Comparison of exact and multiple scales solutions

In the previous section, we calculated the long time limits of the spatial variances generated by the multiple scales analysis for the Maxwellian and delta function initial conditions, and showed that they agreed with those obtained from the exact solutions. Here we compare the complete exact and multiple scales solutions, including the short-time inertial relaxations.

We also examine the relaxation of the Brownian particle from a specified initial condition in the athermal limit (P e→ ∞).

2.4.1 Multiple scales solution for a Maxwellian initial condition

The Maxwellian initial condition is given by

P(x,v,0) = {δ(x)δ(y)}

P e St 2π

exp

−P e St(u²+v²) 2

,

= {δ(x)δ(y)}

P e St 2π

exp

−P e St{(u−y)²+v²} 2

,

where the second step is possible due to the presence ofδ(y). In terms of the rescaled variables (ˆx,w),

P¯(ˆx,w,0) = 1

(2π){δ(ˆx)δ(ˆy)}H¯₀ w₁

2¹²

H¯₀ w₂

2¹²

. (2.45)

Therefore, (2.21) gives the initial conditions for theb⁽⁰⁾m,n’s as⁶

b⁽⁰⁾_0,0(ˆx,y,ˆ 0) = δ(ˆx)δ(ˆy) = δ(x)δ(y) P e St ,

b⁽⁰⁾_m,n(ˆx,y,ˆ 0) = 0 ∀ m+n >0. (2.46)

For small St,

b⁽⁰⁾_m,n(ˆx,y, tˆ 2;St) =b⁽⁰⁾_m,n^I(ˆx,y, tˆ 2) +St b⁽⁰⁾_m,n^II(ˆx,y, tˆ 2) +O(St²), (2.47)

whereb⁽⁰⁾_m,n^Isatisfies the required initial condition, resulting in trivial conditions for all higher order coefficients in the expansion. To leading order, (2.38) for i, m= 0 takes the form

∂b⁽⁰⁾_0,n^I

∂t₂ +y∂b⁽⁰⁾_0,n^I

∂x = 1 P e





∂²b⁽⁰⁾_0,n^I

∂x² +∂²b⁽⁰⁾_0,n^I

∂y²



. (2.48)

Therefore, b⁽⁰⁾_0,0^I =G₀/(P e St) (denoted from here on by ¯G₀), and b⁽⁰⁾_0,n^I(ˆx,y, tˆ ₂) = 0 ∀n≥1.

Since b⁽⁰⁾_0,0^I does not couple to any of the equations for b⁽⁰⁾m,n

I for m ≥ 1, the trivial initial conditions (2.46) imply b⁽⁰⁾m,n

I(ˆx,y, tˆ 2) = 0 ∀m≥1 &n≥0. Thus, b⁽⁰⁾_0,0^I is the only non-zero element at the leading order.

At the next order, the b⁽⁰⁾m,n

II’s satisfy trivial initial conditions and are therefore zero for m≥1 andn≥0;b⁽⁰⁾_0,0^II(also see equation (2.40)) satisfies

Db⁽⁰⁾_0,0^II =− 1 P e

∂²G¯₀

∂x∂y, (2.49)

6The factor of (P e St) in the initial condition is present in all theb⁽ⁱ⁾m,n’s and only serves to normalize the probability density; it does not change the relative orders of the different contributions

where

D= ∂

∂t₂ +y ∂

∂x − 1 P e

∂²

∂x² + ∂²

∂y²

,

and therefore (see (2.42)),

b⁽⁰⁾_0,0^II(ˆx,y, tˆ ₂) =− 1 P e

t²₂ 2

∂²G¯₀

∂x² +t₂∂²G¯₀

∂x∂y

.

We now consider the b⁽¹⁾m,n’s. Using (2.25), the only non-trivial initial conditions are

b⁽¹⁾_0,1(ˆx,y,ˆ 0) = 1 2¹²

∂b⁽⁰⁾_0,0^I

∂ˆy = 1 (2P e St)¹²

δ(x)δ⁰(y) P e St , b⁽¹⁾_1,0(ˆx,y,ˆ 0) = 1

2¹²

∂b⁽⁰⁾_0,0^I

∂ˆx = 1 (2P e St)¹²

δ⁰(x)δ(y) P e St , b⁽¹⁾_1,1(ˆx,y,ˆ 0) =1

4b⁽⁰⁾_0,0^I = 1 4

δ(x)δ(y) P e St ,

where we omit the superscriptI for allb⁽ⁱ⁾_m,n’s (i≥1), it being understood that they represent the leading order coefficients in a small Stexpansion similar to (2.47), and that the higher order corrections do not affect P(x,v, t) to O(St). Using (2.38) for i = 1, the equations governing the non-trivial coefficients are

Db⁽¹⁾_0,1 = 0, Db⁽¹⁾_1,0 = −b⁽¹⁾_0,1, Db⁽¹⁾_1,1 = 0, Db⁽¹⁾_2,0 = −b⁽¹⁾_1,1,

where b⁽¹⁾_2,0 is non-zero despite a trivial initial condition due to b⁽¹⁾_1,1 acting as the forcing

function. The solutions, in order, to the above equations are

b⁽¹⁾_0,1(ˆx,y, tˆ ₂) = 1 (2P e St)¹²

∂G¯₀

∂y +t₂∂G¯₀

∂x

, (2.50)

b⁽¹⁾_1,0(ˆx,y, tˆ ₂) = 1 (2P e St)¹²

(1−t²₂)∂G¯₀

∂x −t₂∂G¯₀

∂y

, (2.51)

b⁽¹⁾_1,1(ˆx,y, tˆ ₂) =G¯₀

4 , (2.52)

b⁽¹⁾_2,0(ˆx,y, tˆ 2) = −t₂G¯₀

4 . (2.53)

In order to determine the completeO(St) correction, we need to consider theb⁽²⁾m,n’s. Of these, only those coefficients that satisfy initial conditions involving second-order derivatives (and therefore jump an order in St when expressed in the original variables) contribute to the O(St) correction. Using (2.33) and (2.25), the initial conditions for these coefficients are

b⁽²⁾_0,0(ˆx,y,ˆ 0) = − 1 (P e St)





∂²b⁽⁰⁾_0,0^I

∂x² +∂²b⁽⁰⁾_0,0^I

∂y²



,

b⁽²⁾_0,2(ˆx,y,ˆ 0) = 1 4(P e St)

∂²b⁽⁰⁾_0,0^I

∂y² , b⁽²⁾_1,1(ˆx,y,ˆ 0) = 1

2(P e St)

∂²b⁽⁰⁾_0,0^I

∂x∂y , b⁽²⁾_2,0(ˆx,y,ˆ 0) = 1

4(P e St)

∂²b⁽⁰⁾_0,0^I

∂x² .

The equations corresponding to these initial conditions are

Db⁽²⁾_0,0 = 0,

Db⁽²⁾_0,2 = 0, Db⁽²⁾_1,1 = −2b⁽²⁾_0,2, Db⁽²⁾_2,0 = −b⁽²⁾_1,1,

with the solutions

b⁽²⁾_0,0(ˆx,y, tˆ ₂) = − 1 P e St

"

∂²G¯₀

∂x² + ∂

∂y+t₂ ∂

∂x 2

G¯₀

#

, (2.54)

b⁽²⁾_0,2(ˆx,y, tˆ ₂) = 1 4(P e St)

∂

∂y+t₂ ∂

∂x 2

G¯₀, (2.55)

b⁽²⁾_1,1(ˆx,y, tˆ ₂) = 1 2(P e St)

(1−t²₂) ∂

∂x −t₂ ∂

∂y

∂G¯₀

∂y +t₂∂G¯₀

∂x

, (2.56)

b⁽²⁾_2,0(ˆx,y, tˆ ₂) = 1 4(P e St)

∂²G¯₀

∂x² + 1 2(P e St)

t₂(t²₂

2 −1) ∂

∂x+t²₂ 2

∂

∂y

∂G¯₀

∂y +t₂∂G¯₀

∂x

. (2.57)

Having calculated the relevant coefficients, we consider the expression for the rescaled probability density ¯P as given by (2.39) upto O(St²) in the rescaled variables. The original and the rescaled probability densities differ by a factor of (P e St)²; this does not, however, alter the relative orders of the different terms, and is therefore not included when comparing contributions from the various terms (below). As mentioned earlier, the b⁽²⁾_m,n’s jump an order due to the presence of second-order derivatives in their initial conditions.

That they contribute to the solution at O(St) also stems from the fact that the non-trivial coefficients at this order, b⁽²⁾_0,0, b⁽²⁾_2,0 and b⁽²⁾_0,2, multiply a product of two even Hermite func- tions ( ¯H₂ and ¯H₀ in this case); theO(St) contribution therefore comes from the presence of theO(1) constant term in the even Hermite functions. Note that each power of the rescaled velocity variable w will render the relevant contribution smaller by O(St¹²) and thus the quadratic term in ¯H₂ would be O(St) smaller than the constant term. For this reason the non-zero coefficients at the first order, St b⁽¹⁾_1,0 and St b⁽¹⁾_0,1, despite being O(St¹²) contribute

only at O(St) to the solution (the additional factor ofSt¹² coming from the velocity variable in ¯H₁), while St b⁽¹⁾_2,0 being O(St) still contributes at the same order. Similarly, the terms containing b⁽¹⁾_1,1 and b⁽²⁾_1,1 will beo(St) since they multiply ¯H₁(w₁/2¹²) ¯H₁(w₂/2¹²). The reason we need to consider them at all is because the equations for the b⁽ⁱ⁾_2,0’s andb⁽ⁱ⁾_1,1’s are coupled, and the former contribute to the O(St) correction.

From (2.24) and (2.32), we observe that the non-trivial elements φ⁽ⁱ⁾_m,n,m+n+j, j ranging m+n−2i to m+n+i, involve at most i^th order derivatives of b⁽⁰⁾m,n. The φ⁽⁴⁾’s would in general contain fourth-order derivatives of b⁽⁰⁾m,n and therefore, terms of the form St⁴φ⁽⁴⁾m,n would only contribute atO(St²) (when taken together with the constant term in the corresponding even ordered Hermite function). On the other hand, the non-zero φ⁽³⁾_m.n’s that contain third order derivatives multiply odd ordered Hermite functions and thus contribute at the same order as the φ⁽⁴⁾m,n’s. We now rewrite (2.39) explicitly including only the terms relevant to O(St).

P(ˆx,y,ˆ u,¯ v;¯ St, P e) (P e St)² =

b⁽⁰⁾_0,0^I+St b⁽⁰⁾_0,0^II

exp

−w₁²+w₂² 2

+St

(2¹²w₁)b⁽¹⁾_0,1e^−t1+(2¹²w₂)b⁽¹⁾_1,0e^−t1

−2b⁽¹⁾_2,0e^−2t1 −

w₁∂b⁽⁰⁾_0,0^I

∂ˆx +w₂∂b⁽⁰⁾_0,0^I

∂yˆ

exp

−w²₁+w₂² 2

+St² n

b⁽²⁾_0,0−2b⁽²⁾_0,2e^−2t1−2b⁽²⁾_2,0e^−2t1o + 2¹²

∂b⁽¹⁾_1,0

∂xˆ +∂b⁽¹⁾_0,1

∂ˆy

e^−t1 + 2¹²

∂b⁽¹⁾_1,0

∂xˆ +∂b⁽¹⁾_0,1

∂yˆ

e^−t1− 1 2

∂²b⁽⁰⁾_0,0^I

∂xˆ² +∂²b⁽⁰⁾_0,0^I

∂yˆ²

exp

−w₁²+w₂² 2

, (2.58)

where we have used ¯H₀(z) = e^−z2, ¯H₁(z) = 2¹²z e^−z2, ¯H₂(z) = 2(z² −1)e^−z2 and retained only the constant term in ¯H₂. It is seen that the terms linear in the velocity variables w involve only b⁽⁰⁾_0,0^I, b⁽¹⁾_0,1 and b⁽¹⁾_1,0, which makes the calculation of the velocity dependent corrections (terms of the form (a:wx)P⁽⁰⁾, where P⁽⁰⁾ is the leading order solution) alone a

much simpler task. On substituting the expressions obtained for the various coefficients, it can be verified that the above series matches up identically to the expansion of the corresponding exact solution (P^m) in Appendix A2.

2.4.2 Multiple scales solution for a delta function initial condition

Here, we briefly present calculations similar to that in the previous section, carried out now for a delta function initial condition. The details are given in Appendix A4. We consider the case where

P(x,v,0) = δ(x)δ(y)δ(u)δ(v),

= δ(x)δ(y)δ(u−y)δ(v),

so that the probability density in rescaled variables can be written as⁷

P¯(ˆx,w,0) =δ(ˆx)δ(ˆy) 2π

X∞ m,n=0

(−1)^m+nH¯_2m(^w1

2¹²) ¯H_2n(^w2

2¹²)

2^2(m+n)m!n! .

For this case, the form of the solution is much more involved, and we restrict ourselves to finding the O(St) velocity dependent corrections to P(x,v, t), which only requires the calculation of the b⁽⁰⁾m,n’s and b⁽¹⁾m,n’s to leading order; the superscripts ‘I’ and ‘II’ used in section 2.4.1 are therefore omitted (the successful comparison of the number densities for this initial condition suggests the correctess of the complete O(St) correction). From (2.21), we obtain the initial conditions for the b⁽⁰⁾m,n’s as

b⁽⁰⁾_2m,2n(ˆx,y,ˆ 0) = (−1)^m+n

2^2(m+n)m!n!δ(ˆx)δ(ˆy),

7We have used the relationδ(z) = ¹

(2π)¹2

P∞ n=0

(−1)ⁿH¯₂n( ^z 21

2 )

2²ⁿn! (see Uhlenbeck & Ornstein 1954).

b⁽⁰⁾_2m+1,2n(ˆx,y,ˆ 0) = 0, b⁽⁰⁾_2m,2n+1(ˆx,y,ˆ 0) = 0, b⁽⁰⁾_2m+1,2n+1(ˆx,y,ˆ 0) = 0.

The structure of the consistency condition (2.36) is such that the sets of coefficients

(b⁽⁰⁾_2m,2n+1,b⁽⁰⁾_2m+1,2n) and (b⁽⁰⁾_2m,2n,b⁽⁰⁾_2m+1,2n+1) form independent subsystems. The trivial initial conditions for the former give

b⁽⁰⁾_2m,2n+1(ˆx,y, tˆ 2) = 0, (2.59)

b⁽⁰⁾_2m+1,2n(ˆx,y, tˆ 2) = 0. (2.60)

For the latter, we first obtain the solution for m = 0 (for which the coupling term in (2.36) is absent) and then solve for increasing mto obtain the general forms

b⁽⁰⁾_2m,2n(ˆx,y, tˆ ₂) = (−1)^m+n (2π)2^2n+mn!

m

X

k=0

t^2m₂ ^−2kQ_m−1

l=k (2n+ 2m−1−2l)

2^kk!(2m−2k)! G¯₀, (2.61) b⁽⁰⁾_2m+1,2n+1(ˆx,y, tˆ ₂) = (−1)^m+n

(2π)2^2n+m+1n!

m

X

k=0

t^2m+1₂ ^−2kQ_m−1

l=k (2n+ 2m+ 1−2l)

2^kk!(2m+ 1−2k)! G¯₀. (2.62)

Using the above expressions in (2.39), it may be verified that the exact and multiple scales solutions are identical to leading order (see Appendix A4.1).

From equations (2.61) and (2.62), the φ⁽¹⁾_m,n,s’s for s6=m+ncan be determined using (2.24). The b⁽¹⁾m,n’s satisfy the same set of equations as theb⁽⁰⁾m,n’s, and from (2.25),

b⁽¹⁾_2m,2n(ˆx,y,ˆ 0) = 0, (2.63)

b⁽¹⁾_2m+1,2n(ˆx,y,ˆ 0) = 1 2¹²

∂b⁽⁰⁾_2m,2n

∂xˆ −2¹²(2m+ 2)∂b⁽⁰⁾_2m+2,2n

∂ˆx , (2.64)

b⁽¹⁾_2m,2n+1(ˆx,y,ˆ 0) = 1 2¹²

∂b⁽⁰⁾_2m,2n

∂ˆy −2¹²(2n+ 2)∂b⁽⁰⁾_2m,2n+2

∂yˆ , (2.65)

b⁽¹⁾_2m+1,2n+1(ˆx,y,ˆ 0) =1

4b⁽⁰⁾_2m,2n. (2.66)

Again, (b⁽¹⁾_2m,2n, b⁽¹⁾_2m+1,2n+1) and (b⁽¹⁾_2m,2n+1, b⁽¹⁾_2m+1,2n) form independent subsystems; they ap- pear in the multiple scales series in the form St b⁽¹⁾_m,ne^−(m+n)t1H¯_mH¯_n. The former will only contribute terms of the form wⁱ₁w^j₂, wherei+j is even. The largest of these corresponds to b⁽¹⁾_1,1(b⁽¹⁾_0,0 = 0) and is O(St²). Therefore, we can restrict our attention to the set (b⁽¹⁾_2m+1,2n, b⁽¹⁾_2m,2n+1) when looking at O(St) corrections. Solving (2.36) for initial conditions given by (2.64) and (2.65), one obtains

b⁽¹⁾_2m+1,2n= 2¹²(−1)^m+n (2π)2^2n+mn!

S₁^(m,n)∂G¯₀

∂ˆx −S₂^(m,n) ∂G¯₀

∂yˆ +t₂∂G¯₀

∂xˆ

, (2.67)

b⁽¹⁾_2m,2n+1= 2¹²(−1)^m+n (2π)2^2n+mn!

S^0(m,n)₂

∂G¯₀

∂yˆ +t₂∂G¯₀

∂xˆ

−S^0(m,n)₁ ∂G¯₀

∂ˆx

, (2.68)

where

S₁^(m,n) =

m

X

k=0

t^2m₂ ^−2kQ_m−1

l=k (2n+ 2m−1−2l) 2^kk!(2m−2k)! , S₂^(m,n) =

m

X

k=0

t^2m+1₂ ^−2kQm

l=k(2n+ 2m+ 1−2l) 2^kk!(2m+ 1−2k)! , S^0(m,n)₁ =

m−1

X

k=0

t^2m₂ ^−2k−1Q_m−2

l=k (2n+ 2m−1−2l) 2^kk!(2m−1−2k)! , S^0(m,n)₂ =

m

X

k=0

t^2m₂ ^−2kQ_m−1

l=k (2n+ 2m+ 1−2l) 2^kk!(2m−2k)! .

Using the relations (2.61), (2.62), (2.67), (2.68) and the expressions for theφ⁽¹⁾’s in (2.39), it can be verified (see Appendix A4.2) that the multiple scales series matches the corresponding exact solution to O(St), for velocity dependent corrections to the leading order solution.

It is seen that the multiple scales series for this initial condition is of the form

P(x,v, t₁, t₂;St, P e) =P⁽⁰⁾(x,v, t₁, t₂;St, P e) +St P⁽¹⁾(x,v, t₁, t₂;St, P e) +. . . (2.69)

Since the P⁽ⁱ⁾’s are themselves functions of St, the term StⁱP⁽ⁱ⁾ will also include higher order contributions of O(St^i+a) (a >0). For instance, b⁽⁰⁾m,nH¯mH¯n in the leading order solu- tion (P⁽⁰⁾=P

b⁽⁰⁾_m,nH¯_mH¯_n) contains terms of the form b⁽⁰⁾_m,nw₁^mw₂ⁿ that are O(St^m+n2 ) when expressed in terms of (u, v), and thereforeo(St) form+n >2. Strictly speaking, they should not be considered when comparing the exact and multiple scales solutions to O(St); it is, however, possible in this case (as illustrated in the appendices) to cast the series in a form which can be identified with terms in the exact solution.

2.4.3 The athermal limit

In the absence of Brownian motion, a particle at rest at the origin of a simple shear flow remains so for all time. Therefore, to illustrate the relaxation of a non-Brownian particle from its initial state, we must choose an initial condition in position space different from that used above. Accordingly, we first derive the (finite Pe) form of the multiple scales solutions when the particle at timet= 0 is at (x, y)≡(0, y₀) with a Maxwellian distribution of velocities. The non-Brownian limit is obtained by letting P e→ ∞in the final expression for the probability density. The initial condition is

P(x,v,0) =δ(x)δ(y−y₀)

P e St 2π

exp

−P e St(u²+v²) 2

, (2.70)

which can be written in the form

P(x,v,0) =δ(x)δ(y−y₀)

P e St 2π

exp

−(P e St)v² 2

exp

−P e St(u−y+y)² 2

,

=δ(x)δ(y−y₀)

P e St 2π

exp

−(P e St)v² 2

X∞

m=0

P e St 2

^m₂ y^m m!

d^m du^m

exp

−P e St(u−y)² 2

,

and in terms of the rescaled variables (ˆx,w), the renormalized probability density becomes

P¯(ˆx,w,0) = 1 2π

X∞ m=0

δ(ˆx)δ(ˆy−yˆ₀)(−1)^myˆ₀^m m! 2^m2

H¯_m w₁

2¹²

H¯₀ w₂

2¹²

, (2.71)

where the delta function allows us to replace ˆyby ˆy₀. The initial conditions for the coefficients b⁽⁰⁾m,n(see (2.21)) are

b⁽⁰⁾_m,0(ˆx,y,ˆ 0) =δ(ˆx)δ(ˆy−yˆ₀)(−1)^myˆ₀^m m! 2^m2 , b⁽⁰⁾_m,n(ˆx,y,ˆ 0) = 0 ∀ n6= 0.

From (2.22), we observe that the system of equations for theb⁽⁰⁾m,n’s (n >0) is independent of theb⁽⁰⁾_m,0’s, and the trivial initial conditions imply that these are zero for all times. With this simplification, the b⁽⁰⁾_m,0’s satisfy

∂b⁽⁰⁾_m,0

∂t2

+ ˆy∂b⁽⁰⁾_m,0

∂ˆx = 0, (2.72)

which gives

b⁽⁰⁾_m,0(ˆx,y, tˆ ₂) = (−1)^mδ(ˆx−ytˆ ₂)δ(ˆy−yˆ₀)ˆy₀^m

m! 2^m2 . (2.73)

From (2.39), to leading order,

P¯⁽⁰⁾(¯x,w, t₁, t₂;St, P e) = X∞ m=0

b⁽⁰⁾_m,0e^−mt1H¯_m w₁

2¹²

H¯₀ w₂

2¹²

,

= X∞ m=0

(−1)^mδ(ˆx−ytˆ₂)δ(ˆy−yˆ₀)ˆy₀^m m! 2^m2

H¯_m w₁

2¹²

H¯₀ w₂

2¹²

. (2.74)

To find the limiting form of the above expression as P e→ ∞, we first consider the limiting forms of the Hermite functions.

P elim→∞

Stfinite

H¯0

w1

2¹²

H¯0

w2

2¹²

= lim

P e St→∞

P e St 2π

exp

−w²₁+w²₂ 2

,

= lim

P e St→∞

P e St 2π

exp

−P e St(u−y)²+v² 2

,

= δ(u−y)δ(v),

P elim→∞

Stfinite

H¯_m w₁

2¹²

H¯_n w₂

2¹²

= 2^m+n2 (−1)^(m+n) lim

P e St→∞

d^(m+n) dw^m₁ dwⁿ₂

H¯₀

w₁ 2¹²

H¯₀

w₂ 2¹²

,

= (−1)^m+n 2

P e St ^m+n₂

δ^(m)(u−y)δ⁽ⁿ⁾(v),

⇒ lim

P e St→∞(−1)^m+n

P e St 2

^m+n₂ H¯_m

w₁ 2¹²

H¯_n

w₂ 2¹²

= δ^(m)(u−y)δ⁽ⁿ⁾(v).

Using the above limits, (2.74) takes the form

P⁽⁰⁾(x,v, t₁, t₂;St) = X∞ m=0

δ(x−y₀t₂)δ(y−y₀)(y₀e^−t1)^m

m! δ^(m)(u−y₀)δ(v), (2.75)

where we have replacedybyy₀. Treating the summation (formally) as a Taylor series expan- sion, we get

P⁽⁰⁾(x,v, t₁, t₂;St) =δ(x−y₀t₂)δ(y−y₀)δ(v)δ(u−y₀+y₀e^−t1), (2.76)

where the term proportional to y₀ in the argument of the last delta function captures, to leading order, the relaxation of the particle from a state of rest at t= 0 to the steady state velocity of y₀1_x in a time of O(τ_p).

The exact solution for this case is readily obtained by solving the Langevin equa- tions of motion, viz. equation (2.3) with F^B = 0, for the same initial conditions. We get

x=y₀(t₂−St) +Sty₀e^−t1, y=y₀,

u=y₀(1−e^−t1), v= 0,

so that the probability density corresponding to this deterministic trajectory can be written as

δ{x−y₀t₂+y₀St(1−e^−t1)}δ(y−y₀)δ(v)δ(u−y₀+y₀e^−t1),

which, to leading order, is identical to (2.76).