1 (−b12 ϕx ψx + b11 ϕx ψy − b22 ϕy ψx + b21 ϕy ψy ), β14 = −1 (b12 ϕx2 − (b11 − b22 )ϕx ϕy − b21 ϕy2 ), β21 = −1 (−ξ1 ψx2 + (ξ2 − ξ3 )ψx ψy + ξ4 ψy2 ), β22 = −1 (ξ1 ϕx ψx + ξ3 ϕx ψy − ξ2 ϕy ψx − ξ4 ϕy ψy ), β23 = −1 (−b12 ψx2 + (b11 − b22 )ψx ψy + b21 ψy2 ), β24 = −1 (b12 ϕx ψx + b22 ϕx ψy − b11 ϕy ψx − b21 ϕy ψy ), β10 = −1 (B10,1 ϕt + B10,2 ψt + b10 ϕx + b20 ϕy − β11 ϕ − β12 ψ ), β20 = −1 (B20,1 ϕt + B20,2 ψt + b10 ψx + b20 ψy − β21 ϕ − β22 ψ ), where

B10,1 = b12 ϕx ψx − b11 ϕx ψy + b22 ϕy ψx − b21 ϕy ψy , B20,1 = b12 ψx2 + (−b11 + b22 )ψx ψy − b21 ψy2 ,

B10,2 = −b12 ϕx2 + (b11 − b22 )ϕx ϕy + b21 ϕy2 ,

B20,2 = −b12 ϕx ψx − b22 ϕx ψy + b11 ϕy ψx + b21 ϕy ψy ,

ξ1 = −2b10y + 2a12 b10 + 2a13 b20 − a15 b11 − a25 b12 , ξ2 = 2b10x − 2a11 b10 − 2a12 b20 + a14 b11 + a24 b12 , ξ3 = −2b20y − a15 b21 + 2a22 b10 + 2a23 b20 − a25 b22 , ξ4 = 2b20x + a14 b21 − 2a21 b10 − 2a22 b20 + a24 b22 . b10y = (2a12 b10 + 2a13 b20 − a15 b11 − a25 b12 + ξ1 )/2, b10x = (2a11 b10 + 2a12 b20 − a14 b11 − a24 b12 + ξ2 )/2, b11y = a13 b21 − a22 b12 ,

b11x = a12 b21 − a21 b12 ,

b12y = a12 b12 − a13 b11 + a13 b22 − a23 b12 , b21y = −a12 b21 + a22 b11 − a22 b22 + a23 b21 , b22y = −a13 b21 + a22 b12 ,

b12x = a11 b12 − a12 b11 + a12 b22 − a22 b12 , b21x = −a11 b21 + a21 b11 − a21 b22 + a22 b21 ,

b11x = −a12 b21 + a21 b12 ,

b20y = (−a15 b21 + 2a22 b10 + 2a23 b20 − a25 b22 + ξ3 )/2, b20x = (−a14 b21 + 2a21 b10 + 2a22 b20 − a24 b22 + ξ4 )/2,

ξ1x = a11 ξ1 − a12 (ξ2 − ξ3 ) − a22 ξ1 , ξ2x = a12 ξ4 − a21 ξ1 , ξ3x = −a12 ξ4 + a21 ξ1 , ξ4x = −a11 ξ4 + a2 1(ξ2 − ξ3 ) + a22 ξ4 , ξ1y = a12 ξ1 − a13 (ξ2 − ξ3 ) − a23 ξ1 , ξ3y = −a13 ξ4 + a22 ξ1 .

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