Differentiation of Solutions of Caputo Boundary Value Problems with respect to Boundary Data

Preprint

Article

Differentiation of Solutions of Caputo Boundary Value Problems with respect to Boundary Data

Altmetrics

Downloads

Views

Comments

A peer-reviewed article of this preprint also exists.

Jeffrey W. Lyons^*

This version is not peer-reviewed

Submitted:

14 May 2024

Posted:

15 May 2024

You are already at the latest version

Alerts

Abstract

Under suitable continuity and uniqueness conditions, solutions of an α order Caputo fractional boundary value problem are differentiated with respect to boundary values and boundary points. This extends well-known results for nth order boundary value problems. The approach used is a standard technique and makes heavy use of recent results for differentiation of solutions of Caputo fractional intial value problems with respect to initial conditions and continuous dependence for Caputo fractional boundary value problems.

Keywords:

Subject: Computer Science and Mathematics - Mathematics

MSC: 26A33, 34A08, 34B15

1. Introduction

Let

n \in N

with

α \in (n - 1, n)

and

a < t_{0} < b

R

. Our concern is characterizing partial derivatives with respect to the boundary data for solutions to the Caputo fractional boundary value problem

D_{* t_{0}}^{α} x (t) = f (t, x (t), x^{'} (t), \dots, x^{(n - 1)} (t)), a < t_{0} < t < b,

(1)

satisfying conjugate boundary conditions

x (t_{i}) = x_{i}

(2)

where

D_{* t_{0}}^{α} x

is the Caputo fractional derivative of order

α

of the function

x (t)

and

a < t_{0} \leq t_{1} < t_{2} < \dots < t_{n} < b

and

x_{i} \in R

for

1 \leq i \leq n

Research into fractional differential equations has seen an explosion of results, [1,2,3,4,6,15,19,21]. In fact, there seem to be a limitless number of different ways to define a fractional derivative. However, two definitions have become the source of focus amongst a broad range of researchers in the field; namely the Riemann-Liouville and Caputo fractional derivatives. For expository material on fractional differential equations, we refer the reader to [5,13,14,17].

In this paper, we impose suitable continuity and uniqueness hypotheses so that given a solution of (1), (2), one may take the derivative with respect to the boundary data. This derivative solves an associated Caputo fractional boundary value problem called the variational equation with interesting boundary data.

This work is an expansion upon well-known previous work for nth order boundary value problems [8,9,10,11,12,18]. In fact, we use the ideas of these works as a guide to help construct our proofs. To that end, we rely heavily upon two recent results for Caputo fractional differential equations. The first [6] establishes differentiation of solutions of Caputo initial value problems with respect to the initial data, and the second [16] establishes the continuous dependence on boundary conditions for Caputo boundary value problems.

Essentially, with a unique solution to a Caputo boundary value problem, we define a difference quotient with respect to the boundary datum. We then view this difference quotient in terms of an initial value problem. This allows us to apply Theorem 3.2 from [6] to show this difference quotient solves the variational equation. Finally, we take a limit by applying the continuous dependence result, Theorem 4.2, from [16] which yields the desired result.

The remainder of the paper is organized as follows. In Section 2, one will find brief definitions of fractional integrals and derivatives. For further study, we refer the reader to [5,13,14,17]. Section 3 introduces us to the variational equation and establishes our sufficient hypotheses. For Section 4, we present the important recent developments in the field of study that have made this paper possible; namely differentiation of Caputo initial value problems [6] and continuous dependence of boundary data for Caputo boundary value problems [16]. To conclude, we have Section 5 that contains the main result and its proof.

2. Fractional Derivatives

Let

α > 0

. The Riemann-Liouville fractional integral of a function x of order

α

, denoted

I_{t_{0}}^{α} x

, is defined as

I_{t_{0}}^{α} x (t) = \frac{1}{Γ (α)} \int_{t_{0}}^{t} {(t - s)}^{α - 1} x (s) d s, t_{0} \leq t,

provided the right-hand side exists. Moreover, let

n \in N

denote a positive integer and assume

n - 1 < α \leq n

. The Riemann-Liouville fractional derivative of order

α

of the function x, denoted

D_{t_{0}}^{α} x

, is defined as

D_{t_{0}}^{α} x (t) = D^{n} I_{t_{0}}^{n - α} x (t),

provided the right-hand side exists. If a function x is such that

D_{t_{0}}^{α} (x (t) - \sum_{i = 0}^{n - 1} x^{(i)} (t_{0}) \frac{{(t - t_{0})}^{i}}{i!})

exists, then the Caputo fractional derivative of order

α

of x is defined by

D_{* t_{0}}^{α} x (t) = D_{t_{0}}^{α} (x (t) - \sum_{i = 0}^{n - 1} x^{(i)} (t_{0}) \frac{{(t - t_{0})}^{i}}{i!}) .

Remark 1.

A sufficient condition to guarantee the existence of the Caputo fractional derivative is the absolute continuity of the

(n - 1)

st derivative of

x (t)

. See Theorem 3.1 in [5] and discussion thereafter.

3. Preliminaries

Throughout this work, we make use of the following assumptions:

(1): $f : (a, b) \times R^{n} \to R$ is continuous;
(2): for $1 \leq i \leq n$ , $\partial f (t, x_{1}, \dots, x_{n}) / \partial x_{i} : (a, b) \times R^{n} \to R$ is continuous; and
(3): solutions to initial value problems for (1) are unique on $(a, b)$ ;

The derivative sought in this manuscript solves a related equation which we define next.

Definition 1.

The α order Caputo fractional variational equation of (1) along a solution

x (t)

is the differential equation

D_{* t_{0}}^{α} z (t) = \sum_{j = 0}^{n - 1} \frac{\partial f}{\partial x_{j}} (t, x (t), x^{'} (t), \dots, x^{(n - 1)} (t)) z^{(j)} .

(3)

Finally, we present two more hypotheses which establish a uniqueness condition for (1) and (3), respectively.

(4): Given points $a < t_{0} \leq t_{1} < t_{2} < \dots < t_{n} < b$ , if y and z are solutions of (1) such that for $1 \leq i \leq n$ , $y (t_{i}) = z (t_{i})$ , then $y (t) = z (t)$ on $(a, b)$ ; and
(5): given points $a < t_{0} \leq t_{1} < t_{2} < \dots < t_{n} < b$ , if u is a solution of (3) along (1) such that for $1 \leq i \leq n$ , $u (t_{i})$ =0, then $u (t) \equiv 0$ on $(a, b)$ .

Next, we present two crucial results that make this work possible.

Let

[c, d] \subset R

and for

x \in C [c, d]

, define

{∥ x ∥}_{0, [c, d]} = max_{t \in [c, d]} | x (t) | .

k \in N

, for

x \in C^{k} [c, d]

, define

{∥ x ∥}_{k, [c, d]} = {max {∥ x ∥}_{0, [c, d]}, ∥ x^{'} ∥_{0, [c, d]}, \dots, ∥ x^{(k)} ∥_{0, [c, d]}} .

We seek a boundary value problem result as an analog of the initial value problem result from Eloe et al [6].

Theorem 1.

Let

f (t, y_{0}, y_{1}, \dots, y_{n - 1})

be continuous and have continuous first partial derivatives with respect to

y_{j}

for

0 \leq j \leq n - 1

on an open, connected, convex set

E \subset R \times R^{n} .

Let

(t_{0}, y_{0}, \dots, y_{n - 1}) \in E

, and let

y (t) : = y (t; t_{0}, y_{0}, \dots, y_{n - 1})

be the unique solution of the initial value problem (1) satisfying

y^{(i)} (t_{0}) = y_{i}, i = 0, \dots, n - 1,

(4)

with maximal interval of existence

[t_{0}, ω) .

Choose

[c, d] \subset [t_{0}, ω) .

Then,

(a): for each $0 \leq j \leq n - 1$ , $α_{j} (t) : = \partial y (t) / \partial y_{j}$ exists and is the solution of the variational equation (3) along $y (t)$ on $[c, d]$ and hence, $[t_{0}, ω)$ satisfying the initial conditions

$α_{j}^{(i)} (t_{0}) = δ_{i j}, 0 \leq i \leq n - 1;$
(b): if, in addition, f has a continuous first derivative with respect to t and

$f (t_{0}, y_{0}, y_{1}, \dots, y_{n - 1}) = 0,$

then $β (t) : = \partial y (t) / \partial t_{0}$ exists and is the solution of the variational equation (3) along $y (t)$ on $[c, d]$ and hence, $[t_{0}, ω)$ satisfying the initial conditions

$β^{(i)} (t_{0}) = - y^{(i + 1)} (t_{0}), 0 \leq i \leq n - 1; a n d$
(c): $β (t) = - \sum_{i = 0}^{n - 1} y^{(i + 1)} (t_{0}) α_{i} (t)$ .

We also use recent continuous dependence on boundary conditions results for Caputo fractional differential equations [16]. The first one is when the left-most boundary condition is to the right of the starting point of the Caputo fractional derivative; namely

t_{0} < t_{1}

, and the second is when they are equal; namely

t_{0} = t_{1}

. Note that the second result has an additional condition to establish continuous dependence to the left of

t_{0}

Theorem 2.

[Case when

t_{0} < t_{1}

] Assume that hypotheses(1), (3), and(4)hold. Let

x (t)

be a solution of (1) on

[t_{0}, b)

[c, d] \subset [t_{0}, b)

with points

c \leq t_{0} \leq t_{1} < t_{2} < \dots < t_{n} < d

, and

ϵ > 0

. Then, there exists a

δ (ϵ, [c, d]) > 0

such that, if for

1 \leq i \leq n

| t_{i} - τ_{i} | < δ

with

c < τ_{1} < τ_{2} < τ_{3} < \dots, τ_{n} < d

and

| x (t_{i}) - y_{i} | < δ

with

y_{i} \in R

, then there exists a solution

y (t)

of (1) satisfying

y (τ_{i}) = y_{i}

. Also,

| | x (t) - {y (t) | |}_{n - 1, [c, d]} < ϵ .

Theorem 3.

[Case when

t_{0} = t_{1}

] Assume that hypotheses(1), (3), and(4)hold. Let

x (t)

be a solution of (1) on

[t_{1}, b)

[c, d] \subset [t_{1}, b)

with points

c = t_{1} < t_{2} < \dots < t_{n} < d

, and

ϵ > 0

. Then, there exists a

δ (ϵ, [c, d]) > 0

such that, if for

2 \leq i \leq n

| t_{i} - τ_{i} | < δ

with

c < τ_{2} < τ_{3} < \dots, τ_{n} < d

and for

1 \leq i \leq n

| x (t_{i}) - y_{i} | < δ

with

y_{i} \in R

, then there exists a solution

y (t)

of (1) satisfying

y (t_{1}) = y_{1}

and for

2 \leq i \leq n

y (τ_{i}) = y_{i}

. Also,

| | x (t) - {y (t) | |}_{n - 1, [c, d]} < ϵ .

Additionally, if

f_{k} : (a, b) \times R^{n} \to R

is a sequence of continuous functions that converge uniformly to f on compact subsets of

[c, d] \times R^{n}

and for

k \geq 1

t_{1}^{k}

is an increasing sequence such that

t_{1}^{k} ↑ t_{1}^{-}

k \to \infty

, then there exists a K such that if

k \geq K

, then

| | x_{k} (t) - x (t) {| |}_{n - 1, [c, d]} \to 0 a s k \to \infty .

4. Main Results

In this section, we present our boundary value problem analog. This is done under the assumption that

t_{0} < t_{1}

. However, with the additional assumption from Theorem 3, the same result is established for

t_{0} = t_{1}

and the proof remains the same. Without this additional assumption, the derivative at

t_{0}

would only be a right-hand derivative but the result still holds.

Theorem 4.

Assume conditions(1)-(5)are satisfied and that

t_{0} < t_{1}

. Let

x (t) : = x (t, t_{1}, \dots, t_{n}, x_{1}, \dots, x_{n})

be a solution of (1) satisfying

x (t_{i}) = x_{i}

for

1 \leq i \leq n

[t_{0}, ω) \subset (a, b)

. Then,

(a): for each $1 \leq j \leq n$ , $z_{j} (t) : = \partial x (t) / \partial x_{j}$ exists and is the solution of the variational equation (3) along $x (t)$ on $[c, d]$ and hence, $[t_{0}, ω)$ satisfying the boundary conditions

$z_{j} (t_{i}) = δ_{i j}, 1 \leq i \leq n;$
(b): if, in addition, f has a continuous first derivative with respect to t and

$f (t_{j}, x (t_{j}), x^{'} (t_{j}), \dots, x^{(n - 1)} (t_{j})) = 0,$

then $w_{j} (t) : = \partial x (t) / \partial t_{j}$ exists and is the solution of the variational equation (3) along $x (t)$ on $[c, d]$ and hence, $[t_{0}, ω)$ satisfying the initial conditions

$w_{j} (t_{i}) = - x^{'} (t_{i}) δ_{i j}, 1 \leq i \leq n; a n d$
(c): for each $1 \leq j \leq n$ , $w_{j} (t) = - x^{'} (t_{j}) z_{j} (t)$ .

Proof.

We will only prove part (a) as the proof of part (b) is similar. Part (c) is immediate consequence from parts (a)and (b) when coupoled with hypothesis (v).

Let

1 \leq j \leq n

, and consider

\partial x (t) / \partial x_{j} .

In the interests of conserving space and lessening the tedious notation, we denote

x (t; t_{1}, \dots, t_{n}, x_{1}, \dots, x_{j}, \dots, x_{n})

x (t; x_{j})

x_{j}

is the boundary value of interest.

Let

δ > 0

be as in Theorem 2,

0 < | h | < δ

be given, and define

z_{j h} (t) = \frac{1}{h} [x (t; x_{j} + h) - x (t; x_{j})] .

Note that for every

h \neq 0

\begin{matrix} z_{j h} (t_{j}) & = & \frac{1}{h} [x (t_{j}, x_{j} + h) - x (t_{j}, x_{j})] \\ = & \frac{1}{h} [(x_{j} + h) - x_{j}] \\ = & \frac{1}{h} [h] \\ = & 1 . \end{matrix}

Also, for every

h \neq 0, 1 \leq k \leq n

with

k \neq j,

\begin{matrix} z_{j h} (t_{k}) & = & \frac{1}{h} [x (t_{k}, x_{j} + h) - x (t_{k}, x_{j})] \\ = & \frac{1}{h} [x_{k} - x_{k}] \\ = & 0 . \end{matrix}

Now that we have established the boundary conditions for

z_{j h} (t)

, we show that

z_{j h} (t)

solves the variational equation. To that end, for

1 \leq i \leq n - 1

, let

v_{i} = x^{(i)} (t_{j}; x_{j})

and

ϵ_{i} = ϵ_{i} (h) = x^{(i)} (t_{j}; x_{j} + h) - v_{i} .

By Theorem 2, for

1 \leq i \leq n - 1, ϵ_{i} = ϵ_{i} (h) \to 0

h \to 0 .

Using the notation of Theorem 1 for solutions of initial value problems for (1), viewing

x (t)

as the solution of an initial value problem, and denoting the solution

x (t) = y (t; t_{j}, x_{j}, v_{1}, \dots, v_{n - 1})

, we have

\begin{matrix} z_{j h} (t) = \frac{1}{h} [y (t; t_{j}, x_{j} + h, v_{1} + ϵ_{1}, \dots, v_{n - 1} + ϵ_{n - 1}) - y (t; t_{j}, x_{j}, v_{1}, \dots, v_{n - 1})] . \end{matrix}

Then, by utilizing telescoping sums, we have

\begin{matrix} z_{j h} (t) = & \frac{1}{h} {[y (t; t_{j}, x_{j} + h, v_{1} + ϵ_{1}, \dots, v_{n - 1} + ϵ_{n - 1}) - y (t; t_{j}, x_{j}, v_{1} + ϵ_{1}, \dots, v_{n - 1} + ϵ_{n - 1})] \\ + [y (t; t_{j}, x_{j}, v_{1} + ϵ_{1}, \dots, v_{n - 1} + ϵ_{n - 1}) - y (t; t_{j}, x_{j}, v_{1}, \dots, v_{n - 1} + ϵ_{n - 1})] \\ + [y (t; t_{j}, x_{j}, v_{1}, \dots, v_{n - 1} + ϵ_{n - 1})) - \dots] \\ + [y (t; t_{j}, x_{j}, v_{1}, \dots, v_{n - 1} + ϵ_{n - 1})) - y (t; t_{j}, x_{j}, v_{1}, \dots, v_{n - 1})]} . \end{matrix}

By Theorem 1 and the Mean Value Theorem, we obtain

\begin{matrix} z_{j h} (t) = & \frac{1}{h} [α_{0} (t, y (t; t_{j}, x_{j} + \bar{h}, v_{1} + ϵ_{1}, \dots, v_{n - 1} + ϵ_{n - 1})) (x_{j} + h - x_{j}) \\ + α_{1} (t, y (t; t_{j}, x_{j}, v_{1} + {\bar{ϵ}}_{1}, \dots, v_{n - 1} + ϵ_{n - 1})) (v_{1} + ϵ_{1} - v_{1}) + \dots \\ + α_{n - 1} (t, y (t; t_{j}, x_{j}, v_{1}, \dots, v_{n - 1} + {\bar{ϵ}}_{n - 1})) (v_{n - 1} + ϵ_{n - 1} - v_{n - 1})] \\ = & α_{0} (t, y (t; t_{j}, x_{j} + \bar{h}, v_{1} + ϵ_{1}, \dots, v_{n - 1} + ϵ_{n - 1})) \\ + \frac{ϵ_{1}}{h} α_{1} (t, y (t; t_{j}, x_{j}, v_{1} + {\bar{ϵ}}_{1}, \dots, v_{n - 1} + ϵ_{n - 1})) + \dots \\ + \frac{ϵ_{n - 1}}{h} α_{n - 1} (t, y (t; t_{j}, x_{j}, v_{1}, \dots, v_{n - 1} + {\bar{ϵ}}_{n - 1})) \end{matrix}

where, for

0 \leq k \leq n - 1, α_{k} (t, y (\cdot))

is the solution of the variational equation (3) along

y (\cdot)

satisfying

α_{k}^{(i)} (t_{j}) = δ_{i k}, 0 \leq i \leq n - 1 .

Furthermore, for

1 \leq i \leq n - 1, v_{i} + {\bar{ϵ}}_{i}

is between

v_{i}

and

v_{i} + ϵ_{i} .

Thus, to show

lim_{h \to 0} z_{j h} (x)

exists, it suffices to show, for

1 \leq i \leq n - 1, lim_{h \to 0} ϵ_{i} / h

exists.

Now, from the construction of

z_{j h} (t),

we have

\begin{matrix} z_{j h} (t_{k}) = 0, 1 \leq k \leq n w i t h k \neq j . \end{matrix}

Hence, for

1 \leq k \leq n

with

k \neq j

, we have a system of

n - 1

linear equations with

n - 1

unknowns:

\begin{matrix} - α_{0} (t_{k}, y (t; t_{j}, x_{j} + \bar{h}, v_{1} + ϵ_{1}, \dots, v_{n - 1} + ϵ_{n - 1})) \\ = \frac{ϵ_{1}}{h} α_{1} (t_{k}, y (t; t_{j}, x_{j}, v_{1} + {\bar{ϵ}}_{1}, \dots, v_{n - 1} + ϵ_{n - 1})) + \dots \\ + \frac{ϵ_{n - 1}}{h} α_{n - 1} (t_{k}, y (t; t_{j}, x_{j}, v_{1}, \dots, v_{n - 1} + {\bar{ϵ}}_{n - 1})) . \end{matrix}

In the system of equations above, we notice that

y (\cdot)

is not always the same. Therefore, we consider the coefficient matrix M based on

y (t)

M : = (\begin{matrix} α_{1} (t_{1}, y (t)) & α_{2} (t_{1}, y (t)) & \dots & α_{n - 1} (t_{1}, y (t)) \\ α_{1} (t_{2}, y (t)) & α_{2} (t_{2}, y (t)) & \dots & α_{n - 1} (t_{2}, y (t)) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ α_{1} (t_{j - 1}, y (t)) & α_{2} (t_{j - 1}, y (t)) & \dots & α_{n - 1} (t_{j - 1}, y (t)) \\ α_{1} (t_{j + 1}, y (t)) & α_{2} (t_{j + 1}, y (t)) & \dots & α_{n - 1} (t_{j + 1}, y (t)) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ α_{1} (t_{n}, y (t)) & α_{2} (t_{n}, y (t)) & \dots & α_{n - 1} (t_{n}, y (t))) \end{matrix}) .

We claim

det (M) \neq 0

. Suppose to the contrary that

det (M) = 0

. Then, there exist

p_{i} \in R

for

1 \leq i \leq n - 1

not all zero such that

p_{1} (\begin{matrix} α_{1} (t_{1}, y (t)) \\ α_{1} (t_{2}, y (t)) \\ ⋮ \\ α_{1} (t_{j - 1}, y (t)) \\ α_{1} (t_{j + 1}, y (t)) \\ ⋮ \\ α_{1} (t_{n}, y (t)) \end{matrix}) + \dots + p_{n - 1} (\begin{matrix} α_{n - 1} (t_{1}, y (t)) \\ α_{n - 1} (t_{2}, y (t)) \\ ⋮ \\ α_{n - 1} (t_{j - 1}, y (t)) \\ α_{n - 1} (t_{j + 1}, y (t)) \\ ⋮ \\ α_{n - 1} (t_{n}, y (t)) \end{matrix}) = (\begin{matrix} 0 \\ 0 \\ ⋮ \\ 0 \\ 0 \\ ⋮ \\ 0 \end{matrix}) .

Set

w (t, y (t)) : = p_{1} α_{1} (t, y (t)) + \dots + p_{n - 1} α_{n - 1} (t, y (t)) .

Then,

w (t, y (t))

is a nontrivial solution of the variational equation (3). However,

w (t, y (t)) = 0

and, for

1 \leq k \leq n - 1

with

k \neq j

w (x_{k}, y (t)) = 0 .

By hypothesis (v),

w (t, y (t)) = 0

. Thus,

p_{1} = p_{2} = \dots = p_{n - 1} = 0

which is a contradiction to the choice of the

p_{i}^{'} s .

Hence,

det (M) \neq 0 .

Thus, as a result of continuous dependence, for

h \neq 0

and sufficiently small,

det (M (h)) \neq 0

implying

M (h)

has an inverse where

M (h)

is the appropriately defined matrix from the system of equations. Therefore, for each

1 \leq i \leq n - 1

, we are able to find

ϵ_{i} / h

using Cramer’s rule.

Note as

h \to 0, det (M (h)) \to det (M),

and so for

1 \leq i \leq n - 1, ϵ_{i} (h) / h \to det (M_{i}) / det M : = B_{i}

h \to 0,

where

M_{i}

is the

n - 1 \times n - 1

matrix found by replacing the appropriate column of the matrix defining M by

\begin{matrix} c o l [- α_{0} (t_{1}, x (t)), - α_{0} (t_{2}, x (t)), \dots, - α_{0} (t_{j - 1}, x (t)), - α (t_{j + 1}, x (t)), \dots, - α_{0} (t_{k}, x (t))] . \end{matrix}

Now, let

z_{j} (t) = lim_{h \to 0} z_{j h} (t),

and by construction of

z_{j h} (t)

z_{j} (t) = \frac{\partial x}{\partial x_{j}} (t) .

Furthermore,

z_{j} (t) = lim_{h \to 0} z_{j h} (t) = α_{0} (t, x (t)) + \sum_{i = 1}^{n - 1} B_{i} α_{i} (t, x (t))

which is a solution of the variational equation (3) along

x (t)

. In addition, for

1 \leq j \leq n

z_{j} (x_{k}) = lim_{h \to 0} z_{j h} (x_{k}) = δ_{j k} .

This completes the argument for

\partial x (t) / \partial x_{j} .

□

Funding

This research received no external funding.

Data Availability Statement

No new data were created or analyzed in this study.

Conflicts of Interest

The author declares no conflicts of interest.

References

S. Abbas, M. Benchohra, J.J. Nieto, Caputo-Fabrizio fractional differential equations with non instantaneous impulses, Rend. Circ. Mat. Palermo (2) 71 (2022), no. 1, 131–144. [CrossRef]
B. Ahmad, M. Alghanmi, S.K. Ntouyas, A. Alsaedi, Ahmed, A study of fractional differential equations and inclusions involving generalized Caputo-type derivative equipped with generalized fractional integral boundary conditions, AIMS Math. 4 (2019), no. 1, 26–42.
M. Bohner, S. Hristova, Stability for generalized Caputo proportional fractional delay integro-differential equations. Bound. Value Probl. 2022, no. 14, 15 pp.
P. Das, S. Rana, H. Ramos, Homotopy perturbation method for solving Caputo-type fractional-order Volterra-Fredholm integro-differential equations, Comput. Math. Methods 1 (2019), no. 5, e1047, 9 pp.
K. Diethelm, The analysis of fractional differential equations. An application-oriented exposition using differential operators of Caputo type, Lecture Notes in Mathematics, 2004, Springer-Verlag, Berlin, 2010, 247 pp.
P.W. Eloe, J.W. Lyons, J.T. Neugebauer, Differentiation of solutions of Caputo initial value problems with respect to initial data, PanAmer. Math. J. 30 (2020), no. 4, 71–80. [CrossRef]
P.W. Eloe, T. Masthay, Initial value problems for Caputo fractional differential equations, J. Fract. Calc. Appl. 9 (2018), no. 2, 178–195.
J. Henderson, Existence of solutions of right focal point boundary value problems for ordinary differential equations, Nonlin. Anal. 5 (1981), 989–1002.
J. Henderson, Disconjugacy, disfocality, and differentiation with respect to boundary conditions, J. Math. Anal. Appl. 121 (1987), no. 1, 1–9.
J. Henderson, B. Hopkins, E. Kim, J.W. Lyons, Boundary data smoothness for solutions of nonlocal boundary value problems for n-th order differential equations, Involve 1 (2008), no. 2, 167–181. [CrossRef]
J. Henderson, B. Karna, C.C. Tisdell, Existence of solutions for three-point boundary value problems for second order equations, Proc. Amer. Math. Soc. 133 (2005), no. 5, 1365–1369.
A.F. Janson, B.T. Juman, J.W. Lyons, The connections between variational equations and solutions of second order nonlocal integral boundary value problems, Dynam. Systems Appl. 23 (2014), no. 2-3, 493–503.
A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B. V., Amsterdam, 2006.
V. Kiryakova, Generalized Fractional Calculus and Applications, Pitman Res. Notes Math. Ser., 301, Longman-Wiley, New York, 1994.
K. Lan, Equivalence of higher order linear Riemann-Liouville fractional differential and integral equations, Proc. Amer. Math. Soc. 148 (2020), no. 12, 5225–5234.
J.W. Lyons, Continuous dependence on boundary conditions for Caputo fractional differential equations, Rocky Mountain J. Math, in press.https://projecteuclid.org/journals/rmjm/rocky-mountain-journal-of-mathematics/ DownloadAcceptedPapers/220531-Lyons.pdf.
I. Podlubny, Fractional Differential Equations. An introduction to fractional derivatives, fractional differential equations to methods of their solution and some of their applications, Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, 1999.
J.D. Spencer, Relations between boundary value functions for a nonlinear differential equation and its variational equations, Canad. Math. Bull. 18 (1975), no. 2, 269–276.
C.C. Tisdell, Basic existence and a priori bound results for solutions to systems of boundary value problems for fractional differential equations, Electron. J. Differential Equations 2016, no. 84, 9 pp.
Y. Wang, X. Li, and Y. Huang, The Green’s function for Caputo fractional boundary value problem with a convection term, AIMS Math. 7 (2022), no. 4, 4887–4897.
Y. Zhou, Existence and uniqueness of solutions for a system of fractional differential equations, Fract. Calc. Appl. Anal. 12 (2009), 195–204.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.

MDPI Initiatives

Important Links

Choose an area of interest and we will send you notifications of new preprints at your preferred frequency.

Disclaimer

Differentiation of Solutions of Caputo Boundary Value Problems with respect to Boundary Data

Abstract

1. Introduction

2. Fractional Derivatives

3. Preliminaries

4. Main Results

Funding

Data Availability Statement

Conflicts of Interest

References

MDPI Initiatives

Important Links

Subscribe