Reply to Kiskinov et al. Comment on “Vatsala et al. Analysis of Sequential Caputo Fractional Differential Equations versus Non-Sequential Caputo Fractional Differential Equations with Applications. Foundations 2022, 2, 1129–1142”

Aghalaya S. Vatsala; Govinda Pageni; V. Anthony Vijesh

doi:10.3390/foundations4040032

In our article [1], we are not trying to provide a counter-example to Theorems 1–3 of [2]. In fact, Theorem 1 is Theorem 2.2 of [3] (p. 93). In order to solve a Caputo fractional differential equation of order q, where

0 < q < 1

, we do not require the solution to be a

C^{1}

function on the interval

[a, b]

. It is enough that the solution is a

C^{q}

function, which essentially means the Caputo

q^{t h}

order derivative exists on the needed interval.

Here, we provide the following definitions of left and right Caputo fractional derivatives for clarity.

Definition 1.

The Caputo (left-sided) fractional derivative of

f (t)

of order α, when

n - 1 < α < n

, is given by equation

{}^{c}D_{a^{+}}^{α} f (t) = \frac{1}{Γ (n - α)} \int_{a}^{t} {(t - s)}^{n - α - 1} f^{(n)} (s) d s, t > a,

(1)

and the (right-sided) fractional derivative is given by

{}^{c}D_{b^{-}}^{α} f (t) = \frac{{(- 1)}^{n}}{Γ (n - α)} \int_{t}^{b} {(s - t)}^{n - α - 1} f^{(n)} (s) d s, t < b,

(2)

where

f^{(n)} (t) = \frac{d^{n} (f)}{d t^{n}}

.

In this work, we use Caputo fractional derivatives of order q, where

0 < q < 1

.

Remark 1.

In order for Theorem 2.2 of [2], (p. 93) to hold true, the Caputo left fractional derivative can be computed when the function can be expressed as a function of

(t - a)

. Then, for the same function, the Caputo right fractional derivative has to be computed as a function of

(b - t)

. Furthermore, if

f \in C^{1} [a, b]

, then part (a) and (b) of Theorem 2.2 of [2] holds true.

However, it is to be noted that, from Theorem 2.2 of [3], both the Caputo left derivative and the Caputo right derivative have been defined to yield the integer derivative result

q \to n

as a special case. In particular, when

0 < q < 1

, then

q \to 1

yields the solution of the first order differential equations.

Remark 2.

Observe that the solution of the Caputo fractional differential equation of order α is not a

C^{n}

solution on

[a, b]

, when

(n - 1) < α < n

. See Equation (4.1.62) of [3], which is the solution of an

α^{t h}

-order Caputo fractional differential equation when

n - 1 < α < n

.

In short, when we are seeking the solution of a left Caputo fractional differential equation of order q, where

0 < q < 1

, the solution is not

C^{1}

on

(a, b]

. Similarly, when we are seeking the solution of a right Caputo fractional differential equation of order q, where

0 < q < 1

, the solution is not

C^{1}

on

[a, b)

.

The next comment is about the equivalence of the Caputo left derivative and Caputo right derivative of a function at any point on the interval

(a, b)

.

Consider the linear left Caputo fractional differential equation

{}^{c}D_{a^{+}}^{q} u = λ u, a < t < b, u (a) = u_{a} .

Then, the solution of this left Caputo fractional differential equation is given by

u_{l} (t) = u_{a} E_{q, 1} (λ {(t - a)}^{q}) = \sum_{k = 0}^{\infty} \frac{{(λ {(t - a)}^{q})}^{k}}{Γ (q k + 1)} .

Now, let us consider the right Caputo fractional differential equation,

{}^{c}D_{b^{-}}^{q} u = λ u, a < t < b, u (b) = u_{a},

It is easy to check that the solution of this right Caputo fractional differential equation is

u_{r} (t) = u_{a} E_{q, 1} (λ {(b - t)}^{q}) = \sum_{k = 0}^{\infty} \frac{{(λ {(b - t)}^{q})}^{k}}{Γ (q k + 1)} .

Although the above solutions are solutions of Caputo fractional initial value problems, we want to show the equivalence of the left and right Caputo derivative of

u_{l}

and

u_{r}

at any point on

(a, b)

. Observe that the two solutions are equal at

t = \frac{a + b}{2}

. Now, we claim that the two solutions are equal for any t in

(a, b)

, say

t = t_{1}

. Then, we need to substitute

t_{1} = a + α

in the left Caputo fractional derivative solution and

t_{1} = b - α

in the right Caputo fractional derivative solution. Then, it is easy to check that the two solutions are one and the same for any

t_{1}

on

(a, b)

when

α \in (0, (b - a)]

. If

α = 0

, then

u_{l} (a) = u_{a} = u_{r} (b)

.

As an example, consider

a = 1, b = 2

, and

t_{1} = 1.25

; then, in this case,

α = 0.25

. Then, the above left and right solution can be written as

u_{l} (1 + 0.25) = u_{a} E_{q, 1} (λ {(0.25)}^{q}) = \sum_{k = 0}^{\infty} \frac{{(λ {(0.25)}^{q})}^{k}}{Γ (q k + 1)},

u_{r} (2 - 0.25) = u_{a} E_{q, 1} (λ {(0.25)}^{q}) = \sum_{k = 0}^{\infty} \frac{{(λ {(0.25)}^{q})}^{k}}{Γ (q k + 1)} .

Next, we consider a few examples to demonstrates that Caputo fractional derivative of order q, where

0 < q < 1

exists, even if it is not a

C^{1}

function.

Let

f (t) = {(t - a)}^{ω} .

(3)

It is easy to see that the left Caputo fractional derivative of order q is

{}^{c}D_{a^{+}}^{q} {(t - a)}^{ω} = \frac{Γ (ω + 1) {(t - a)}^{(ω - q)}}{Γ (ω - q + 1)} .

(4)

Similarly, the right Caputo fractional derivative of

{(b - t)}^{ω}

is

{}^{c}D_{b^{-}}^{q} {(b - t)}^{ω} = \frac{Γ (ω + 1) {(b - t)}^{(ω - q)}}{Γ (ω - q + 1)} .

(5)

Observe that

f (t) = {(t - a)}^{ω}

is not

C^{1}

at

t = a

, and

f (t) = {(b - t)}^{ω}

, is not

C^{1}

at

t = b

, when

ω < 1

. However, the Caputo left fractional derivative of

{(t - a)}^{ω}

exists on

[a, b]

and zero at

t = a

. Similarly, the Caputo right fractional derivative of

{(b - t)}^{ω}

exists on

[a, b]

and zero at

t = b

. This satisfies part (a) of Theorem 2.2 of [3]. If

ω = 1

, then part (a) of Theorem 2 of [3] holds true.

If

ω = 1

, then (4) and (5) become

{}^{c}D_{a^{+}}^{q} (t - a) = \frac{{(t - a)}^{(1 - q)}}{Γ (2 - q)},

(6)

and

{}^{c}D_{b^{-}}^{q} (b - t) = \frac{{(b - t)}^{(1 - q)}}{Γ (2 - q)} .

(7)

If

q = 1

, then the left derivative equals

f^{'} (t) = 1

and the right derivative equals

- (- f^{'} (t)) = 1

. This validates Remark 1, as well as Theorem 2.2 of [3]

In particular, if

ω = q

, then

{}^{c}D_{a^{+}}^{q} {(t - a)}^{q} = Γ (q + 1) = {}^{c}D_{b^{-}}^{q} {(b - t)}^{q} .

It appears that the authors of [2] have computed the Caputo derivative of

f (x) = {(x - a)}^{q} {(b - x)}^{q},

at a specific point numerically by substituting the

x_{1}

value in place of the s value in the integral. The integral of this function will not be in closed form.

In fact, for any function

f = f (t - a)

, where

f \in C^{q}

on

(a, b]

, we have

{}^{c}D_{a^{+}}^{q} (f (t - a)) {|_{t = a + α} = F (t - a) |}_{t = a + α} = F (α),

and

{}^{c}D_{b^{-}}^{q} (f (b - t)) {|_{t = b - α} = F (b - t) |}_{t = b - α} = F (α) .

This proves that

{}^{c}D_{a^{+}}^{q} (f (t - a)) {|_{t = a + α} = {}^{c}D_{b^{-}}^{q} (f (b - t)) |}_{t = b - α} = F (α) .

This is our claim in [1].

The reference [4] is devoted to Caputo fractional differential equations.

Conclusions: We conjecture that this result can be extended to Caputo sequential fractional derivative of order

n q

, which is sequential of order q. Note that the integer derivative is sequential.

References

Vatsala, A.S.; Pageni, G.; Vijesh, V.A. Analysis of Sequential Caputo Fractional Differential Equations versus Non-Sequential Caputo Fractional Differential Equations with Applications. Foundations 2022, 2, 1129–1142. [Google Scholar] [CrossRef]
Kiskinov, H.; Petkova, M.; Zahariev, A. Comment on Vatsala et al. Analysis of sequential Caputo fractional differential equations versus non-sequential Caputo fractional differential equations with applications. Foundations 2022, 2, 1129–1142. Foundations 2024, 4, 488–490. [Google Scholar]
Kilbas, A.A.; Srivsatava, H.M.; Trujillo, J.J. Theory and Apllications of Fractional Differential Equations; North Holland: Amsterdam, The Netherlands, 2006. [Google Scholar]
Diethelm, K. The Analysis of Fractional Differential Equations; Springer: Berlin/Heidelberg, Germany, 2004. [Google Scholar]

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