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This study presents a novel and efficient iterative approach to approximating the fixed points of contraction mappings in Banach spaces, specifically approximating the solutions of nonlinear fractional differential equations of the Caputo type. We establish two theorems proving the stability and convergence of the proposed method, supported by numerical examples and graphical comparisons, which indicate a faster convergence rate compared to existing methods, including those by Agarwal, Gursoy, Thakur, Ali and Ali, and $D^{* *}$ . Additionally, a data dependence result for approximate operators using the proposed method is provided. This approach is applied to achieve the solutions for Caputo-type fractional differential equations with boundary conditions, demonstrating the efficacy of the method in practical applications.

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1. Introduction

Fixed point theory is a dynamic field of mathematics with deep connections to functional analysis and topology. It serves as a fundamental tool in the study of nonlinear operators and nonlinear analysis. Over time, fixed point results for contraction mappings have found extensive applications in diverse areas such as topology, differential equations, economics, game theory, optimal control, dynamical systems, and functional analysis. Many researchers have contributed to the development of this field by proposing various extensions and generalizations of contraction mappings [1,2,3,4].

Various iterative procedures are widely employed to approximate fixed points of contraction mappings. Notable examples include the Picard [5], Mann [6], Ishikawa [7], and Noor [8] iterations. In recent years, many researchers have focused on refining these methods. Ullah et al. [9] introduced the AK iteration, while Thakur et al. [10] developed a three-step iteration method in 2016. Later, Piri et al. [11] proposed a more efficient iteration in 2018, outperforming Thakur’s approach [10]. Ali et al. [12] demonstrated that their iteration process exhibits superior convergence speed for Zamfirescu-type mappings. In a subsequent work [13], the stability and convergence properties were explored for weak contractions, as well as generalized non-expansive mappings introduced by Hardy and Rogers, using the F-iteration method. Moreover, a study published in [14] analyzed the JF iteration algorithm to approximate fixed points of nonlinear operators satisfying condition (E). Recently, a new iteration method, referred to as $D^{* *}$ [15], was introduced, exhibiting faster convergence compared to the methods proposed by Agarwal et al. [16], Gursoy et al. [17], Noor [8], Ullah et al. [9], Piri et al. [11], and Thakur et al. [10], as well as several other approaches. Very recently, a new fixed-point iteration method was introduced by Alam and Rohen [18] to approximate the fixed points of contraction mappings in a Banach space.

The exploration of fixed point theory encompasses distinct aspects, including the development and existence of fixed points. A notable area within this field is the investigation of data dependence, a topic with a long-standing history that continues to be a focal point in fixed point research. Foundational results on data dependence for the Picard iteration are available in [19,20]. Solutz, Harder, and Hicks investigated the data dependence of various iterations processes for nonlinear contraction mappings [21,22,23].

It is not an easy task to cover every iteration method developed for approximating a fixed point. We discuss a few of them here. Agarwal et al. [16] introduced the following S iteration method for sequences ${α_{n}}, {β_{n}} \subset (0, 1)$ :

(1) $\{\begin{matrix} u_{0} \in C, \\ u_{n + 1} = (1 - α_{n}) S u_{n} + α_{n} S v_{n}, \\ v_{n} = (1 - β_{n}) u_{n} + β_{n} S u_{n}, n = 0, 1, 2, 3, \dots \end{matrix}$

An iteration method known as Picard-S, proposed by Gursoy et al. [17], which improves upon the approach of Agarwal et al. [16], is defined for sequences ${α_{n}}, {β_{n}} \subset (0, 1)$ as follows:

(2) $\{\begin{matrix} u_{0} \in C, \\ u_{n + 1} = S v_{n}, \\ v_{n} = (1 - α_{n}) S w_{n} + α_{n} S u_{n}, \\ w_{n} = (1 - β_{n}) u_{n} + β_{n} S u_{n}, n = 0, 1, 2, 3, \dots \end{matrix}$

In 2016, Thakur et al. [10] defined an iteration method for sequences ${α_{n}}, {β_{n}} \subset (0, 1)$ , defined as follows:

(3) $\{\begin{matrix} u_{0} \in C, \\ u_{n + 1} = S v_{n}, \\ v_{n} = S ((1 - α_{n}) u_{n} + α_{n} w_{n}), \\ w_{n} = (1 - β_{n}) u_{n} + β_{n} S u_{n}, n = 0, 1, 2, 3, \dots \end{matrix}$

They showed that their iteration method converges more rapidly for certain classes of mappings than the methods defined by Picard [5], Mann [6], Noor [8], Agarwal et al. [16], and Abbas et al. [24] through specific numerical tests.

In 2020, Ali and Ali [12] introduced the F iteration process that converges faster than those of Agarwal et al. [16], Gursoy et al. [17], Vatan’s two-step iteration [25], and Thakur et al. [10]. This method is defined for a sequence, ${α_{n}} \subset (0, 1)$ , as follows:

(4) $\{\begin{matrix} u_{0} \in C, \\ u_{n + 1} = S v_{n}, \\ v_{n} = S w_{n}, \\ w_{n} = S ((1 - α_{n}) u_{n} + α_{n} S u_{n}), n = 0, 1, 2, 3, \dots \end{matrix}$

Recently, Ali et al. [15] presented and analyzed their $D^{* *}$ iteration procedure, which is defined as follows:

(5) $\{\begin{matrix} u_{0} \in C, \\ u_{n + 1} = S^{2} v_{n}, \\ v_{n} = S ((1 - α_{n}) S u_{n} + α_{n} S w_{n}), \\ w_{n} = S ((1 - β_{n}) u_{n} + β_{n} S u_{n}), n = 0, 1, 2, 3, \dots, \end{matrix}$

where

{α_{n}}

{β_{n}} \subset (0, 1)

Motivated by these factors, we propose a new iteration technique that exhibits an accelerated convergence rate for contraction mappings compared to the iterations by Agarwal et al. [16], Gursoy et al. [17], Thakur et al. [10], Ali and Ali [12], and Ali et al. [15]. Our new iteration method is defined as follows:

Let C be a Banach space and $S : C \to C$ an operator. Then, the sequence of iterates ${u_{n}}$ is generated via an initial approximation, $u_{0}$ , in the following manner:

(6) $\{\begin{matrix} u_{0} \in C, \\ u_{n + 1} = S ((1 - α_{n}) S w_{n} + α_{n} S v_{n}), \\ v_{n} = S ((1 - β_{n}) S w_{n} + β_{n} S x_{n}), \\ w_{n} = S^{2} x_{n}, \\ x_{n} = S ((1 - γ_{n}) u_{n} + γ_{n} S u_{n}), n = 0, 1, 2, 3, \dots, \end{matrix}$

where the sequences

{α_{n}}, {β_{n}}, {γ_{n}} \subset (0, 1)

On the other hand, fractional differential equations (FDEs) extend classical differential equations by incorporating derivatives of non-integer (fractional) order. This generalization provides a powerful framework for modeling systems with memory and hereditary properties, making FDEs particularly useful in various fields such as physics, biology, finance, and engineering. Unlike traditional integer-order derivatives, fractional derivatives account for the entire history of a process, allowing for more accurate and realistic descriptions of complex phenomena. The most commonly used definitions of fractional derivatives include the Riemann–Liouville and Caputo derivatives. Among these, the Caputo derivative is preferred in applications involving initial value problems, as it allows for more physically meaningful initial conditions expressed in terms of integer-order derivatives. FDEs have found applications in areas such as viscoelasticity, anomalous diffusion, control theory, and signal processing. The growing interest in these equations has spurred the development of numerical methods and iterative schemes to approximate their solutions, addressing the challenges posed by the nonlocal nature of fractional derivatives. Research on FDEs focuses on both theoretical aspects, such as the existence, uniqueness, and stability of solutions, and practical computational methods to solve them efficiently; e.g., see [13,26,27,28].

This study builds upon prior work by proposing a novel iteration technique to approximate fixed points of contraction operators in Banach spaces. A stability theorem is established to validate the reliability of the proposed method. Through numerical examples, we demonstrate that our approach achieves faster convergence compared to existing methods, including those by Agarwal et al. [16], Gursoy et al. [17], Thakur et al. [10], Ali and Ali [12], and Ali et al. [15]. Moreover, we derive a result on data dependence for an approximate operator using our iteration framework. Finally, the method is applied to obtain solutions for a nonlinear fractional differential equation (FDE), highlighting its practical utility.

2. Preliminaries

In this section, relevant definitions, lemmas, and results are presented that are useful for the paper.

Definition 1

([2]). Consider two iteration sequences, ${u_{n}}$ and ${v_{n}}$ , that both converge to the same point, $u^{*}$ . If there exist two real-valued sequences, ${ζ_{n}}$ and ${η_{n}}$ , such that $∥ u_{n} - u^{*} ∥ \leq ζ_{n}$ and $∥ v_{n} - u^{*} ∥ \leq η_{n}$ for all $n = 1, 2, 3, \dots$ , then the sequence ${u_{n}}$ is said to converge more rapidly than ${v_{n}}$ if

$lim_{n \to \infty} \frac{ζ_{n}}{η_{n}} = 0 .$

Definition 2

([29,30]). Let C be a Banach space, and consider an iteration process given by $u_{n + 1} = F (u_{n}, S)$ that converges to $u^{*}$ , a fixed point of S. This process is said to be stable with respect to S, or S-stable, if the following condition holds for a given sequence ${ζ_{n}}$ in C:

$lim_{n \to \infty} ∥ ζ_{n} - F (ζ_{n}, S) ∥ = 0 if and only if lim_{n \to \infty} ζ_{n} = u^{*} .$

Lemma 1

([31]). Let ${ζ_{n}}$ and ${δ_{n}}$ be two non-negative real sequences. If the inequality

$ζ_{n + 1} \leq (1 - η_{n}) ζ_{n} + δ_{n}$

holds for all $n = 1, 2, 3, \dots$ , where $0 < η_{n} < 1$ , with the conditions $\sum_{n = 0}^{\infty} η_{n} = \infty$ and ${lim}_{n \to \infty} \frac{δ_{n}}{η_{n}} = 0$ , then it follows that

$lim_{n \to \infty} ζ_{n} = 0 .$

One of the fundamental fixed-point theorems in metric space was introduced by Banach [32] in 1922. It asserts the following:

Theorem 1

([32]). If C is a Banach space, then an operator $S : C \to C$ satisfying the contraction condition

(7) $∥ S y - S z ∥ \leq β ∥ y - z ∥ for all y, z \in C and β \in [0, 1),$

has a fixed point, $u^{*} \in C$ . Moreover, the sequence of Picard iterations defined by $u_{n + 1} = S u_{n}$ converges to $u^{*}$ , a fixed point of S, for any initial guess, $u_{0} \in C$ .

We now introduce a significant lemma that plays a fundamental role in our analysis.

Lemma 2

([33]). Consider two non-negative real sequences, ${ζ_{n}}$ and ${δ_{n}}$ . If the following inequality holds,

(8) $ζ_{n + 1} \leq (1 - η_{n}) ζ_{n} + η_{n} δ_{n},$

where $0 < η_{n} < 1$ , and the conditions $\sum_{n = 0}^{\infty} η_{n} = \infty$ and $δ_{n} \geq 0$ is a bounded sequence are satisfied, then we have

$0 \leq \underset{n \to \infty}{lim sup} ζ_{n} \leq \underset{n \to \infty}{lim sup} δ_{n} .$

Definition 3

([34]). If $u (y)$ is a continuous function in $[0, \infty)$ , and $n \in N$ , $0 < t < y < \infty$ . Then, the derivative

(9) ${}^{c}D^{γ} u (y) = \frac{1}{Γ (n - γ)} \int_{0}^{y} \frac{u^{(n)} (t)}{{(y - t)}^{γ + 1 - n}} d t,$

where $(n - 1) < γ < n$ , is known as the Caputo fractional derivative of order γ.

3. Main Results

This section provides analytical proof establishing that the iterative sequences defined by (1)–(5) converge at a slower rate compared to the newly introduced iteration method (6). To support our theoretical analysis, we also include numerical examples.

We begin by demonstrating the convergence of our iteration method (6) to a unique fixed point of contraction mapping.

Theorem 2.

Let $S : C \to C$ be a contraction mapping with a constant $β \in [0, 1)$ , where C is a Banach space. Then, the iterative sequence ${u_{n}}$ generated through the procedure described in (6) converges to the unique fixed point of S.

Proof.

Let $u^{*}$ be the unique fixed point of the contraction S so that $S u^{*} = u^{*}$ . We aim to prove that the iterative sequence generated via (6) converges to $u^{*}$ . From the four inequalities, we analyze each step as follows.

(10) $\begin{matrix} ∥ x_{n} - u^{*} ∥ & = ∥S ((1 - γ_{n}) u_{n} + γ_{n} S u_{n}) - u^{*}∥ \\ = ∥S ((1 - γ_{n}) u_{n} + γ_{n} S u_{n}) - S (u^{*})∥ \\ \leq β ∥(1 - γ_{n}) u_{n} + γ_{n} S u_{n} - u^{*}∥ (\sin ce S is a contraction) \\ = β ∥(1 - γ_{n}) (u_{n} - u^{*}) + γ_{n} (S u_{n} - u^{*})∥ \\ \leq β ((1 - γ_{n}) ∥ u_{n} - u^{*} ∥ + γ_{n} ∥ S u_{n} - u^{*} ∥) \\ \leq β ((1 - γ_{n}) ∥ u_{n} - u^{*} ∥ + γ_{n} β ∥ u_{n} - u^{*} ∥) \\ = β ((1 - γ_{n} + γ_{n} β) ∥ u_{n} - u^{*} ∥) \\ = β (1 - γ_{n} (1 - β)) ∥ u_{n} - u^{*} ∥ \\ \leq β ∥ u_{n} - u^{*} ∥ (\sin ce 1 - γ_{n} (1 - β) < 1) . \end{matrix}$

Next, using (10), we have

(11) $\begin{matrix} ∥ w_{n} - u^{*} ∥ & = ∥ S^{2} x_{n} - u^{*} ∥ \\ = β ∥ S x_{n} - u^{*} ∥ \\ \leq β^{2} ∥ x_{n} - u ∥ \leq β^{3} ∥ u_{n} - u^{*} ∥ . \end{matrix}$

And using (10) and (11), we obtain

(12) $\begin{matrix} ∥ v_{n} - u^{*} ∥ & = ∥ S ((1 - β_{n}) S w_{n} + β_{n} S x_{n}) - u^{*} ∥ \\ \leq β ∥ (1 - β_{n}) S w_{n} + β_{n} S x_{n} - u^{*} ∥ \\ \leq β ((1 - β_{n}) ∥ S w_{n} - u^{*} ∥ + β_{n} ∥ S x_{n} - u^{*} ∥) \\ \leq β (β (1 - β_{n}) ∥ w_{n} - u^{*} ∥ + β β_{n} ∥ x_{n} - u^{*} ∥) \\ = β^{2} (1 - β_{n}) ∥ w_{n} - u^{*} ∥ + β^{2} β_{n} ∥ x_{n} - u^{*} ∥ \\ \leq β^{5} (1 - β_{n}) ∥ u_{n} - u^{*} ∥ + β^{3} β_{n} ∥ u_{n} - u^{*} ∥ \\ = β^{3} (β^{2} (1 - β_{n}) + β_{n}) ∥ u_{n} - u^{*} ∥ \\ = β^{3} (β^{2} (1 - β_{n}) + β_{n}) ∥ u_{n} - u^{*} ∥ \\ = β^{3} (β^{2} + (1 - β^{2}) β_{n}) ∥ u_{n} - u^{*} ∥ . \end{matrix}$

Hence, using (10)–(12), we get

$\begin{matrix} ∥ u_{n + 1} - u^{*} ∥ & = ∥ S ((1 - α_{n}) S w_{n} + α_{n} S v_{n}) - u^{*} ∥ \\ \leq β ∥ (1 - α_{n}) S w_{n} + α_{n} S v_{n} - u^{*} ∥ \\ = β ∥ (1 - α_{n}) S w_{n} + α_{n} S v_{n} - (1 - α_{n}) S u^{*} - α_{n} S u^{*} ∥ \\ \leq β ((1 - α_{n}) ∥ S w_{n} - u^{*} ∥ + α_{n} ∥ S v_{n} - u^{*} ∥) (by the triangle inequality) \\ \leq β^{2} (1 - α_{n}) ∥ w_{n} - u^{*} ∥ + β^{2} α_{n} ∥ v_{n} - u^{*} ∥ \\ \leq β^{5} (1 - α_{n}) ∥ u_{n} - u^{*} ∥ + β^{5} α_{n} (β^{2} + (1 - β^{2}) β_{n} ∥ u_{n} - u^{*} ∥) \\ \leq β^{5} [1 - α_{n} + β^{2} α_{n} + α_{n} β_{n} (1 - β^{2})] ∥ u_{n} - u^{*} ∥ \\ = β^{5} [1 - α_{n} (1 - β^{2}) + α_{n} β_{n} (1 - β^{2})] ∥ u_{n} - u^{*} ∥ \\ = β^{5} [1 - α_{n} (1 - β^{2}) (1 - β_{n})] ∥ u_{n} - u^{*} ∥ \\ = β^{5} ∥ u_{n} - u^{*} ∥ (\sin ce 1 - α_{n} (1 - β^{2}) (1 - β_{n})) < 1) \\ \leq β^{5 n} ∥ u_{0} - u^{*} ∥ \end{matrix}$

Therefore, the sequence ${u_{n}}$ generated via the iteration method (6) converges to the unique fixed point $u^{*}$ of S. This completes the proof. □

We now present a result that establishes the stability of our iteration process (6).

Theorem 3.

Let C be a Banach space, and let $S : C \to C$ be a contraction mapping with $β \in [0, 1)$ . Then, the sequence ${u_{n}}$ generated via the iteration method (6) is S-stable, provided that the sequence ${γ_{n}}$ is bounded away from zero.

Proof.

Assume that the iterative sequence ${u_{n}}$ , defined by (6), follows the form

$u_{n + 1} = F (u_{n}, S),$

and it converges to a unique fixed point,

u^{*}

, of the mapping S. Now, consider an arbitrary sequence,

{p_{n}}

. According to the triangle inequality, we have

$∥ p_{n + 1} - u^{*} ∥ \leq ∥ p_{n + 1} - F (p_{n}, S) ∥ + ∥ F (p_{n}, S) - u^{*} ∥,$

where the function F is defined as

$\begin{matrix} F (p_{n}, S) & = S (α_{n} S v_{n} + (1 - α_{n}) S w_{n}), \\ v_{n} & = S (β_{n} S w_{n} + (1 - β_{n}) S x_{n}), \\ w_{n} & = S^{2} x_{n}, \\ x_{n} & = S (γ_{n} S p_{n} + (1 - γ_{n}) p_{n}), n \in N . \end{matrix}$

Following the steps from Theorem 2, we obtain the following:

$∥ p_{n + 1} - u^{*} ∥ \leq ∥ p_{n + 1} - F (p_{n}, S) ∥ + β^{5} (1 - α_{n} (1 - β_{n}) (1 - β^{2})) (1 - γ_{n} (1 - β)) ∥ p_{n} - u^{*} ∥ .$

Since $1 - α_{n} (1 - β_{n}) (1 - β^{2}) < 1$ , the inequality simplifies to the following:

$∥ p_{n + 1} - u^{*} ∥ \leq ∥ p_{n + 1} - F (p_{n}, S) ∥ + β^{5} (1 - γ_{n} (1 - β)) ∥ p_{n} - u^{*} ∥ .$

Define the following:

$δ_{n} = ∥ p_{n + 1} - F (p_{n}, S) ∥, η_{n} = γ_{n} (1 - β), ζ_{n} = ∥ p_{n} - u^{*} ∥ .$

When Lemma 1 is used, if ${lim}_{n \to \infty} δ_{n} = 0$ and ${γ_{n}}$ is bounded away from 0, it follows that ${lim}_{n \to \infty} ζ_{n} = 0$ . This implies the following:

$lim_{n \to \infty} ∥ p_{n} - u^{*} ∥ = 0 or lim_{n \to \infty} p_{n} = u^{*} .$

Conversely, if ${lim}_{n \to \infty} p_{n} = u^{*}$ , i.e., ${lim}_{n \to \infty} ∥ p_{n} - u^{*} ∥ = 0$ and ${lim}_{n \to \infty} ∥ p_{n + 1} - u^{*} ∥ = 0$ , we have the following:

$∥ p_{n + 1} - F (p_{n}, S) ∥ \leq ∥ p_{n + 1} - u^{*} ∥ + ∥ F (p_{n}, S) - u^{*} ∥ .$

By the contraction property,

$∥ F (p_{n}, S) - u^{*} ∥ \leq β^{5} (1 - α_{n} (1 - β_{n}) (1 - β^{2})) (1 - γ_{n} (1 - β)) ∥ p_{n} - u^{*} ∥ .$

Thus,

$lim_{n \to \infty} ∥ p_{n + 1} - F (p_{n}, S) ∥ = 0 .$

This completes the proof that the iterative procedure ${u_{n}}$ defined by (6) is stable with respect to S, or S-stable. □

To demonstrate that the iteration scheme defined in (6) achieves a faster rate of convergence compared to the iteration methods given in (4) and (5), we present the following theoretical findings.

Theorem 4.

Let C be a Banach space, and let $S : C \to C$ be a contraction mapping with $β \in [0, 1)$ defined on a non-empty, closed, and convex subset, C. The iteration processes described by (4) and (5) exhibit a slower rate of convergence compared to the iteration process in (6). Our proposed scheme in (6) converges faster to the unique fixed point, $u^{*}$ , of the contraction mapping S, provided that the sequences ${α_{n}}$ , ${β_{n}}$ , and ${γ_{n}}$ stay bounded away from 0 and 1.

Proof.

Since the sequences ${α_{n}}$ , ${β_{n}}$ , and ${γ_{n}}$ are bounded away in the interval $(0, 1)$ , there exist constants $α^{'}, α^{″}, β^{'}, β^{″}, γ^{'}, γ^{″} \in (0, 1)$ , such that

$0 < α^{'} \leq α_{n} \leq α^{″} < 1, 0 < β^{'} \leq β_{n} \leq β^{″} < 1, 0 < γ^{'} \leq γ_{n} \leq γ^{″} < 1 .$

For our iterative procedure defined in (6), and using the proof technique from Theorem 2, we can express the following:

$∥ u_{n + 1} - u^{*} ∥ \leq β^{5} (1 - α_{n} (1 - β_{n}) (1 - β^{2})) (1 - γ_{n} (1 - β)) ∥ u_{n} - u^{*} ∥ .$

Substituting the bounds on the sequences gives us the following:

(13) $\begin{matrix} ∥ u_{n + 1} - u^{*} ∥ & \leq β^{5} (1 - α^{'} (1 - β^{'}) (1 - β^{2})) (1 - γ^{'} (1 - β)) ∥ u_{n} - u^{*} ∥ \\ \leq β^{5 n} {[(1 - α^{'} (1 - β^{'}) (1 - β^{2})) (1 - γ^{'} (1 - β))]}^{n} ∥ u_{0} - u^{*} ∥ . \end{matrix}$

For the iterative procedure defined in (4), we similarly find the following:

(14) $\begin{matrix} ∥ u_{n + 1} - u^{*} ∥ & \leq β^{3} (1 - α_{n} (1 - β)) ∥ u_{n} - u^{*} ∥ \\ \leq β^{3} (1 - α^{'} (1 - β)) ∥ u_{n} - u^{*} ∥ \\ \leq β^{3 n} {(1 - α^{'} (1 - β))}^{n} ∥ u_{0} - u^{*} ∥ . \end{matrix}$

Letting $ζ_{n} =$ $β^{5 n} {[(1 - α^{'} (1 - β^{'}) (1 - β^{2})) (1 - γ^{'} (1 - β))]}^{n}$ from (13) and $δ_{n} = β^{3 n} {(1 - α^{'} (1 - β))}^{n}$ from (14), we compute the following:

$\frac{ζ_{n}}{δ_{n}} = {(\frac{β^{2} (1 - α^{'} (1 - β^{'}) (1 - β^{2})) (1 - γ^{'} (1 - β))}{(1 - α^{'} (1 - β))})}^{n},$

where

\frac{β^{2} (1 - α^{'} (1 - β^{'}) (1 - β^{2})) (1 - γ^{'} (1 - β))}{(1 - α^{'} (1 - β))} < 1

. This leads to

{lim}_{n \to \infty} \frac{ζ_{n}}{δ_{n}} = 0

, confirming that our iteration process (6) converges faster than the iterative procedure (4).

Next, for the iterative procedure defined in (5), we find the following:

$\begin{matrix} ∥ u_{n + 1} - u^{*} ∥ & \leq β^{4} (1 - α_{n} (1 - β) - α_{n} β_{n} β (1 - β)) ∥ u_{n} - u^{*} ∥ \\ \leq β^{4} (1 - α^{'} (1 - β) - α^{'} β^{'} β (1 - β)) ∥ u_{n} - u^{*} ∥ . \end{matrix}$

This gives

(15) $∥ u_{n + 1} - u^{*} ∥ \leq β^{4 n} {(1 - α^{'} (1 - β) - α^{'} β^{'} β (1 - β))}^{n} ∥ u_{0} - u^{*} ∥ .$

Defining $δ_{n} = β^{4 n} {(1 - α^{'} (1 - β) - α^{'} β^{'} β (1 - β))}^{n}$ from (15), we find the following:

$\frac{ζ_{n}}{δ_{n}} = {(\frac{β (1 - α^{'} (1 - β^{'}) (1 - β^{2})) (1 - γ^{'} (1 - β))}{(1 - α^{'} (1 - β) - α^{'} β^{'} β (1 - β))})}^{n},$

where

\frac{β (1 - α^{'} (1 - β^{'}) (1 - β^{2})) (1 - γ^{'} (1 - β))}{(1 - α^{'} (1 - β) - α^{'} β^{'} β (1 - β))} < 1

. Thus,

{lim}_{n \to \infty} \frac{ζ_{n}}{δ_{n}} = 0

indicates that our iterative procedure (6) converges more rapidly than the iteration method (5). □

3.1. Numerical Examples

We will now provide numerical evidence demonstrating that the iteration process defined in (6) converges more rapidly than the five earlier iteration processes proposed by Agarwal et al. [16], Gursoy et al. [17], Thakur et al. [10], Ali and Ali [12], and Ali et al. [15].

Example 1

([18]). Consider a contraction mapping $S : C \to C$ defined by

(16) $S u = \frac{4}{5} u + \frac{6}{5}, for all u \in C,$

where $C = [3, 9]$ . Let us define the sequences

${α_{n} = \frac{2 n}{4 n + 1}}, {β_{n} = \frac{4 n^{2} + n}{{(3 n + 1)}^{2}}}, {γ_{n} = \frac{2 n}{7 n + 3}} \subset (0, 1)$

and set the initial guess $u_{0} = 5$ . Table 1 and Figure 1 and Figure 2 illustrate the iterations of the methods defined by (1)–(5), alongside our iterative procedure defined by (6). All methods converge to the unique fixed point, $u^{*} = 6$ , of S using the stopping criterion $∥ u_{n} - u ∥ < 10^{- 6}$ . It is evident that the newly defined iterative procedure converges to the fixed point more quickly than the other methods.

Example 2

([18]). Consider a contraction mapping, $S : C \to C$ , defined by

(17) $S u = \sqrt{u^{2} - 3 u + 9}, for all u \in C,$

where $C = [1, 100]$ . Let the sequences be defined as

${α_{n} = \frac{2 n + 1}{9 n + 3}}, {β_{n} = \frac{3 n}{5 n + 1}}, {γ_{n} = \frac{n}{5 n + 3}} \subset (0, 1)$

with an initial guess $u_{0} = 7$ . Table 2 and Figure 3 and Figure 4 below illustrate the iterations of the methods defined in (1)–(5) and our iterative procedure (6), all of which converge to the unique fixed point, $u * = 3$ , of S using the stopping criterion $∥ u_{n} - u ∥ < 10^{- 6}$ . It is evident that the newly defined iterative procedure converges to the fixed point more quickly than the other methods.

Remark 1.

The data presented in the tables and figures illustrate that the newly defined iterative method achieves a notably faster convergence to the fixed point compared to other established methods, including those by Agarwal, Gursoy, Thakur, Ali and Ali, and $D^{* *}$ . This is evident from the rapid reduction in iteration values toward the fixed point. Specifically, the new method reaches approximate stability within fewer iterations, underscoring its efficiency and potential for solving nonlinear equations with increased accuracy. The graphical comparisons further confirm the superiority of this approach by showing a steeper convergence curve relative to other methods.

3.2. Data Dependence

We now establish a result on the data dependence. An operator, $S^{'} : C \to C$ , is said to be an approximate version of the operator $S : C \to C$ if there exists a constant, $η > 0$ , such that

$∥ S y - S^{'} y ∥ \leq η, \forall y \in C .$

Theorem 5.

Let $S, S^{'} : C \to C$ be two contraction operators defined on a Banach space, C, where $0 < β < 1$ . Assume that $S^{'}$ is an approximate version of S with a constant, $η > 0$ , such that $∥ S u - S^{'} u ∥ \leq η$ for all $u \in C$ .

Consider the iterative sequence ${u_{n}}$ generated via the method (6) for the operator S and the sequence ${u_{n}^{'}}$ for the operator $S^{'}$ , defined as follows:

(18) $\{\begin{matrix} u_{0}^{'} \in C, \\ u_{n + 1}^{'} = S^{'} ((1 - α_{n}) S^{'} w_{n}^{'} + α_{n} S^{'} v_{n}^{'}), \\ v_{n}^{'} = S^{'} ((1 - β_{n}) S^{'} w_{n}^{'} + β_{n} S^{'} x_{n}^{'}), \\ w_{n}^{'} = S^{' 2} x_{n}^{'}, \\ x_{n}^{'} = S^{'} ((1 - γ_{n}) u_{n}^{'} + γ_{n} S^{'} u_{n}^{'}), n = 0, 1, 2, \dots \end{matrix}$

where the sequences ${α_{n}}, {β_{n}}, {γ_{n}} \subset (0, 1)$ satisfy the condition $\frac{1}{2} \leq α_{n} \leq γ_{n}$ for all $n \in N$ , and $\sum_{n = 1}^{\infty} α_{n} (1 - β) = \infty$ .

Then the following inequality holds:

$∥ y - y^{'} ∥ \leq \frac{15 η}{1 - β},$

where y and $y^{'}$ are the unique fixed points of the operators S and $S^{'}$ , respectively.

Proof.

From the iteration methods described in the problem, we aim to analyze the data dependence between two sequences, ${u_{n}}$ and ${u_{n}^{'}}$ , generated for the operators S and $S^{'}$ in (6) and (18), respectively.

Using the iteration methods, we obtain the following:

Bounding $∥ x_{n}^{'} - x_{n} ∥$ :
$\begin{matrix} ∥ x_{n}^{'} - x_{n} ∥ & = ∥ S^{'} ((1 - γ_{n}) u_{n}^{'} + γ_{n} S^{'} u_{n}^{'}) - S ((1 - γ_{n}) u_{n} + γ_{n} S u_{n}) ∥ \\ \leq ∥ S^{'} ((1 - γ_{n}) u_{n}^{'} + γ_{n} S^{'} u_{n}^{'}) - S ((1 - γ_{n}) u_{n}^{'} + γ_{n} S^{'} u_{n}^{'}) ∥ \\ + ∥ S ((1 - γ_{n}) u_{n}^{'} + γ_{n} S^{'} u_{n}^{'}) - S ((1 - γ_{n}) u_{n} + γ_{n} S u_{n}) ∥ \\ \leq η + β ∥ (1 - γ_{n}) u_{n}^{'} + γ_{n} S^{'} u_{n}^{'} - (1 - γ_{n}) u_{n} - γ_{n} S u_{n} ∥ \\ \leq η + β ((1 - γ_{n}) ∥ u_{n}^{'} - u_{n} ∥ + γ_{n} ∥ S^{'} u_{n}^{'} - S u_{n} ∥) \\ \leq η + β ((1 - γ_{n}) ∥ u_{n}^{'} - u_{n} ∥ + γ_{n} η + γ_{n} β ∥ u_{n}^{'} - u_{n} ∥) \\ = η + β η γ_{n} + β (1 - γ_{n} (1 - β)) ∥ u_{n}^{'} - u_{n} ∥ . \end{matrix}$
Bounding $∥ w_{n}^{'} - w_{n} ∥$ :
$\begin{matrix} ∥ w_{n}^{'} - w_{n} ∥ & = ∥ S^{' 2} x_{n}^{'} - S^{2} x_{n} ∥ \\ \leq ∥ S^{'} (S^{'} x_{n}^{'}) - S (S^{'} x_{n}^{'}) ∥ + ∥ S (S^{'} x_{n}^{'}) - S (S x_{n}) ∥ \\ \leq η + β ∥ S^{'} x_{n}^{'} - S x_{n} ∥ \\ \leq η + β η + β^{2} ∥ x_{n}^{'} - x_{n} ∥ \\ \leq η + β η + β^{2} (η + β η γ_{n} + β (1 - γ_{n} (1 - β)) ∥ u_{n}^{'} - u_{n} ∥) . \end{matrix}$
Bounding $∥ v_{n}^{'} - v_{n} ∥$ :
$\begin{matrix} ∥ v_{n}^{'} - v_{n} ∥ & = ∥ S^{'} (β_{n} S^{'} w_{n}^{'} + (1 - β_{n}) S^{'} x_{n}^{'}) - S (β_{n} S w_{n} + (1 - β_{n}) S x_{n}) ∥ \\ \leq ∥ S^{'} (β_{n} S^{'} w_{n}^{'} + (1 - β_{n}) S^{'} x_{n}^{'}) - S (β_{n} S^{'} w_{n}^{'} + (1 - β_{n}) S^{'} x_{n}^{'}) ∥ \\ + ∥ S (β_{n} S^{'} w_{n}^{'} + (1 - β_{n}) S^{'} x_{n}^{'}) - S (β_{n} S w_{n} + (1 - β_{n}) S x_{n}) ∥ \\ \leq η + β ∥ β_{n} S^{'} w_{n}^{'} + (1 - β_{n}) S^{'} x_{n}^{'} - β_{n} S w_{n} - (1 - β_{n}) S x_{n} ∥ \\ \leq η + β β_{n} ∥ S^{'} w_{n}^{'} - S w_{n} ∥ + β (1 - β_{n}) ∥ S^{'} x_{n}^{'} - S x_{n} ∥ \\ \leq η + β β_{n} (η + β ∥ w_{n}^{'} - w_{n} ∥) + β (1 - β_{n}) (η + β ∥ x_{n}^{'} - x_{n} ∥) \\ \leq η + β β_{n} (η + β η) + β (1 - β_{n}) (η + β (η + β (γ_{n} η + (1 - γ_{n}) (1 - β) ∥ u_{n}^{'} - u_{n} ∥))) \\ = η + β η + β^{2} η + β^{3} (1 - β_{n}) (γ_{n} η + (1 - γ_{n}) (1 - β) ∥ u_{n}^{'} - u_{n} ∥) . \end{matrix}$
Bounding $∥ u_{n + 1}^{'} - u_{n + 1} ∥$ :
$\begin{matrix} ∥ u_{n + 1}^{'} - u_{n + 1} ∥ & = ∥ S^{'} ((1 - α_{n}) S^{'} w_{n}^{'} + α_{n} S^{'} v_{n}^{'}) - S ((1 - α_{n}) S w_{n} + α_{n} S v_{n}) ∥ \\ \leq ∥ S^{'} ((1 - α_{n}) S^{'} w_{n}^{'} + α_{n} S^{'} v_{n}^{'}) - S ((1 - α_{n}) S^{'} w_{n}^{'} + α_{n} S^{'} v_{n}^{'}) ∥ \\ + ∥ S ((1 - α_{n}) S^{'} w_{n}^{'} + α_{n} S^{'} v_{n}^{'}) - S ((1 - α_{n}) S w_{n} + α_{n} S v_{n}) ∥ \\ \leq η + β (1 - α_{n}) (η + β ∥ w_{n}^{'} - w_{n} ∥) + β α_{n} (η + β ∥ v_{n}^{'} - v_{n} ∥) \\ \leq η + β η + β^{2} η + β^{3} η (1 - α_{n}) + β^{4} η (1 - α_{n}) \\ + β^{5} (1 - α_{n}) α_{n} (γ_{n} η + (1 - γ_{n}) (1 - β) ∥ u_{n}^{'} - u_{n} ∥) . \end{matrix}$

Using the assumptions $\frac{1}{2} \leq α_{n} \leq γ_{n}$ and $\sum_{n = 1}^{\infty} α_{n} (1 - β) = \infty,$ we can summarize that

$∥ u_{n + 1}^{'} - u_{n + 1} ∥ \leq 7 η + η α_{n} + (1 - α_{n} (1 - β)) ∥ u_{n}^{'} - u_{n} ∥ .$

Now, define

$ζ_{n} = ∥ u_{n}^{'} - u_{n} ∥, η_{n} = α_{n} (1 - β), and δ_{n} = \frac{15 η}{1 - β} .$

Simplifying gives the following:

$\begin{matrix} ζ_{n + 1} & \leq 7 (1 - α_{n} + α_{n}) η + η α_{n} + (1 - α_{n} (1 - β)) ∥ u_{n}^{'} - u_{n} ∥ \\ \leq 7 (α_{n} + α_{n}) η + η α_{n} + (1 - α_{n} (1 - β)) ∥ u_{n}^{'} - u_{n} ∥ . \end{matrix}$

Thus, we have

$ζ_{n + 1} \leq α_{n} (1 - β) \frac{15 η}{1 - β} + (1 - η_{n}) ζ_{n} = η_{n} δ_{n} + (1 - η_{n}) ζ_{n} .$

Consequently, by applying Lemma 2, we conclude

$0 \leq \underset{n \to \infty}{lim sup} ζ_{n} \leq \underset{n \to \infty}{lim sup} δ_{n},$

which yields

$0 \leq \underset{n \to \infty}{lim sup} ∥ u_{n}^{'} - u_{n} ∥ \leq \underset{n \to \infty}{lim sup} \frac{15 η}{1 - β} .$

This implies

$∥ y^{'} - y ∥ \leq \frac{15 η}{1 - β} .$

This completes the proof of the theorem. □

4. Application

In this context, we seek to utilize our iterative procedure (6) to demonstrate the existence of a unique solution for the following Caputo-type nonlinear fractional differential equations with boundary conditions:

(19) $\begin{matrix} \{\begin{matrix} {}^{c}D^{γ} u (y) + h (y, u (y)) = 0, 1 < γ < 2, \\ u (0) = u (1) = 0, 0 \leq y \leq 1, \end{matrix} \end{matrix}$

where

h : [0, 1] \times R \to R

is a continuous function, and

{}^{c}D^{γ}

represents the Caputo-type fractional derivative of order

γ

Consider $C = C [0, 1]$ , a Banach space equipped with the standard supremum norm, which comprises continuous real functions defined over the interval $[0, 1]$ . Within this framework, we investigate the following integral equation:

(20) $u (y) = \int_{0}^{1} g (y, z) h (y, u (y)) d z,$

where

g (y, z)

is the Green’s function, defined as follows:

(21) $g (y, z) = \{\begin{matrix} \frac{y {(1 - z)}^{γ - 1} - {(y - z)}^{γ - 1}}{Γ (γ)}, & 0 \leq z \leq y \leq 1, \\ \frac{y {(1 - z)}^{γ - 1}}{Γ (γ)}, & 0 \leq y \leq z \leq 1 . \end{matrix}$

Theorem 6.

Let $C = C [0, 1]$ denote a Banach space, and define the mapping $S : C \to C$ by

(22) $S u (y) = \int_{0}^{1} g (y, z) h (y, u (y)) d z,$

for every $u \in C$ and $y \in [0, 1]$ . Assume that the continuous function $h : [0, 1] \times R \to R$ satisfies the condition

$| h (y, u_{1}) - h (y, u_{2}) | \leq β | u_{1} - u_{2} |,$

for all $u_{1}, u_{2} \in C$ and $y \in [0, 1]$ , with $0 < β < 1$ . Under these conditions, the problem (19) has a unique solution. Moreover, the iterative sequence ${u_{n}}$ defined in (6), converges to the solution of problem (19).

Proof.

A function, $u \in C$ , is considered a solution to the Caputo-type nonlinear fractional differential equation if and only if it satisfies the corresponding integral Equation (20).

To show this, let $u_{1}, u_{2} \in C$ for $y \in [0, 1]$ . We can express the difference as follows:

$| S u_{1} (y) - S u_{2} (y) | = |\int_{0}^{1} g (y, z) h (y, u_{1} (y)) d z - \int_{0}^{1} g (y, z) h (y, u_{2} (y)) d z| .$

This simplifies to

$| S u_{1} (y) - S u_{2} (y) | \leq \int_{0}^{1} g (y, z) | h (y, u_{1} (y)) - h (y, u_{2} (y)) | d z .$

By applying the Lipschitz condition for h, we obtain

$| S u_{1} (y) - S u_{2} (y) | \leq \int_{0}^{1} g (y, z) β | u_{1} (y) - u_{2} (y) | d z .$

Using the supremum norm in $C = C [0, 1]$ , this leads to

$| S u_{1} (y) - S u_{2} (y) | \leq β ∥ u_{1} - u_{2} ∥ .$

Thus, in the Banach space $C [0, 1]$ , the operator S acts as a contraction for $β \in (0, 1)$ . By applying the Banach contraction principle, we conclude that problem (19) admits a unique solution. Furthermore, according to Theorem 2, the iterative sequence ${u_{n}}$ generated via (6) converges to the fixed point of S, ensuring that ${u_{n}}$ converges to a unique solution of problem (19). This confirms the existence and approximation of a unique solution to the Caputo-type nonlinear fractional differential Equation (19). □

We now present the following example to support the above result.

Example 3

([13]). Consider the fractional differential equation:

(23) $\{\begin{matrix} {}^{c}D^{γ} u (y) + y^{2} = 0, & 0 \leq y \leq 1, γ = 1.5, \\ u (0) = u (1) = 0 . \end{matrix}$

The exact solution of the problem (23) is expressed as follows:

(24) $u (y) = \frac{1}{Γ (γ)} \int_{0}^{y} (y {(1 - z)}^{γ - 1} - {(y - z)}^{γ - 1}) z^{2} d z + \frac{y}{Γ (γ)} \int_{y}^{1} {(1 - z)}^{γ - 1} z^{2} d z .$

We define the operator $S : C [0, 1] \to C [0, 1]$ as follows:

(25) $S u (y) = \frac{1}{Γ (γ)} \int_{0}^{y} (y {(1 - z)}^{γ - 1} - {(y - z)}^{γ - 1}) z^{2} d z + \frac{y}{Γ (γ)} \int_{y}^{1} {(1 - z)}^{γ - 1} z^{2} d z .$

Or, equivalently,

(26) $S u (y) = \int_{0}^{1} g (y, z) z^{2} d z,$

where

g (y, z)

is the Green’s function defined in (21). In view of the lines of the proof of Theorem 6, one can easily prove that S is a contraction mapping with contraction constant

β \in (0, 1)

. Thus, according to Theorem 2, the iterative sequence

{u_{n}}

generated via (6) converges to the solution of problem (23).

Now, using the initial guess $u_{0} (y) = y (1 - y), y \in [0, 1]$ and the control sequences $α_{n} = 0.95, β_{n} = 0.85, γ_{n} = 0.55, n = 0, 1, 2, 3, \dots$ , we observed that the iteration method (6) successfully approximates the exact solution of problem (23) for the operator given in (25). The results are presented in Table 3.

Innovation and Relevance of the Proposed Iteration Method

Innovation: This study introduces a new iterative process for approximating fixed points of contraction mappings in Banach spaces. The proposed iterative method is supported by a rigorous stability and convergence analysis, which includes new theorems that demonstrate faster convergence and better numerical stability compared to existing methods. The inclusion of a data dependence theorem provides a deeper insight into the sensitivity of solutions concerning initial conditions or parameter changes. While most iterative methods focus on integer-order differential equations, the proposed method is specifically tailored to nonlinear fractional differential equations of the Caputo type, filling a critical gap in the literature.

Relevance: Fractional calculus is increasingly used in fields like signal processing, viscoelasticity, and control systems due to its ability to model memory and hereditary properties. The proposed method addresses these emerging needs by providing a robust tool for solving fractional differential equations. The method demonstrates improved computational efficiency, making it suitable for large-scale and real-time systems with which traditional methods may fall short due to slower convergence rates. Graphical comparisons and numerical results in the study highlight that the proposed method outperforms existing iterative methods, such as the Agarwal, Picard-S, Ali and Ali, and $D^{* *}$ iteration methods, in terms of accuracy and efficiency.

5. Conclusions

In this paper, we have introduced a novel iteration method and applied it to solve a class of Caputo-type nonlinear fractional differential equations. The existence and uniqueness of solutions were established through the associated integral equation and the Banach contraction principle. We demonstrated that the proposed iteration scheme converges effectively to the fixed point of the contraction mapping, S. Our approach ensures faster convergence compared to other known methods, which we validated through both theoretical analysis and supporting numerical examples. Additionally, the study confirms that the data dependence of the solution is well structured under specific conditions. The iteration method developed in this work opens up new possibilities for solving more complex nonlinear problems in Banach spaces. Future research may explore the applicability of this technique to higher-order fractional equations and other operator classes, further expanding its utility across fields such as fluid dynamics, biology, and robotics. Can the proposed method in this paper be applied to analyze the Caputo fractional order systems in [28]?

Author Contributions

Conceptualization, N.H.E.E., E.A. and F.A.K.; methodology, A.A. and E.A.; software, A.A. and F.A.K.; formal analysis, N.H.E.E. and M.S.A.; investigation, A.A.; resources, N.H.E.E. and M.S.A.; data curation, E.A.; writing—original draft, F.A.K.; writing—review and editing, D.F. and M.S.A.; supervision, F.A.K.; project administration, D.F.; funding acquisition, D.F. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The original contributions presented in the study are included in the article. Further inquiries can be directed to the corresponding authors.

Acknowledgments

The authors express their gratitude to the anonymous reviewers for their valuable comments and suggestions, which significantly enhanced the quality of the paper.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

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.

Figures and Tables

Figure 1. Graphical comparison of the convergence of different iterative methods for Example 1.

Figure 2. Three-dimensional view of convergence of different iterative methods for Example 1.

Figure 3. Graphical comparison of the convergence of different iterative methods for Example 2.

Figure 4. Three-dimensional view of convergence of different iterative methods for Example 2.

Table 1

Numerical comparison of different iteration methods for Example 1.

Iterations	S (1)	Picard-S (2)	Tahkur (3)	F (4)	D** (5)	New (6)
1	5.000000	5.000000	5.000000	5.000000	5.000000	5.000000
2	5.220000	5.376000	5.376000	5.528960	5.631360	5.716570
3	5.396376	5.613680	5.613680	5.780265	5.866371	5.920455
4	5.534485	5.761656	5.761656	5.897881	5.951894	5.977748
5	5.641691	5.853237	5.853237	5.952636	5.982747	5.993785
6	5.724550	5.909741	5.909741	5.978059	5.993827	5.998266
7	5.788430	5.944538	5.944538	5.989845	5.997795	5.999516
8	5.837598	5.965942	5.965942	5.995303	5.999213	5.999865
9	5.875400	5.979096	5.979096	5.997828	5.999720	5.999962
10	5.904440	5.987174	5.987174	5.998996	5.999900	5.999990
11	5.926734	5.992133	5.992133	5.999536	5.999964	5.999997
12	5.943841	5.995176	5.995176	5.999786	5.999987	5.999999
13	5.956963	5.997043	5.997043	5.999901	5.999995	6
14	5.967025	5.998187	5.998187	5.999954	5.999998	6
15	5.974739	5.998889	5.998889	5.999979	5.999999	6
16	5.980650	5.999319	5.999319	5.999990	6	6
17	5.985181	5.999583	5.999583	5.999996	6	6
18	5.988651	5.999744	5.999744	5.999998	6	6
19	5.991310	5.999843	5.999843	5.999999	6	6
20	5.993346	5.999904	5.999904	6	6	6
⋮	⋮	⋮	⋮	⋮	⋮	⋮
31	5.999649	6	6	6	6	6
⋮	⋮	⋮	⋮	⋮	⋮	⋮
56	6	6	6	6	6	6

Table 2

Numerically comparison of different iteration methods for Example 2.

Iterations	S (1)	Picard-S (2)	Tahkur (3)	F (4)	D** (5)	New (6)
1	7.000000	7.000000	7.000000	7.000000	7.000000	7.000000
2	5.979938	5.178788	5.178230	4.430961	3.872568	3.526189
3	5.089974	3.918512	3.917441	3.270875	3.067185	3.017741
4	4.362783	3.289318	3.288519	3.033445	3.003680	3.000473
5	3.818576	3.074904	3.074616	3.003750	3.000196	3.000013
6	3.453317	3.017977	3.017901	3.000416	3.000010	3
7	3.234627	3.004225	3.004206	3.000046	3.000001	3
8	3.115886	3.000988	3.000983	3.000005	3	3
9	3.055674	3.000231	3.000230	3.000001	3	3
10	3.026355	3.000054	3.000054	3	3	3
11	3.012385	3.000013	3.000013	3	3	3
12	3.005799	3.000003	3.000003	3	3	3
13	3.002711	3.000001	3.000001	3	3	3
14	3.001266	3	3	3	3	3
15	3.000591	3	3	3	3	3
16	3.000276	3	3	3	3	3
17	3.000129	3	3	3	3	3
18	3.000060	3	3	3	3	3
19	3.000028	3	3	3	3	3
20	3.000013	3	3	3	3	3
21	3.000006	3	3	3	3	3
22	3.000003	3	3	3	3	3
23	3.000001	3	3	3	3	3
24	3	3	3	3	3	3

Table 3

Comparison between exact and approximate solutions.

S. No.	y	$u (y)$	$u_{4}$	$u_{8}$	$u_{15}$	$u_{50}$
1	0	0.00000000	0.00000000	0.00000000	0.00000000	0.00000000
2	0.1	0.01713998	0.01713998	0.01713998	0.01713998	0.01713998
3	0.2	0.03377353	0.03377353	0.03377353	0.03377353	0.03377353
4	0.3	0.04904026	0.04904026	0.04904026	0.04904026	0.04904026
5	0.4	0.06181761	0.06181761	0.06181761	0.06181761	0.06181761
6	0.5	0.07077394	0.07077394	0.07077394	0.07077394	0.07077394
7	0.6	0.07439773	0.07439773	0.07439773	0.07439773	0.07439773
8	0.7	0.07101707	0.07101707	0.07101707	0.07101707	0.07101707
9	0.8	0.05881384	0.05881384	0.05881384	0.05881384	0.05881384
10	0.9	0.03583472	0.03583472	0.03583472	0.03583472	0.03583472
11	1.0	0.00000000	0.00000000	0.00000000	0.00000000	0.00000000

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A Novel and Efficient Iterative Approach to Approximating Solutions of Fractional Differential Equations

Content area

Abstract

Full text

2. Preliminaries

3. Main Results

3.1. Numerical Examples

3.2. Data Dependence

4. Application