The composition rule

4.2 The composition rule

Since Ia^βX(t) and D^βaX(t), when they exist, are second-order stochastic pro- cesses, we can consider expressions such as I_a^αIa^βX(t) and D^α_aIa^βX(t). These expressions, and those like it, make up what we will call the composition rule.

They will play an important role in later chapters. This section deals with parts of the composition rule.

The following theorem is used to derive many of the remaining properties in this chapter. In deterministic fractional calculus the proof of this theorem is often left out. We include the proof for this m.s. version – as do Hafiz et al. (2001) and Hafiz (2004), albeit in a less detailed manner – for the sake of completeness.

Theorem 4.6. Let β > 0 and α > 0 and letX(t) be a second-order stochastic process such thatI_a^αIa^βX(t)exists fort∈[a, b]⊂T. Then

I_a^αI_a^βX(t) =I_a^α+βX(t). Proof:

I_a^αI_a^βX(t) = Z t

a

(t−θ)^α−1 Γ(α)

Z θ a

(θ−s)^β−1

Γ(β) X(s)ds dθ

= Z t

a

X(s) Z t

s

(t−θ)^α−1 Γ(α)

(θ−s)^β−1 Γ(β) dθ

ds . Consider

I= Z t

s

(t−θ)^α−1 Γ(α)

(θ−s)^β−1 Γ(β) dθ . Lettingυ=θ−swe have

I= 1 Γ(α)Γ(β)

Z t−s 0

(t−υ−s)^α−1υ^β−1dυ . Now lettingυ= (t−s)uwe have

I = 1

Γ(α)Γ(β) Z 1

0

(t−s)^α−1(1−u)^α−1(t−s)^β−1u^β−1(t−s)du

= (t−s)^α+β−1 Γ(α)Γ(β)

Z 1 0

(1−u)^α−1u^β−1du

= (t−s)^α+β−1

Γ(α)Γ(β) ·Γ(α)Γ(β) Γ(α+β)

= (t−s)^α+β−1 Γ(α+β) .

4.2. The composition rule

So

I_a^αI_a^βX(t) = Z t

a

(t−s)^α+β−1

Γ(α+β) X(s)ds

= I_a^α+βX(t).

In Theorem 4.6 we considered the m.s. fractional integral of a m.s. frac- tional integral. In the next few theorems we find expressions forI_a^αD_a^βX(t)and D_a^αIa^βX(t). The proofs used when finding these expressions will rely heavily on Leibniz’s rule and IBP. This is not surprising as Leibniz’s rule deals with the derivative of an integral and IBP deals with the integral of a derivative (deriva- tives and integrals being defined in the m.s. sense for second-order stochastic processes). We will see that the proofs are very similar and will often rely on parts of previously proved theorems. It would be cumbersome to constantly re- fer to previous theorems when they are used so, in most situations, we will not do so.

Theorem 4.7. Let α > 0 and β ∈ (m−1, m] where m ∈ N. Let X(t) be a second-order stochastic process such thatD^βaX(t)exists fort∈[a, b]⊂T. Then

I_a^αD_a^βX(t) =I_a^α−βX(t)−

m−1

X

j=0

(t−a)^α−β+j

Γ(α−β+j+ 1)X^(j)(a), α≥β.

Proof:

Using integration by parts we have I_a¹X⁽¹⁾(t) =

Z t a

X⁽¹⁾(s)ds

= [X(s) ]^t_a+ 0

= X(t)−X(a),

I_a²X⁽²⁾(t) = Z t

a

(t−s)

Γ(2) X⁽²⁾(s)ds

=

(t−s)

Γ(2) X⁽¹⁾(s) t

a

+ Z t

a

X⁽¹⁾(s)ds

| {z }

I_a¹X⁽¹⁾(t)

= −(t−a)¹

Γ(2) X⁽¹⁾(a)−X(a) +X(t),

4.2. The composition rule

and

I_a³X⁽³⁾(t) = Z t

a

(t−s)²

Γ(3) X⁽³⁾(s)ds

=

(t−s)²

Γ(3) X⁽²⁾(s) t

a

+ Z t

a

(t−s)

Γ(2) X⁽²⁾(s)ds

| {z }

I_a²X⁽²⁾(t)

= −(t−a)²

Γ(3) X⁽²⁾(a)− (t−a)¹

Γ(2) X⁽¹⁾(a)−X(a) +X(t). Continuing in the same manner we have, forα=β =m,

I_a^mX^(m)(t) = X(t)−

m−1

X

j=0

(t−a)^j

Γ(j+ 1)X^(j)(a). (4.2)

Forα=β ∈(m−1, m)

I_a^βD^β_aX(t) = I_a^β I_a^m−βX^(m)(t) =I_a^mX^(m)(t)

= X(t)−

m−1

X

j=0

(t−a)^j

Γ(j+ 1)X^(j)(a)

where we have used equation (4.2) to find the last line.

Letn∈N. Whenα=nandβ =m I_aⁿD^m_aX(t) = I_a^n−mI_a^mX^(m)(t)

= I_a^n−m

X(t)−

m−1

X

j=0

(t−a)^j

Γ(j+ 1)X^(j)(a)

= I_a^n−mX(t)−

m−1

X

j=0

(t−a)^n−m+j

Γ(n−m+j+ 1)X^(j)(a). Whenα∈(n−1, n)andβ =m

β =m ≤ n−1 < α < n

so thatα−m >0. Under these conditions we have I_a^αX^(m)(t) = I_a^m−mI_a^αX^(m)(t)

= I_a^α−mI_a^mX^(m)(t)

= I_a^α−m

X(t)−

m−1

X

j=0

(t−a)^j

Γ(j+ 1)X^(j)(a)

= I_a^α−mX(t)−

m−1

X

j=0

(t−a)^α−m+j

Γ(α−m+j+ 1)X^(j)(a).

4.2. The composition rule

Forα∈(n−1, n]andβ∈(m, m−1) I_a^αD_a^βX(t) = I_a^αI_a^m−βX^(m)(t)

= I_a^α−βI_a^mX^(m)(t)

= I_a^α−β

X(t)−

m−1

X

j=0

(t−a)^j

Γ(j+ 1)X^(j)(a)

= I_a^α−βX(t)−

m−1

X

j=0

(t−a)^α−β+j

Γ(α−β+j+ 1)X^(j)(a).

In Theorem 4.7 we have considered the fractional integral of a fractional derivative when the integral is of an order greater than, or equal to, that of the fractional derivative. In the following theorem, Theorem 4.8, we will look at the case when the order of the integral is smaller than that of the derivative.

Theorem 4.8. Letn∈Nandm∈Nand let0< α < β whereα∈(n−1, n]and β ∈ (m−1, m]. Let X(t)be a second-order stochastic process such that Da^βX(t) exists fort∈[a, b]⊂T. Then

(a)

I_a^αD_a^βX(t) =D_a^β−αX(t)

whenβ−α∈(m−1, m)andβ 6=m.

(b)

I_a^αD_a^βX(t) =D_a^β−αX(t)−(t−a)^{m−β−1+α}

Γ(m−β+α) X^(m−1)(a) whenα∈(0,1),β−α∈(m−2, m−1]andβ 6=m.

(c)

I_aⁿD^β_aX(t) =D^β−n_a X(t)−

n−1

X

j=0

(t−a)^m−β+j

Γ(m−β+j+ 1)X^(m−n+j)(a) forβ∈(m−1, m]

(d)

I_a^αD_a^βX(t) =D_a^β−αX(t)−

n−1

X

j=1

(t−a)^{m−β−n+α+j}

Γ(m−β−n+α+j+ 1)X^(m−n+j)(a) whenα∈(n−1, n),α >1.

4.2. The composition rule

Proof

(a)

I_a^αD_a^βX(t) = I_a^αI_a^m−βX^(m)(t)

= I_a^{m−(β−α)}X^(m)(t)

= D^β−α_a X(t). (b) Letβ−α=m−1.

I_a^αD_a^βX(t) = I_a^αI_a^m−βX^(m)(t)

= I_a^{m−(β−α)}X^(m)(t)

= I_a¹X^(m)(t)

= X^(m−1)(t)−X^(m−1)(a)

= D^β−α_a X(t)−(t−a)^{m−1−β+α}

Γ(m−β+α) X^(m−1)(a). Now let β−α∈(m−2, m−1).

I_a^αD_a^βX(t) = I_a^{m−(β−α)}X^(m)(t)

= I(m−1)−(β−α)

a I¹X^(m)(t)

= I(m−1)−(β−α)

a X^(m)(t)−I(m−1)−(β−α)

a X^(m−1)(a)

= I(m−1)−(β−α)

a X^(m−1)(t)−(t−a)(m−1)−(β−α)

Γ(m−β+α) X^(m−1)(a)

= D^β−α_a X(t)−(t−a)^{m−β−1+α}

Γ(m−β+α) X^(m−1)(a).

(c) Using equation (4.2) and recalling that we are working under the assump- tion thatβ > α, we have, forβ =mandα=n,

I_aⁿD^m_aX(t) = I_aⁿdⁿ dtⁿ

X^(m−n)(t)

= X^(m−n)(t)−

n−1

X

j=0

(t−a)^j

Γ(j+ 1)X^(m−n+j)(a). (4.3)

4.2. The composition rule

Using equation (4.3) we have, forβ ∈(m−1, m)andα=n, I_aⁿD^β_aX(t) = I_aⁿI_a^m−βX^(m)(t)

= I_a^m−βI_aⁿX^(m)(t)

= I_a^m−β

X^(m−n)(t)−

n−1

X

j=0

(t−a)^j

Γ(j+ 1)X^(m−n+j)(a)

= I_a^m−βX^(m−n)(t)−

n−1

X

j=0

(t−a)^m−β+j

Γ(m−β+j+ 1)X^(m−n+j)(a)

= D_a^β−nX(t)−

n−1

X

j=0

(t−a)^m−β+j

Γ(m−β+j+ 1)X^(m−n+j)(a). (d) Using equation (4.3) we have forβ =mandα∈(n−1, n)whereα >1,

I_a^αX^(m)(t) = I_a^α−n+1I_aⁿ⁻¹X^(m)(t)

= I_a^α−n+1

X^(m−n+1)−

n−2

X

j=0

(t−a)^j

Γ(j+ 1)X^(m−n+1+j)(a)

= I_a^α−n+1X^(m−n+1)−

n−2

X

j=0

(t−a)^α−n+1+j

Γ(α−n+j+ 2)X^(m−n+1+j)(a)

= D_a^m−αX(t)−

n−1

X

j=1

(t−a)^α−n+j

Γ(α−n+j+ 1)X^(m−n+j)(a). Let β ∈ (m−1, m) and α ∈ (n−1, n) where α > 1. Ifn = m, then β−α∈(0,1). Under these conditions we have

I_a^αD_a^βX(t) = I_a^α+m−βX^(m)(t)

= I_a^{1−(β−α)}I_a^m−1X^(m)(t)

= I_a^{1−(β−α)}

X^m−m+1(t)−

m−2

X

j=0

(t−a)^j

Γ(j+ 1)X^(m−m+1+j)(a)

= I_a^{1−(β−α)}X¹(t)−

m−2

X

j=0

(t−a)^1−β+α+j

Γ(j+ 2−β+α)X^(j+1)(a)

= D^β−α_a X(t)−

m−1

X

j=1

(t−a)^{m−β−n+α+j}

Γ(m−β−n+α+j+ 1)X^(m−n+j)(a). Let β ∈ (m−1, m) and α ∈ (n−1, n) where α > 1 and n 6= m. Using

4.2. The composition rule

equation (4.3) we have

I_a^αD^β_aX(t) = I_a^α I_a^m−βX^(m)(t)

= I_a^{m−β+α−n+1}I_aⁿ⁻¹X^(m)(t)

= I_a^{m−β+α−n+1}

X^(m−n+1)(t)−

n−2

X

j=0

(t−a)^j

Γ(j+ 1)X^(m−n+j+1)(a)

= I_a^{m−β+α−n+1}X^(m−n+1)(t)

−

n−2

X

j=0

(t−a)m−β−n+α+1+j

Γ(m−β+α−n+j+ 2)X^(m−n+j+1)(a)

= D_a^β−αX(t)−

n−1

X

j=1

(t−a)^{m−β−n+α+j}

Γ(m−β−n+α+j+ 1)X^(m−n+j)(a). Hafiz et al. (2001) and Hafiz (2004) are only able to consider a situation like that in Part (a) of the theorem that we have just completed because they only define the fractional derivative for orders β ∈ (0,1]. Parts (b), (c) and (d) are extensions that result from our mean-square fractional derivative being defined forβ > 0. In Part (a) we have used the exact method that is used to prove Theorem 4.5(i) of Hafiz et al. (2001) and Theorem 3.3(1) of Hafiz (2004) where it states – incorrectly – that I_a^αD^βaX(t) = D^β−αa X(t) for α ∈ (0,1] and β ∈ (0,1] when α ≤ β. The reason this is incorrect is that it includes the case whenα = β = 1. This leads back to the discussion at the end of Section 4.1 where we looked at the necessity of defining Da^βX(t) to be X^(m)(t) when β =m ∈ N. In Hafiz et al. (2001), Hafiz (2004) and El-Sayed et al. (2005) the RH m.s. fractional derivative (referred to as the Caputo fractional derivative in said papers) is defined as

D_a^βX(t) =I_a^1−βX(t)˙ , β ∈(0,1]. (4.4) Note the inclusion ofβ = 1. If we simply allowβ to be 1 in equation (4.4) we get

D_a^βX(t) =I_a^1−βX(t) =˙ I_a⁰X(t) = ˙˙ X(t).

So the definition given in equation (4.4)seemsreasonable. Using this definition we find that, forα∈(0,1]andβ ∈(0,1]whereα≤β,

I_a^αD^β_aX(t) =I_a^αI_a^1−βX(t) =˙ I_a^{1−(β−α)}X(t) =˙ D^β−α_a X(t). (4.5) If we letα=β = 1in (4.5) we get

I_a^αD^β_aX(t) =D^β−α_a X(t) =D_a¹⁻¹X(t) =D_a⁰X(t) =X(t).

4.2. The composition rule

However, using IBP we know that

I_a^αD_a^βX(t) =I_a¹D¹_aX(t) =X(t)−X(a).

This does not, in general, equalX(t). The mistake that Hafiz et al. (2001) and Hafiz (2004) have made has been to treat the cases whenαand/orβare positive integers the same as the cases when they are not integers – a mistake that is probably, in part, caused by not using the definition D^βaX(t) = X^(m)(t) when β =m∈N. Our definition ofDa^βX(t)gives separate expressions forβ =m∈N andβ /∈Nand so for the remainder of the chapter we will do as we have done for Theorem 4.7 and Theorem 4.8 – we will, when necessary, treat the cases whenα and/orβ are positive integers separately from the cases when they are not.

Theorem 4.9. Letβ > 0, n ∈ Nand m ∈ N. LetX(t) be a stochastic process such thatIa^βX(t)exists fort∈[a, b]⊂T. Then

D^α_aI_a^βX(t) =I_a^β−αX(t) whenα∈(n−1, n]and

(a) β =mandn≤β, or (b) β ∈(m−1, m)andn < β.

Proof

(a) Letα =n.

Using Leibniz’s rule forβ = 1we have d

dtI_a¹X(t) = X(t). Forβ = 2we have

d

dtI_a²X(t) = d dt

Z t a

(t−s)

Γ(2) X(s)ds= Z t

a

X(s)ds+ 0 =I_a¹X(t). d²

dt²I_a²X(t) = d dt

d

dtI_a²X(t) = d

dtI_a¹X(t) =X(t). Forβ = 3we have

d

dtI_a³X(t) = d dt

Z t a

(t−s)²

Γ(3) X(s)ds

= Z t

a

(t−s)

Γ(2) X(s)ds+ 0

= I_a²X(t). d²

dt²I_a³X(t) = d dt

d

dtI_a³X(t) = d

dtI_a²X(t) =I_a¹X(t) d³

dt³I_a³X(t) = d dt

d²

dt²I_a³X(t) = d

dtI_a¹X(t) =X(t).

4.2. The composition rule

Continuing in this manner we have, forβ =mandα=n

Dⁿ_aI_a^mX(t) = I_a^m−nX(t) forn≤m. (4.6) Now let α ∈ (n−1, n) and β = m. Using equation (4.6) we have for n=m

D^α_aI_a^βX(t) = I_a^n−α dⁿ

dtⁿI_a^mX(t)

= I_a^m−α d^m

dt^mI_a^mX(t)

= I_a^m−αX(t) and forn < m

D^α_aI_a^βX(t) = I_a^n−α dⁿ

dtⁿI_a^mX(t)

= I_a^n−αI_a^m−nX(t)

= I_a^m−αX(t). (b) Here we have the conditions

n−1 < α ≤ n ≤ m−1 < β < m . Sincen≥1we haveβ >1.

Letα =n. Using Leibniz’s rule we have d

dtI_a^βX(t) = d dt

Z t a

(t−s)^β−1

Γ(β) X(s)ds

= Z t

a

(t−s)^β−2

Γ(β−1) X(s)ds+

(t−s)^β−1 Γ(β) X(s)

s=t

= I_a^β−1X(t), β >1. (4.7)

Taking the derivative of (4.7) we have d²

dt²I_a^βX(t) = d dt

d

dtI_a^βX(t)

= d

dtI_a^β−1X(t)

= I_a^β−2X(t), β >2 (4.8)

where (4.7) has been used – withβ replaced by(β−1)– in the last step.

4.2. The composition rule

Taking the derivative of (4.8) we have d³

dt³I_a^βX(t) = d dt

d²

dt²I_a^βX(t)

= d

dtI_a^β−2X(t)

= I_a^β−3X(t), β >3

where (4.7) has been used – withβ replaced by(β−2)– in the last step.

Continuing in this manner we have dⁿ

dtⁿI_a^βX(t) =I_a^β−nX(t), β > n. (4.9) Now letα∈(n−1, n)andβ ∈(m−1, m)whereβ > n. Using equation (4.9) we have

D^α_aI_a^βX(t) = I_a^n−α dⁿ

dtⁿI_a^βX(t)

= I_a^n−αI_a^β−nX(t)

= I_a^β−αX(t).

Corollary 4.2. Letβ∈(m−1, m],m∈N, and letX(t)be a second order stochas- tic process such thatIa^βX(t)exists fort∈[a, b]⊂T. Then forj∈ {0,1,2, ..., m−1}

d^j

dt^jI_a^βX(t)|_t=a= 0. Proof:

Using the definition ofD⁰_aX(t)we haveD⁰_aIa^βX(t)|_t=0=Ia^βX(t)|_t=0 = 0.

Using Parts (a) and (b) of Theorem 4.9 we have d^j

dt^jI_a^βX(t) =I_a^β−jX(t) j∈ {1,2, ..., m−1}. Using Part (c) of Theorem 4.1 we have

d^j

dt^jI_a^βX(t)|_t=a=I_a^β−jX(t)|_t=a= 0 j∈ {1,2, ..., m−1}.

Using Corollary 4.2 we see that the result in Theorem 4.7 becomes

I_a^αDa^βX(t) = Ia^α−βX(t) if X(t) is itself a mean-square fractional integral or derivative.

4.2. The composition rule

Theorem 4.10. Letn∈Nandm∈Nwherem < n. LetX(t)be a second-order stochastic process such thatX^(n−m)(t) exists and is mean-square continuous on [a, b]⊂T. Then, forα∈(n−1, n],

D_a^αI_a^mX(t) =D^α−m_a X(t). Proof

Letα=n. Using Leibniz’s rule we have form= 1 d²

dt²I_a¹X(t) = d dt

d

dtI_a¹X(t) = ˙X(t) d³

dt³I_a¹X(t) = d dt

d²

dt²I_a¹X(t) = d

dtX(t) = ¨˙ X(t) d⁴

dt⁴I_a¹X(t) = d dt

d³

dt³I_a¹X(t) = d dt

X(t) =¨ X⁽³⁾(t) ... ...

dⁿ

dtⁿI_a¹X(t) = X⁽ⁿ⁻¹⁾(t), n≥1.

Using equation (4.6) we have form= 2 d³

dt³I_a²X(t) = d dt

d²

dt²I_a²X(t) = d

dtX(t) = ˙X(t) d⁴

dt⁴I_a²X(t) = d dt

d³

dt³I_a²X(t) = d

dtX(t) = ¨˙ X(t) d⁵

dt⁵I_a²X(t) = d dt

d⁴

dt⁴I_a²X(t) = d

dtX(t) =¨ X⁽³⁾(t) ... ...

dⁿ

dtⁿI_a²X(t) = X⁽ⁿ⁻²⁾(t), n≥2.

Continuing in this manner we have forn≥m dⁿ

dtⁿI_a^mX(t) = X^(n−m)(t). (4.10)

Whenα∈(n−1, n)andn > mwe have D^α_aI_a^mX(t) = I_a^n−α dⁿ

dtⁿI_a^mX(t)

= I_a^n−αX^(n−m)(t)

where we have used equation (4.10) to find the last line.

Sinceα−m∈(n−m−1, n−m)we have I_a^n−αX^(n−m)(t) =I(n−m)−(α−m)

a X^(n−m)(t) =D^α−m_a X(t)

and soD_a^αI_a^mX(t) =D_a^α−mX(t)whenα∈(n−1, n).

4.2. The composition rule

The result we derive next is an extension of Theorem 3.4 of Hafiz et al.

(2001). It, like our Theorems 4.10 and 4.9, deals with m.s. derivatives of m.s.

fractional integrals.

LetX(t)be a stochastic process such thatX⁽ⁿ⁾(t),n∈N, exists and is mean- square continuous. SinceX⁽ⁿ⁾(t) exists, allX^(j)(t),j ∈ {0,1,2, ..., n−1}, exist and are mean-square continuous. ThusI_a^γX(t)andI_a^γX^(j)(t)exist and are mean square continuous for allγ ≥1andj∈ {0,1,2, ..., n−1}.

Letβ >1. Using IBP we have I_a^βX(t)˙ = (t−s)^β−1

Γ(β) X(s)

t a

+ Z t

a

(t−s)^β−2

Γ(β−1) X(s)ds

= I_a^β−1X(t)−(t−a)^β−1

Γ(β) X(a). (4.11)

Using Leibniz’s rule we have d

dtI_a^βX(t) = d dt

Z t a

(t−s)^β−1

Γ(β) X(s)ds

= Z t

a

(t−s)^β−2

Γ(β−1) X(s)ds+

(t−s)^β−1 Γ(β) X(s)

s=t

= I_a^β−1X(t). (4.12)

Substituting (4.11) into (4.12) we have d

dtI_a^βX(t) = I_a^βX(t) +˙ (t−a)^β−1

Γ(β) X(a). (4.13)

Taking the derivative of (4.13) we have d²

dt²I_a^βX(t) = d

dtI_a^βX(t) +˙ d dt

(t−a)^β−1 Γ(β) X(a)

= I_a^βX(t) +¨ (t−a)^β−1 Γ(β)

X(a) +˙ (t−a)^β−2

Γ(β−1) X(a) (4.14) where we have used (4.13) – withX(t)replaced byX(t)˙ – to find an expression for _dt^dIa^βX(t).˙

Taking the derivative of (4.14) we have d³

dt³I_a^βX(t) = d

dtI_a^βX(t) +¨ d dt

(t−a)^β−1

Γ(β) X(a) +˙ (t−a)^β−2 Γ(β−1) X(a)

=I_a^βX⁽³⁾(t) +(t−a)^β−1

Γ(β) X(a) +¨ (t−a)^β−2 Γ(β−1) X(a)˙ +(t−a)^β−3

Γ(β−2) X(a)

4.2. The composition rule

where we have used (4.13) – withX(t)replaced byX(t)¨ – to find an expression for _dt^dIa^βX(t).¨

Continuing in this manner we have the following result.

Theorem 4.11. Let n ∈ N and let X(t) be a second-order stochastic process such thatX⁽ⁿ⁾(t)exists and is mean-square continuous ont ∈[a, b]⊂T. Then, forβ >1,

I_a^βX⁽ⁿ⁾(t) = dⁿ

dtⁿI_a^βX(t)−

n−1

X

j=0

(t−a)^β−n+j

Γ(β−n+j+ 1)X^(j)(a).

Note that if _dt^dj_jX(t)|_t=a = 0for j ∈ {0,1,2, . . . , n−1}, as is the case when X(t)is itself a m.s. fractional integral or derivative, the result will be

Ia^βX⁽ⁿ⁾(t) = _dt^dnnIa^βX(t).

Theorem 4.11 gives an expression for _dt^dnnI_a^βX(t) when β ∈ {1,2,3, . . . , n} – something for which an expression had not previously been found. It also gives alternative expressions for _dt^dnnIa^βX(t)whenβ≥nand forIa^βX⁽ⁿ⁾(t)whenβ >1.

In Theorem 4.11 β = 1 has not been included. This is because the result does not, in general, hold forβ = 1as we show next. The method we will use is the same as that used to prove Theorem 4.11.

Letβ = 1. Using IBP we have

I_a¹X(t)˙ = X(t)−X(a) (4.15)

and using Leibniz’s rule we have d

dtI_a¹X(t) = d dt

Z t a

X(s)ds=X(t). (4.16)

Substituting equation (4.15) into equation (4.16) we have d

dtI_a¹X(t) = I_a¹X(t) +˙ X(a). (4.17) Now, taking the derivative of equation (4.17) we get

d²

dt²I_a¹X(t) = d

dtI_a¹X(t) +˙ d dtX(a)

= I_a¹X(t) + ˙¨ X(a) + 0 (4.18)

where we have used equation (4.17) to find an expression for _dt^dI_a¹X(t).˙ Taking the derivative of equation (4.18) we have

d³

dt³I_a¹X(t) = d

dtI_a¹X(t) +¨ d dtX(a)˙

= I_a¹X⁽³⁾(t) +X⁽²⁾(a) + 0

4.2. The composition rule

where we have used equation (4.17) to find an expression for _dt^dI_a¹X(t). Follow-¨ ing this pattern we see that

dⁿ

dtⁿI_a¹X(t) =I_a¹X⁽ⁿ⁾(t) +X⁽ⁿ⁻¹⁾(a).

Clearly, only whenn= 1will the expression in Theorem 4.11 hold forβ = 1.

IBP is used in the proof of Theorem 4.11. Due to this Theorem 4.11 will not hold forβ∈(0,1). Hafiz et al. (2001) include theβ∈(0,1)range despite stating that IBP is used to prove the result.

Since Theorem 4.11 is consistent with previous work for non-integer values ofβ that are greater than 1, we will formally extend it to include the rangeβ∈(0,1).

Theorem 4.11 and its formal extension, can be used to derive many other properties – one of which we show next.

Theorem 4.12. Let n ∈ N and let X(t) be a second-order stochastic process such thatX⁽ⁿ⁾(t) exists and is mean-square continuous on [a, b] ⊂ T. Then for α∈(n−1, n)

D^α_aI_a^αX(t) =X(t). Proof

D^α_aI_a^αX(t) = I_a^n−α dⁿ

dtⁿ I_a^αX(t)

= I_a^n−α

I_a^αX⁽ⁿ⁾(t) +

n−1

X

j=0

(t−a)^α−n+j

Γ(α−n+j+ 1)X^(j)(a)

= I_aⁿX⁽ⁿ⁾(t) +

n−1

X

j=0

(t−a)^j

Γ(j+ 1)X^(j)(a)

=

X(t)−

n−1

X

j=0

(t−a)^j

Γ(j+ 1)X^(j)(a)

+

n−1

X

j=0

(t−a)^j

Γ(j+ 1)X^(j)(a)

= X(t).

In Theorem 4.9 we proved thatDⁿ_aI_aⁿX(t) =X(t). Therefore, by combining Theorem 4.9 with Theorem 4.12 we see thatD^α_aI_a^αX(t) =X(t)forα >0.

In the following two theorems we find expressions forD^α_aDa^βX(t). In Theo- rem 4.13 we will consider the case whenβ = m,m ∈ N, and in Theorem 4.14 we will consider the case whenβ∈(m−1, m).

Theorem 4.13. Letn∈Nandm ∈Nand letX(t)be a second order stochastic process such thatX^(m+n)(t)exists fort∈[a, b]⊂T. Then forα >0

D_a^αX^(m)(t) =D_a^α+mX(t).

4.2. The composition rule

Proof:

Whenα=nthe result clearly holds.

If n−1< α < n then n−1 +m < α+m < n+m and so

D^α+m_a X(t) =I(n+m)−(α+m)

a X^(n+m)(t) =I_a^n−αX^(n+m)(t). Using this we have forn−1< α < n

D_a^αX^(m)(t) =I_a^n−αdⁿ

dtⁿX^(m)(t) =I_a^n−αX^(n+m)(t) =D^α+m_a X(t).

Theorem 4.14. Letα ∈(n−1, n]and β ∈(m−1, m)wheren ∈Nand m∈ N. LetX(t) be a second-order stochastic process such that X^(m+n)(t) exists and is mean-square continuous on[a, b]⊂T. Then

(a) D_a^αD_a^βX(t) =D^α+β_a X(t)

forα∈(0,1)andα+β ∈(m−1, m].

(b) D_a^αD_a^βX(t) =D^α+β_a X(t) + (t−a)^m−β−α

Γ(m−β−α+ 1)X^(m)(a) forα∈(0,1)andα+β ∈(m, m+ 1).

(c) D_a^αD_a^βX(t) =D^α+β_a X(t) +

n−1

X

j=0

(t−a)^m−β−n+j

Γ(m−β−n+ 1 +j)X^(m+j)(a) forα=n.

Proof:

(a)

Forα∈(0,1)

D^α_aD^β_aX(t) = I_a^1−α d

dt I_a^m−βX^(m)(t)

= I_a^1−α

I_a^m−βX^(m+1)(t) +(t−a)^m−β−1

Γ(m−β) X^(m)(a)

= I_a^m−(α+β)I_a¹X^(m+1)(t) + (t−a)^m−β−α

Γ(m−β−α+ 1)X^(m)(a).

(4.19)

4.2. The composition rule

Thus, whenα+β=mwe have

D^α_aD^β_aX(t) = I_a¹X^(m+1)(t) +X^(m)(a)

= X^(m)(t)−X^(m)(a) +X^(m)(a)

= X^(m)(t)

= D^α+β_a X(t).

When α +β ∈ (m −1, m) the first term on the right hand side of (4.19) becomes

I_a^{1−α+m−β}X^(m+1)(t) = I_a^m−(α+β)[X^(m)(t)−X^(m)(a)]

= I_a^m−(α+β)X^(m)(t)− (t−a)^m−β−α

Γ(m−β−α+ 1)X^(m)(a)

= D^α+β_a X(t)− (t−a)^m−β−α

Γ(m−β−α+ 1)X^(m)(a). Substituting this into equation (4.19) we have, forα+β ∈(m−1, m),

D^α_aD^β_aX(t) = D^α+β_a X(t)− (t−a)^m−β−α

Γ(m−β−α+ 1)X^(m)(a) + (t−a)^m−β−α

Γ(m−β−α+ 1)X^(m)(a)

= D^α+β_a X(t). (b)

Ifα∈(0,1)andα+β ∈(m, m+ 1)then D^α_aD^β_aX(t) = I_a^1−α

d

dtI_a^m−βX^(m)(t)

= I_a^1−α

I_a^m−βX^(m+1)(t) + (t−a)^m−β−1

Γ(m−β) X^(m)(a)

= I(m+1)−(α+β)

a X^(m+1)(t) + (t−a)^m−β−α

Γ(m−β−α+ 1)X^(m)(a)

= D^α+β_a X(t) + (t−a)^m−β−α

Γ(m−β−α+ 1)X^(m)(a).

4.2. The composition rule

(c)

Whenα=nwe have D^α_aD^β_aX(t) = dⁿ

dtⁿI_a^m−βX^(m)(t)

= I_a^m−β dⁿ

dtⁿX^(m)(t) +

n−1

X

j=0

(t−a)^m−β−n+j

Γ(m−β−n+j+ 1)X^(m+j)(a)

= I(m+n)−(β+n)

a X^(m+n)(t)

+

n−1

X

j=0

(t−a)^m−β−n+j

Γ(m−β−n+j+ 1)X^(m+j)(a)

= D^n+β_a X(t) +

n−1

X

j=0

(t−a)^m−β−n+j

Γ(m−β−n+j+ 1)X^(m+j)(a).

Theorem 4.14 completes our exploration of the composition rule. The prop- erties in Sections 4.1 and 4.2 are only a selection of the properties of m.s. frac- tional integrals and derivatives. In the following chapter we will consider one last property.

Chapter 5

Integrals and derivatives of