Existence and uniqueness of solutions for fuzzy fractional integro-differential equations with boundary conditions

Abstract

The analytical behavior of fractional-order differential equations under uncertainty is often difficult to investigate. To address this challenge, this study considers Caputo-type fuzzy fractional Volterra integro-differential equations (FFVIDEs) with boundary conditions. An iterative numerical approach based on the Adomian decomposition method (ADM) is proposed to obtain approximate fuzzy solutions. The existence and uniqueness of the solution are established using Banach’s fixed point theorem. Numerical simulations are carried out in MATLAB to illustrate the symmetry between the lower and upper $\rho$-cut representations of the fuzzy solutions. The graphical results demonstrate the accuracy and efficiency of the proposed method in handling uncertainty in fractional systems.

Introduction

Aside from pure mathematics, there is a strong desire to learn more about the subject applications in fluid flow, biology, fractal theory, control theory, electrochemistry, and viscoelasticity stimulate the study of fractional calculus (see^1,2,3). Fixed point theorems are commonly used to investigate the existence and uniqueness of solutions in the domain of fractional calculus (see^4,5,6). The solution properties were discovered in^7,8, while several numerical approaches, including quadrature, product integration, variational iteration, and fuzzy transforms, were discovered in (see^5,9). The use of fuzzy sets is well-suited for models with uncertainty, leading to a focus on fuzzy fractional calculus for addressing these types of problems. The study of fuzzy fractional differential equations (FFDEs) is a new area of fuzzy mathematics that is expanding steadily. To investigate a novel sort of dynamical system, the notion of fractional order fuzzy differential equations (FDEs) was recently developed¹⁰. The authors Agarwal, Lakshmikantham, and Nieto¹¹ originated the fractional case generalization of the H-differentiability. In¹² authors discovered the existence and uniqueness of results for FFDEs. Several studies and solutions of order $0<\eta \le 1$, (see^13,14) FFDEs have been published in recent years. However, uncertainty can take a heavy toll on real-world issues at any time. Incomplete data, measurement errors, or identifying initial conditions can all lead to uncertainty.

The area of fuzzy fractional integro-differential equations (FFIDEs) has past few years piqued the interest of many researchers since it is recognized as a powerful tool for presenting imprecise parameters and handling their dynamical systems in real fuzzy conditions. Analyzing the majority of FFIDE’s problems is difficult. Some real-world scenarios deal with this, including the golden mean, quantum optics, gravity, engineering phenomena, practical systems, and medical science. The existence of solutions for various fuzzy fractional integral equations is important, particularly in the context of developing efficient numerical methods for approximating these solutions¹⁵. By employing the generalized Hukuhara derivative¹⁶, the notion of fractional Caputo H-differentiability was utilized to study fuzzy fractional differential¹⁷ and integro-differential¹⁸ equations. In¹⁹, weakly contractive mappings were employed to establish the existence of a unique solution for FFVIDEs without Liptchitz conditions. In²⁰ the existence and uniqueness of the solution for FFVIDEs in Caputo’s sense were proven, as well as the usage of Adomian decomposition to approximate the solution. In^21,22,23, stability properties for FFVIDEs were explored, including the Ulam-Hyers stability. The variational iteration method was used in²⁴ to get numerical solutions for FFDEs. The fuzzy Caputo derivative was implemented in²⁵ using the modified fractional Euler (MFE) technique.

The investigation for interval-valued functions begun in²⁶ continued in²⁷, which describes the MFE and Adams generalization methods. Since the initial value problem (IVP) for the FFDEs and the corresponding FFVIDEs are not equivalent in general, a desirable condition is presented in²⁸ to make this equivalence valid. The fractional Euler method is used²⁹ to prove the solution of FFDEs under Caputo differentiability using generalized Taylor’s expansion as fuzzy-valued functions.

Very recently, Alexandru Mihai Bica et al.,³⁰ studied the numerical method for fractional fuzzy integral equations, and Ehsan Ul Haq et al.,³¹ investigated a reliable method for a system of fuzzy solutions. The method of Adomian decomposition has been used in this seller to provide approximate results for the system of fuzzy fractional initial value problems. A new fractional-order method based on Jacobi polynomials was introduced by Bidari et al.³² for finding the solution to FFIDEs. They developed a set of equations to represent the suggested problem using operational matrices. They proved that under certain conditions, the proposed method converges. The fuzzy product integration approach was utilized by Bica et al.³³ to develop an iterative method for solving fuzzy fractional Volterra integral equations. The fuzzy Volterra integral equation with weak singularity was investigated in³⁴ using a piece-wise spline collocation approach. Additionally, they have shown that the solution exists and converges to content validity. FFIDE of the Volterra and Fredholm types can be solved using the Chebyshev spectral method according to a recent proposal by Kumar et al.,³⁵.

Caputo-type fuzzy fractional Volterra integro-differential equations (FFVIDEs) emerge naturally in real-life situations where memory, heredity, and uncertainty play a significant role in the system’s behavior. These equations combine three key concepts-fractional calculus, fuzzy logic, and integro-differential operators-making them suitable for modeling complex real-world processes that cannot be accurately described using classical differential equations.

In real life, many physical, biological, and engineering systems exhibit memory effects (dependence on past states) and imprecise information due to environmental fluctuations or measurement errors. The fractional derivative in the Caputo sense captures memory and hereditary properties, while fuzziness handles uncertainty or vagueness in system parameters and initial conditions. For instance: In engineering, they model viscoelastic materials where stress depends not only on the current strain but also on the entire deformation history. In biology, fuzzy fractional models describe population dynamics or enzyme kinetics under uncertain environmental influences. In electrical circuits, they represent RLC circuits with fractional elements and uncertain parameters. In control theory, they help design controllers for systems with uncertain or partially known dynamics.

The usage of fuzzy fractional calculus, also known as fuzzy fractional integro-differential equations, has grown significantly in recent years due to their extensive use in simulating a variety of physical industrial processes, including heat and mass transport, bio-mechanics, electromagnetic fields, etc. For the approximate solutions of fractional fuzzy Volterra integro-differential equations, several researchers have developed some numerical approaches using different types of fractional derivatives^{36,37,38,39,40,41,42,43,44,45,46}.

Several authors have contributed significantly to recent developments in fuzzy fractional calculus. For example,⁴⁷ investigated a novel method for solving fractional Abel k-integral equations and linear FDEs in a fuzzy environment. This study shows the effectiveness of applying fractional calculus with fuzzy systems to improve solution accuracy. Similarly,⁴⁸ examined fuzzy Langevin fractional delay differential equations using the granular derivative, demonstrating the possibility for combining both mathematical frameworks. Furthermore,⁴⁹ investigated fuzzy fractional delay integro-differential equations using the extended Atangana-Baleanu fractional derivative, giving a unique approach to solving these complicated equations. Additionally,⁵⁰ investigated the existence and stability of solutions for fuzzy neutral fractional integro-differential equations involving the Caputo fractional generalized Hukuhara derivative.

The existence of the FFVIDEs has been further investigated using Schauder’s fixed point theorem and the uniqueness has been studied using the Banach contraction principle in⁵¹. In addition, recently the Adomian decomposition method (ADM) with initial condition⁵² and ADM with boundary conditions^38,39, Shehu Adomian decomposition method⁵³, successive approximation method⁵⁴, monotone iterative technique⁵⁵ two-dimensional legendre wavelet method⁵⁶ have all been used to examine the solutions of FFVIDEs.

Reference	Method Used	Type of Problem Studied	Remarks / Contributions
1. Savla, S. et al. (2024)	Shehu Adomian decomposition method	Solving linear and nonlinear fuzzy fractional Volterra -Fredholm integro-differential equations (IVP)	Provided approximate analytical solutions; Via Adomian Polynimoals.
2. Vu, H. et al. (2019)	Successive approximation method	Stability for initial value problems of fuzzy Volterra integro-differential equation with fractional order derivative	Discussed existence and uniqueness; limited to simple approximation method.
3. Shabestari, M.R.M. et al. (2018)	Two-dimensional Legendre wavelet method	Numerical solution of fuzzy fractional integro-differential equation via (IVP)	Focused on deterministic systems and investigated existence and uniqueness under Hukuhara differentiability
Proposed Work	Adomian Decomposition Method	Fuzzy fractional Volterra integro -differential equations with boundary conditions (BVP)	ADM improves convergence and handles uncertainty, efficiently, and requires no discretization or linearization.

In this work, we try to solve FFVIDEs under Caputo derivative using the ADM. In order to solve FFVIDEs with our proposed approach. To the best of our knowledge, this is the first time in the literature that numerical solution of the FFVIDEs are examined under Caputo derivative employing ADM with the fractional order $0 <\eta \le 1$ and fuzzy initial conditions are investigated. The goal of this work is to look into it.

Motivation

Numerous authors have focused on exploring numerical solutions for FFVIDEs, as many of these equations are difficult to solve analytically. Therefore, researchers are starting to use several numerical approaches. Some examples of these numerical methods include the Laplace transform method, variational iteration method, semi-analytical method, and Adomian decomposition method (ADM) to obtain better approximation solutions for FFIDEs. All of these studies encouraged us to develop a numerical approach to solving FFVIDE. Additionally, our analysis of the literature revealed that numerous investigations of numerical solutions of FFIDEs were solved only with initial conditions. Hamoud⁵⁷ proposed the ADM to investigate the fractional Volterra integro-differential equation’s solution without a fuzzy concept. Also in²⁰ N. Ahmad et al. only addressed the FFVIDEs’ initial value problem. As a result, this study inspires us to adapt research for FFVIDE with boundary conditions. Therefore, FFVIDEs must be developed with boundary conditions.

Novelty

The following is a summary of the work’s primary contribution:

• 1. The authors provide a new framework for the Adomian decomposition approach to solve FFVIDEs with generalized boundary conditions.

• 2. Comparing to the existing studies in (see^20,57) we present boundary conditions for the FFVIDEs investigated.

• 3. To demonstrate the existence of the unique solution using the Banach fixed point theory to the considered equation. To provide lower/uppercut solutions for FFVIDEs with the help of the Adomian decomposition technique.

• 4. To establish the effectiveness and reliability of the ADM, we provided a sufficient number of examples to verify the theory and proposed an algorithm for solving these types of equations.

Overview of the paper’s sections

The paper is organized as follows: The introduction is included in the first part. Additionally, the motivation, novelty, and structure of the article are described. Section 2 presents some basic ideas about fuzzy numbers and the basics of fractional calculus, whereas Section 3 is focused on establishing the existence and uniqueness of the FFBVPs under consideration. Section 4 presents the methodology of Adomian decomposition. In Section 5, we analyze the suggested method’s accuracy using a numerical example and Section 6 concludes with a brief overview of this investigation.

Preliminaries

The definitions of fuzzy calculus and its key notions are provided in this section.

Definition 1

²⁰ A fuzzy set A is said to be a fuzzy number, if the following conditions are satisfied.

• 1. A is normal; there exist, $y_{0} \in {\mathcal {R}}$ such that $A\left( y_{0}\right) =1$,

• 2. $A(\lambda {y_1}+(1-\lambda ) {y_2}) \ge \min (A(y_1), A(y_2))$ holds, for all, ${y_1}, {y_2} \in {\mathcal {R}}$ and $0 \le \lambda \le 1$ i.e, A is convex,

• 3. A is upper semi-continuous,

• 4. A is compactly supported that is $cl \{y \in {\mathcal {R}} \mid A(y)>0\} \subseteq {\mathcal {R}}$ is compact.

Definition 2

⁵⁷ A fuzzy number A is $\xi$-level set characterised as

$$[A]_{\xi } = \{y\in R: A(y)\ge \xi \},$$

where $\xi \in (0,1]$ and $y\in R$ For a fuzzy number A, its $\xi$-cuts are closed intervals in R and we denoted by $[A]_{\xi }=[\underline{\xi }, \overline{\xi }].$

Definition 3

²⁹ The RL fractional integral of order $\eta$ of fuzzy function ${\tilde{f}}(y,\xi )$, based on its $\xi$-level illustrations:

$\left[ I^{\eta } {\tilde{f}} (y,\xi )\right] =\left[ I^{\eta } {\underline{f}}(y,\xi ), I^{\eta } {\overline{f}}(y,\xi )\right]$ where,

$$\begin{aligned} I^{\eta } {\underline{f}}(y,\xi )=\frac{1}{\Gamma (\eta )} \int _0^{y}(y-t)^{\eta -1} {\underline{f}}(t,\xi ){dt},\\ I^{\eta } {\overline{f}}(y,\xi )=\frac{1}{\Gamma (\eta )} \int _0^{y}(y-t)^{\eta -1} {\overline{f}}(t,\xi ){dt}. \end{aligned}$$

Note: In this paper, we consider (i) differentiable type to find the solution for FFVIDEs (from the theorem 2.1 in³⁸). However, a type (ii) differentiable solution is the same as type (i) differentiable.

Definition 4

³⁸ The Caputo fractional derivative of order $\eta$ of fuzzy function ${\tilde{f}}(y,\xi )$, it can be stated as follows, based on its $\xi$-level illustrations:

$\left[ D^{\eta } {\tilde{f}} (y,\xi )\right] =\left[ D^{\eta } {\underline{f}}(y,\xi ), D^{\eta } {\overline{f}}(y,\xi )\right]$ where,

$$\begin{aligned} D^{\eta }[{\underline{f}}(y,\xi )]= {\left\{ \begin{array}{ll} & \frac{1}{\Gamma (r-\eta )}\int \limits _0^y(y-t)^{r-\eta -1}{\underline{f}}^{r}(y,\xi )dt, ~~ r-1< \eta < r\\ & \frac{d^r}{dt^r} {\underline{f}}(y,\xi ), ~~ \eta =r, r\in N, \end{array}\right. } \end{aligned}$$

$$\begin{aligned} D^{\eta }[{\overline{f}}(y,\xi )]= {\left\{ \begin{array}{ll} & \frac{1}{\Gamma (r-\eta )}\int \limits _0^y(y-t)^{r-\eta -1}{\overline{f}}^{r}(y,\xi )dt, ~~ r-1<\eta < r \\ & \frac{d^r}{dt^{r}} {\overline{f}}(y,\xi ), ~~ \eta =r, r\in N. \end{array}\right. } \end{aligned}$$

Problem formulation

In this study, we use the fuzzy idea to extend the concept to FVIDEs across the Caputo-type of order $\eta \in (0,1]$.

$$\begin{aligned} D^{\eta }[\tilde{\vartheta } (y,\xi )] = {\left( {\tilde{f}}(y,\xi )+\int _{0}^y \Upsilon _1 (y,\varrho ) \Lambda _{1} (\tilde{\vartheta }(\varrho ,\xi ))d\varrho +\int _{0}^T \Upsilon _{2} (y,\varrho ) \Lambda _{2} (\tilde{\vartheta }(\varrho ,\xi ))d\varrho \right) } \end{aligned}$$

(1)

with the boundary condition

$$\begin{aligned} p_{1}\tilde{\vartheta }(0,\xi )+p_{2}\tilde{\vartheta }(T,\xi )=\tilde{p_3}, ~~ p_{1},p_{2} \in {\mathcal {R}},~ p_{1}+p_{2} \ne 0, ~ \tilde{p_3} \in {\mathbb {R}}_f. \end{aligned}$$

(2)

Where $\tilde{\vartheta }(y,\xi )$ is a fuzzy function, $D^{\eta }(.), \eta \in (0,1]$, is the Caputo fractional derivative, ${\tilde{f}}: X\rightarrow {\mathcal {R}}_f, \Upsilon _i:X \times X \rightarrow {\mathcal {R}}$ for $i=1,2$ are continuous functions, $\Lambda _i: {\mathcal {R}} \rightarrow {\mathcal {R}}$ are Lipschitz continuous function, where $X= [0,1]$.

Main result

The focus of this section will be on the discussion of the existence and uniqueness of the solution to Eq. (1) with boundary condition (2), based on the hypothesis.

${\mathbb {H}}1:$ For any $\tilde{\vartheta }_{1}, \tilde{\vartheta }_{2} \in C (X,{\mathcal {R}}_f)$, there exists two constants $c_1,c_2 > 0$ such that,

$$|\Lambda _{1}(\tilde{\vartheta }_{1}(y,\xi ))-\Lambda _{1}(\tilde{\vartheta }_{2}(y,\xi ))|\le c_1 |\tilde{\vartheta }_{1} - \tilde{\vartheta }_{2}|,$$

$$|\Lambda _{2}(\tilde{\vartheta }_{1}(y,\xi ))-\Lambda _{2}(\tilde{\vartheta }_{2}(y,\xi ))|\le c_2 |\tilde{\vartheta }_{1} - \tilde{\vartheta }_{2}|.$$

${\mathbb {H}}2:$ For the set of all positive continuous function on $J = \{ (y,\varrho )\in {\mathcal {R}} \times {\mathcal {R}} : 0 \le \varrho \le y \le 1 \},$ there exists the functions ${\Upsilon _1}^*, {\Upsilon _{2}}^*$ such that

$${\Upsilon _1}^* = \sup _{y\in [0,1]} \int _{0}^y |\Upsilon _1(y,\varrho )|d\varrho <\infty$$

and

$$\begin{aligned} {\Upsilon _{2}}^* = \sup _{y\in [0,1]} \int _{0}^1 |\Upsilon _{2}(y,\varrho )|d\varrho <\infty . \end{aligned}$$

${\mathbb {H}}3:$ The function ${\tilde{f}}:[0,1]\rightarrow {\mathcal {R}}_f$ is continuous, there exists a constant $L > 0$, such that $L=\sup \{|{\tilde{f}}(y,\xi )|:0\le y \le T\}.$

Theorems and Lemma

Lemma 1

If a function $\tilde{\vartheta }(y,\xi ) \in C[0, T]$ satisfies $({\mathbb {H}}1) - ({\mathbb {H}}3)$ in [0, T], then the problems (1)-(2) are similar to obtaining a continuous solution of the FFVIDEs.

$$\begin{aligned} {\tilde{\vartheta } (y,\xi )} =&\int _{0}^y \frac{(y-t)^{\eta -1}}{\Gamma (\eta )}\Big ( {\tilde{f}}(t,\xi )+\int _{0}^t \Upsilon _1 (t,\varrho ) \Lambda _{1}(\tilde{\vartheta }(\varrho ,\xi ))d\varrho \nonumber \\ &+\int _{0}^T \Upsilon _{2} (t,\varrho ) \Lambda _{2}(\tilde{\vartheta }(\varrho ,\xi ))d\varrho \Big )dt \nonumber \\ &- {\frac{1}{p_{1}+p_{2}}}\Big [ \int _{0}^T\frac{p_{2}(T-t)^{\eta -1}}{\Gamma (\eta )} \Big ( {\tilde{f}}(t,\xi )+\int _{0}^t \Upsilon _1 (t,\varrho ) \Lambda _{1} (\tilde{\vartheta }(\varrho ,\xi ))d\varrho \nonumber \\&+\int _{0}^T \Upsilon _{2} (t,\varrho ) \Lambda _{2} (\tilde{\vartheta }(\varrho ,\xi ))d\varrho \Big )dt-\tilde{p_3} \Big ]. \end{aligned}$$

(3)

The above equation is equivalent to

$$\begin{aligned} \begin{aligned} {\tilde{\vartheta } (y,\xi )} =&{\tilde{h}}(y,\xi )- {\frac{1}{p_{1}+p_{2}}} \int _{0}^T\frac{p_{2}(T-t)^{\eta -1}}{\Gamma (\eta )} \Big ( \int _{0}^t \Upsilon _1 (t,\varrho ) \Lambda _{1} (\tilde{\vartheta }(\varrho ,\xi ))d\varrho \\ &+\int _{0}^T \Upsilon _{2} (t,\varrho ) \Lambda _{2} (\tilde{\vartheta }(\varrho ,\xi ))d\varrho \Big )dt \\ &+ \int _{0}^y \frac{(y-t)^{\eta -1}}{\Gamma (\eta )} {\left( \int _{0}^t \Upsilon _1 (t,\varrho ) \Lambda _{1} (\tilde{\vartheta }(\varrho ,\xi ))d\varrho +\int _{0}^T \Upsilon _{2} (t,\varrho ) \Lambda _{2} (\tilde{\vartheta }(\varrho ,\xi ))d\varrho \right) dt } \end{aligned} \end{aligned}$$

(4)

where,

$$\begin{aligned} \begin{aligned} {\tilde{h}}(y,\xi ) =&{\frac{\tilde{p_3}}{p_{1}+p_{2}}} - {\frac{1}{p_{1}+p_{2}}} \int _{0}^T\frac{{p_{2}}(T-t)^{\eta -1}}{\Gamma (\eta )} ({\tilde{f}}(t,\xi ) )dt+ \int _{0}^y \frac{(y-t)^{\eta -1}}{\Gamma (\eta )} {\left( {\tilde{f}}(t,\xi )\right) dt }. \end{aligned} \end{aligned}$$

(5)

Theorem 1

(Existence theorem) Let the hypotheses $({\mathbb {H}}1)$ and $({\mathbb {H}}2)$ be satisfied and $\Lambda _1(\tilde{\vartheta }(y,\xi )\le \mu _{1}, \Lambda _2(\tilde{\vartheta }(y,\xi )\le \mu _{2}$ for all $y\in [0, T]$ and for all $\tilde{\vartheta }(y,\xi ) \in {\mathcal {R}}_f$, Under this hypothesis our considered equation has at least one solution in [0, T].

Proof

Let the operator $\Upsilon : C(X, {\mathcal {R}}_f ) \rightarrow C(X, {\mathcal {R}}_f )$ characterized by

$$\begin{aligned} (\Upsilon {\tilde{\vartheta }) (y,\xi )} =&\int _{0}^y \frac{(y-t)^{\eta -1}}{\Gamma (\eta )}\Big ( {\tilde{f}}(t,\xi )+\int _{0}^t \Upsilon _1 (t,\varrho ) \Lambda _{1} (\tilde{\vartheta }(\varrho ,\xi ))d\varrho \nonumber \\ \nonumber&+\int _{0}^T \Upsilon _{2} (t,\varrho ) \Lambda _{2} (\tilde{\vartheta }(\varrho ,\xi ))d\varrho \Big )dt \\ \nonumber&- {\frac{1}{p_{1}+p_{2}}}\Big [\int _{0}^T\frac{p_{2}(T-t)^{\eta -1}}{\Gamma (\eta )}\Big ({\tilde{f}}(t,\xi )+\int _{0}^t \Upsilon _1 (t,\varrho ) \Lambda _{1}(\tilde{\vartheta }(\varrho ,\xi ))d\varrho \\ &+\int _{0}^T \Upsilon _{2} (t,\varrho ) \Lambda _{2} (\tilde{\vartheta }(\varrho ,\xi ))d\varrho \Big )dt-\tilde{p_3} \Big ],\ for\ y\in [0,1]. \end{aligned}$$

(6)

We begin by showing that the operator $\Upsilon$ is completely continuous.

(1) To prove $\Upsilon$ is continuous:  Let $\tilde{\vartheta }_n$ be a sequence converges to $\tilde{\vartheta }$ in $C(X,{\mathcal {R}}_f)$, for each $\tilde{\vartheta }_n,\tilde{\vartheta } \in C(X,{\mathcal {R}}_f)$ and for any $y\in X$, we have

$$\begin{aligned} |\Upsilon {\tilde{\vartheta _n}(y,\xi )} - \Upsilon {\tilde{\vartheta } (y,\xi )}|&\le \frac{1}{\Gamma (\eta )}\int _{0}^y (y-t)^{\eta -1}\Big (\int _{0}^{t}|\Upsilon _1(t,\varrho )||\Lambda _{1}(\tilde{\vartheta _n}(\varrho ,\xi ))-\Lambda _{1}(\tilde{\vartheta }(\varrho ,\xi ))|d\varrho \\ &\quad + \int _{0}^{T}|\Upsilon _{2}(t,\varrho )||\Lambda _{2}(\tilde{\vartheta _n}(\varrho ,\xi ))-\Lambda _{2}(\tilde{\vartheta }(\varrho ,\xi ))|d\varrho \Big )dt \\&\quad + \frac{|p_{2}|}{|p_{1}+p_{2}|\Gamma (\eta )} \int _{0}^T (T-t)^{\eta -1}\Big (\int _{0}^{t}|\Upsilon _1(t,\varrho )||\Lambda _{1}(\tilde{\vartheta _n}(\varrho ,\xi ))\\ &\quad -\Lambda _{1}(\tilde{\vartheta }(\varrho ,\xi ))|d\varrho + \int _{0}^{T}|\Upsilon _{2}(t,\varrho )||\Lambda _{2}(\tilde{\vartheta _n}(\varrho ,\xi ))-\Lambda _{2}(\tilde{\vartheta }(\varrho ,\xi ))|d\varrho \Big ) dt. \end{aligned}$$

Taking supremum on both sides,

$$\begin{aligned} \Vert \Upsilon {\tilde{\vartheta _n}(y,\xi )} - \Upsilon {\tilde{\vartheta } (y,\xi )}\Vert&\le \frac{1}{\Gamma (\eta )}\Big ({\Upsilon _{1}}^{*}\Vert \Lambda _{1}(\tilde{\vartheta _n}(\varrho ,\xi ))-\Lambda _{1}(\tilde{\vartheta }(\varrho ,\xi ))\Vert \\ &\quad + {\Upsilon _{2}}^{*}\Vert \Lambda _{2}(\tilde{\vartheta _n}(\varrho ,\xi ))-\Lambda _{2}(\tilde{\vartheta }(\varrho ,\xi ))\Vert \Big ) \int _{0}^y (y-t)^{\eta -1}dt \\&\quad + \frac{|p_{2}|}{|p_{1}+p_{2}|\Gamma (\eta )} \Big ({\Upsilon _{1}}^{*}\Vert \Lambda _{1}(\tilde{\vartheta _n}(\varrho ,\xi ))-\Lambda _{1}(\tilde{\vartheta }(\varrho ,\xi ))\Vert \\ &\quad + {\Upsilon _{2}}^{*}\Vert \Lambda _{2}(\tilde{\vartheta _n}(\varrho ,\xi ))-\Lambda _{2}(\tilde{\vartheta }(\varrho ,\xi ))\Vert \Big ) \int _{0}^T (T-t)^{\eta -1}dt \end{aligned}$$

Since, $\Lambda _{1}$ and $\Lambda _{2}$ are continuous, $\int _{0}^y (y-t)^{\eta -1}dt$ is bounded. As a result, we can deduce that $\Vert \Upsilon {\tilde{\vartheta }_n(y,\xi )} - \Upsilon {\tilde{\vartheta } (y,\xi )}\Vert _{\infty } \rightarrow 0$ as $n\rightarrow \infty$. Thus $\Upsilon$ is continuous.

(2) To prove $\Upsilon$ is bounded: $\Upsilon$ maps bounded sets into bounded sets in $C\left( X, {\mathcal {R}}_{f}\right) .$ We demonstrate that a positive constant $\lambda$ exists, for every $\rho > 0$ such that for any $\tilde{\vartheta } \in {\mathcal {B}}_{\rho } =\{ \tilde{\vartheta } \in C(X,{\mathcal {R}}_f): \Vert \tilde{\vartheta }\Vert _{\infty } \leqslant \eta \}$. One has $\Vert \tilde{\vartheta }\Vert _{\infty } \leqslant \lambda$. Let $\mu _{1}=\sup _{\tilde{\vartheta } \in X \times [0, \eta ]} \Lambda _{1}(\tilde{\vartheta }(y,\xi ))+1$ and $\mu _{2}=\sup _{\tilde{\vartheta } \in X \times [0, \eta ]} \Lambda _{2}(\tilde{\vartheta }(y,\xi ))+1 .$ Also for each $y \in X$ and for any $\tilde{\vartheta } \in {\mathcal {B}}_{r^{\prime }}$, we have

$$\begin{aligned} 0 \le |\Upsilon {\tilde{\vartheta }(y,\xi )|} =&\Big |\int _{0}^y \frac{(y-t)^{\eta -1}}{\Gamma (\eta )} \Big ( {\tilde{f}}(t,\xi )+\int _{0}^t \Upsilon _1 (t,\varrho ) \Lambda _{1} (\tilde{\vartheta }(\varrho ,\xi ))d\varrho \\ &+\int _{0}^T \Upsilon _{2} (t,\varrho ) \Lambda _{2}(\tilde{\vartheta }(\varrho ,\xi ))d\varrho \Big )dt \\ &- {\frac{1}{p_{1}+p_{2}}}\Big [\int _{0}^T\frac{p_{2}(T-t)^{\eta -1}}{\Gamma (\eta )}\Big ( {\tilde{f}}(t,\xi )+\int _{0}^t \Upsilon _1 (t,\varrho ) \Lambda _{1} (\tilde{\vartheta }(\varrho ,\xi ))d\varrho \\ &+\int _{0}^T \Upsilon _{2} (t,\varrho ) \Lambda _{2} (\tilde{\vartheta }(\varrho ,\xi ))d\varrho \Big )dt-\tilde{p_3} \Big ] \Big |,\ for\ y\in [0,T] \\ \le&\int _{0}^y \frac{(y-t)^{\eta -1}}{\Gamma (\eta )} \Big ( |{\tilde{f}}(t,\xi ) |+\int _{0}^t |\Upsilon _1 (t,\varrho )||\Lambda _{1} (\tilde{\vartheta }(\varrho ,\xi ))|d\varrho \\ &+\int _{0}^T |\Upsilon _{2} (t,\varrho )||\Lambda _{2} (\tilde{\vartheta }(\varrho ,\xi ))|d\varrho \Big )dt \\ &+ {\frac{1}{|p_{1}+p_{2}|}} \int _{0}^T\frac{|p_{2}|(T-t)^{\eta -1}}{\Gamma (\eta )} \Big ( |{\tilde{f}}(t,\xi )|+\int _{0}^t |\Upsilon _1 (t,\varrho )||\Lambda _{1} (\tilde{\vartheta }(\varrho ,\xi )|)d\varrho \\ &+\int _{0}^T |\Upsilon _{2} (t,\varrho )||\Lambda _{2} (\tilde{\vartheta }(\varrho ,\xi )|)d\varrho \Big )dt+\frac{|\tilde{p_3}|}{|p_{1}+p_{2}|}. \end{aligned}$$

Taking the supremum on both sides, we obtain

$$\begin{aligned} \Vert \Upsilon \tilde{\vartheta }(y,\xi )\Vert _{\infty }\le&\int _{0}^y\frac{(y-t)^{\eta -1}}{\Gamma (\eta )}\Big (L+[\Upsilon ^{*}_1(\mu _{1})+\Upsilon ^{*}_{2}(\mu _{2})]\Big ) dt\\ &+{\frac{1}{|p_{1}+p_{2}|}}\int _{0}^T\frac{|p_{2}|(T-t)^{\eta -1}}{\Gamma (\eta )} \Big (L+[\Upsilon ^{*}_1(\mu _{1})+\Upsilon ^{*}_{2}(\mu _{2})]\Big )dt+\frac{|\tilde{p_3}|}{|p_{1}+p_{2}|} \\ \le&\frac{T^{\eta }\Big (L+[\Upsilon ^{*}_1(\mu _{1})+\Upsilon ^{*}_{2}(\mu _{2})]\Big )}{\Gamma (\eta +1)}\left( 1+\frac{|{p_{2}}|}{|p_{1}+p_{2}|} \right) + \frac{|\tilde{p_3}|}{|p_{1}+p_{2}|} \\ \le&\frac{\Big (L+[\Upsilon ^{*}_1(\mu _{1})+\Upsilon ^{*}_{2}(\mu _{2})]\Big )}{\Gamma (\eta +1)}\left( 1+\frac{|{p_{2}}|}{|p_{1}+p_{2}|} \right) + \frac{|\tilde{p_3}|}{|p_{1}+p_{2}|} =\lambda . \end{aligned}$$

Thus, for every $\Vert \Upsilon \tilde{\vartheta }\Vert \le \lambda$ for every $\tilde{\vartheta }\subset {\mathcal {B}}_{r^{\prime }}$, which that $\Upsilon {\mathcal {B}}_{r^{\prime }}\subset {\mathcal {B}}_{\lambda }$.

(3) To prove $\Upsilon$ is equicontinuous:  $\Upsilon$ maps bounded sets into equicontinuous sets of $C\left( X, {\mathcal {R}}_{f}\right) .$ Let $y_{1}, y_{2} \in [0,1]$ with $y_{1}<y_{2}$. For $\tilde{\vartheta } \in {\mathcal {B}}_{\rho }$, (defined in last step ${\mathcal {B}}_{\rho }$) we have

$$\begin{aligned} |(\Upsilon \tilde{\vartheta })(y_{2},\xi )-(\Upsilon \tilde{\vartheta }) (y_{1},\xi )|&\le \Big |\int _{0}^{y_{2}} \frac{({y_{2}}-t)^{\eta -1}}{\Gamma (\eta )} \Big ( |{\tilde{f}}(t,\xi )|+\int _{0}^t |\Upsilon _1 (t,\varrho )||\Lambda _{1} (\tilde{\vartheta }(\varrho ,\xi ))|d\varrho \end{aligned}$$

$$\begin{aligned}&\quad +\int _{0}^T |\Upsilon _{2} (t,\varrho )||\Lambda _{2} (\tilde{\vartheta }(\varrho ,\xi ))|d\varrho \Big )dt - \int _{0}^{y_{1}} \frac{({y_{1}}-t)^{\eta -1}}{\Gamma (\eta )}\Big ( |{\tilde{f}}(t,\xi )|\\ &\quad +\int _{0}^t |\Upsilon _1 (t,\varrho )||\Lambda _{1} (\tilde{\vartheta }(\varrho ,\xi ))|d\varrho +\int _{0}^T |\Upsilon _{2} (t,\varrho )||\Lambda _{2} (\tilde{\vartheta }(\varrho ,\xi ))|d\varrho \Big )dt \Big |\\&\le \Big |\int _{0}^{y_{1}} \frac{({y_{2}}-t)^{\eta -1}-({y_{1}}-t)^{\eta -1}}{\Gamma (\eta )}\Big ( {\tilde{f}}(t,\xi )+\int _{0}^t \Upsilon _1 (t,\varrho ) \Lambda _{1} (\tilde{\vartheta }(\varrho ,\xi ))d\varrho \\ &\quad +\int _{0}^T \Upsilon _{2} (t,\varrho ) \Lambda _{2} (\tilde{\vartheta }(\varrho ,\xi ))d\varrho \Big )dt \Big |+ \Big |\int _{y_{1}}^{y_{2}} \frac{({y_{2}}-t)^{\eta -1}}{\Gamma (\eta )}\Big ( {\tilde{f}}(t,\xi ) \\ &\quad +\int _{0}^t \Upsilon _1 (t,\varrho ) \Lambda _{1} (\tilde{\vartheta }(\varrho ,\xi ))d\varrho +\int _{0}^T \Upsilon _{2} (t,\varrho ) \Lambda _{2} (\tilde{\vartheta }(\varrho ,\xi ))d\varrho \Big )dt\Big |\\&\le \frac{1}{\Gamma (\eta )}\int _{0}^{y_{1}}[({y_{1}}-t)^{\eta -1}-({y_{2}}-t)^{\eta -1}]\Big (|{\tilde{f}}(t,\xi )|\\ &\quad +\int _{0}^t |\Upsilon _1(t,\varrho )||\Lambda _{1}(\tilde{\vartheta }(\varrho ,\xi ))|d\varrho +\int _{0}^T |\Upsilon _{2} (t,\varrho )||\Lambda _{2} (\tilde{\vartheta }(\varrho ,\xi ))|d\varrho \Big )dt\\ &\quad +\frac{1}{\Gamma (\eta )} \int _{y_{1}}^{y_{2}} ({y_{2}}-t)^{\eta -1}\Big ( |{\tilde{f}}(t,\xi )|+\int _{0}^t |\Upsilon _1 (t,\varrho )||\Lambda _{1} (\tilde{\vartheta }(\varrho ,\xi ))|d\varrho \\&\quad +\int _{0}^T |\Upsilon _{2} (t,\varrho ) ||\Lambda _{2} (\tilde{\vartheta }(\varrho ,\xi ))|d\varrho \Big )dt. \end{aligned}$$

Hence,

$$\begin{aligned} |\Upsilon {\tilde{\vartheta }(y_{2},\xi )}-\Upsilon {\tilde{\vartheta } (y_{1},\xi )}|&\le \frac{1}{\Gamma (\eta )}\int _{0}^{y_{1}}\left[ ({y_{1}}-t)^{\eta -1}-({y_{2}}-t)^{\eta -1}\right] \Big [L+\Upsilon _1(\mu _{1})+\Upsilon _{2}(\mu _{2})\Big ]dt\\ &\le \frac{1}{\Gamma (\eta )}\int _{y_{1}}^{y_{2}}\left[ ({y_{2}}-t)^{\eta -1}\right] \Big [L+\Upsilon _1(\mu _{1})+\Upsilon _{2}(\mu _{2})\Big ]dt \\ &\le \frac{[L+\Upsilon _1(\mu _{1})+\Upsilon _{2}(\mu _{2})]}{[\eta +1]} \Big [ {y_{1}}^\eta -{y_{2}}^\eta +{2(y_{2}-y_{1})}^\eta \Big ] \\ |\Upsilon {\tilde{\vartheta }(y_{2},\xi )}-\Upsilon {\tilde{\vartheta } (y_{1},\xi )}|&\le {\frac{2[L+\Upsilon _1(\mu _{1})+\Upsilon _{2}(\mu _{2})]}{[\eta +1]}} {|y_{2}-y_{1}|}^\eta . \end{aligned}$$

$\Upsilon {\tilde{\vartheta }(y_{2},\xi )}-\Upsilon {\tilde{\vartheta } (y_{1},\xi )} \rightarrow 0$ as $y_{2}-y_{1} \rightarrow 0$. This implies that,

$$|(\Upsilon \tilde{\vartheta })\left( y_{2}, \xi \right) -(\Upsilon \tilde{\vartheta })\left( y_{1}, \xi \right) |\rightarrow 0.$$

The set $\Upsilon {\mathcal {B}}_{\rho }$ is equicontinuous. Using the Arzelà–Ascoli theorem, we determine that $\Upsilon$ is completely continuous.

For the final step, let us consider the set defined by

$$E=\{\tilde{\vartheta }\in C([0,T],{\mathcal {R}}_f):\tilde{\vartheta }=\delta \Upsilon (\tilde{\vartheta }) \quad for \quad 0<\delta <1\}.$$

Now, we show that the above set E is bounded.

Let us take $\tilde{\vartheta }\in E$. For every $y \in [0, T]$, we have, from Eq. (6)

$$\begin{aligned} \tilde{\vartheta }(y,\xi ) =&\delta \Bigg [\int _{0}^y \frac{(y-t)^{\eta -1}}{\Gamma (\eta )} \Big ( {\tilde{f}}(t,\xi )+\int _{0}^t \Upsilon _1 (t,\varrho ) \Lambda _{1} (\tilde{\vartheta }(\varrho ,\xi ))d\varrho \\ &+\int _{0}^T \Upsilon _{2} (t,\varrho ) \Lambda _{2}(\tilde{\vartheta }(\varrho ,\xi )) \Big )dt \\ &- {\frac{1}{p_{1}+p_{2}}}\Big [\int _{0}^T\frac{p_{2}(T-t)^{\eta -1}}{\Gamma (\eta )}\Big ( {\tilde{f}}(t,\xi )+\int _{0}^t \Upsilon _1 (t,\varrho ) \Lambda _{1} (\tilde{\vartheta }(\varrho ,\xi ))d\varrho \\ &+\int _{0}^T \Upsilon _{2} (t,\varrho ) \Lambda _{2} (\tilde{\vartheta }(\varrho ,\xi ))d\varrho \Big )dt-\tilde{p_3} \Big ] \Bigg ] ,\ for\ y\in [0,T] \\ |\tilde{\vartheta }(y,\xi )|=&\Bigg |\delta \Bigg [\int _{0}^y \frac{(y-t)^{\eta -1}}{\Gamma (\eta )} \Big ( {\tilde{f}}(t,\xi )+\int _{0}^t \Upsilon _1 (t,\varrho ) \Lambda _{1} (\tilde{\vartheta }(\varrho ,\xi ))d\varrho \\ &+\int _{0}^T \Upsilon _{2} (t,\varrho ) \Lambda _{2}(\tilde{\vartheta }(\varrho ,\xi ))s \Big )dt \\ &-{\frac{1}{p_{1}+p_{2}}}\Big [\int _{0}^T\frac{p_{2}(T-t)^{\eta -1}}{\Gamma (\eta )}\Big ( {\tilde{f}}(t,\xi )+\int _{0}^t \Upsilon _1 (t,\varrho ) \Lambda _{1} (\tilde{\vartheta }(\varrho ,\xi ))d\varrho \\ &+\int _{0}^T \Upsilon _{2} (t,\varrho ) \Lambda _{2} (\tilde{\vartheta }(\varrho ,\xi ))d\varrho \Big )dt-\tilde{p_3} \Big ] \Bigg ]\Bigg |,\ for\ y\in [0,T] \\ \le&\Bigg |\delta \Bigg [ {\tilde{h}}(y,\xi ) +\int _{0}^y \frac{(y-t)^{\eta -1}}{\Gamma (\eta )} \Big [ \int _{0}^t \Upsilon _1 (t,\varrho ) \Lambda _{1} (\tilde{\vartheta }(\varrho ,\xi ))d\varrho \\ &+\int _{0}^T \Upsilon _{2} (t,\varrho ) \Lambda _{2}(\tilde{\vartheta }(\varrho ,\xi ))d\varrho \Big ]dt \\ &- {\frac{1}{p_{1}+p_{2}}}\Big (\int _{0}^T\frac{p_{2}(T-t)^{\eta -1}}{\Gamma (\eta )}\Big [\int _{0}^t \Upsilon _1 (t,\varrho ) \Lambda _{1} (\tilde{\vartheta }(\varrho ,\xi ))d\varrho \\ &+\int _{0}^T \Upsilon _{2} (t,\varrho ) \Lambda _{2} (\tilde{\vartheta }(\varrho ,\xi ))d\varrho \Big ]dt \Big ) \Bigg ]\Bigg |\end{aligned}$$

where, ${\tilde{h}}(y,\xi )$ is defined in Lemma (1).

$$\begin{aligned} {\tilde{h}}(y,\xi ) =&{\frac{\tilde{p_3}}{p_{1}+p_{2}}} - {\frac{1}{p_{1}+p_{2}}} \int _{0}^T\frac{{p_{2}}(T-t)^{\eta -1}}{\Gamma (\eta )} ({\tilde{f}}(t,\xi ) )dt + \int _{0}^y \frac{(y-t)^{\eta -1}}{\Gamma (\eta )} {\left( {\tilde{f}}(t,\xi )\right) dt }\\ |{\tilde{h}}(y,\xi )|\le&A. \end{aligned}$$

We already obtain $|\Lambda _1(\tilde{\vartheta }(y,\xi )|\le \mu _{1}, |\Lambda _2(\tilde{\vartheta }(y,\xi )|\le \mu _{2}$, for every $y\in [0,T].$

$$\begin{aligned} |\tilde{\vartheta }(y,\xi ) |&= |{\tilde{h}}(y,\xi )|+\frac{1}{\Gamma (\eta )} \int _{0}^y (y-t)^{\eta -1} \Big [ \int _{0}^t |\Upsilon _1 (t,\varrho ) ||\Lambda _{1} (\tilde{\vartheta }(\varrho ,\xi ))|d\varrho \\ &\quad +\int _{0}^T|\Upsilon _{2} (t,\varrho )||\Lambda _{2}(\tilde{\vartheta }(\varrho ,\xi ))|d\varrho \Big ]dt \\ &\quad - {\frac{|p_{2}|}{|p_{1}+p_{2}|}}\frac{1}{\Gamma (\eta )}\int _{0}^T(T-t)^{\eta -1}\Big [\int _{0}^t |\Upsilon _1 (t,\varrho )||\Lambda _{1} (\tilde{\vartheta }(\varrho ,\xi ))|d\varrho \\ &\quad +\int _{0}^T |\Upsilon _{2} (t,\varrho )||\Lambda _{2} (\tilde{\vartheta }(\varrho ,\xi ))|d\varrho \Big ] dt \\&\le A+\Bigg (\frac{[\Upsilon _1\mu _{1}+\Upsilon _{2}\mu _{2}]}{\Gamma (\eta )}\int _{0}^y(y-t)^{\eta -1}dt \Bigg )\left( 1+\frac{|{p_{2}}|}{|p_{1}+p_{2}|} \right) \\&\le A+\Bigg (\frac{[\Upsilon _1\mu _{1}+\Upsilon _{2}\mu _{2}]y^{\eta }}{\Gamma (\eta +1)} \Bigg )\left( 1+\frac{|{p_{2}}|}{|p_{1}+p_{2}|} \right) \\&\le A+\Bigg (\frac{[\Upsilon _1\mu _{1}+\Upsilon _{2}\mu _{2}]T^{\eta }}{\Gamma (\eta +1)} \Bigg )\left( 1+\frac{|{p_{2}}|}{|p_{1}+p_{2}|} \right) \\ &\le d. \end{aligned}$$

Define $A+\Bigg (\frac{[\Upsilon _1\mu _{1}+\Upsilon _{2}\mu _{2}]T^{\eta }}{\Gamma (\eta +1)}\Bigg )\left( 1+\frac{|{p_{2}}|}{|p_{1}+p_{2}|} \right) =d$. Therefore $\Vert \tilde{\vartheta }\Vert \le d$. This shows that any $\tilde{\vartheta }\in E$ is bounded. Therefore, the set E is bounded.

Furthermore, the operator $\Upsilon$ has a fixed point, according to Schaefer’s fixed point theorem. This shows that at least one solution $\tilde{\vartheta }(y)$ for all $y \in [0, T]$ exists.

Now, we finish up the consequence of the hypothesis dependent on the contraction mapping rule. $\Upsilon$ has a unique fixed point. This shows that the problems (1)-(2) has unique solution. $\square$

Theorem 2

(Uniqueness theorem) Assume that ${\mathbb {H}}1$ to ${\mathbb {H}}3$ holds. If

$$\frac{\Big ( [\Upsilon ^{*}_{1}(c_{1})+\Upsilon ^{*}_{2}(c_{2})]\Big )}{\Gamma (\eta +1)}\left( 1+\frac{|{p_{2}}|}{|p_{1}+p_{2}|} \right) < 1,$$

then there exists a unique solution of the Eq.(1).

Proof

$\tilde{\vartheta }(y,\xi )$ is the solution of the Eq. (1), with given boundary condition, if $\tilde{\vartheta }(y,\xi )$ satisfies

$$\begin{aligned} {\tilde{\vartheta } (y,\xi )} =&\int _{0}^y \frac{(y-t)^{\eta -1}}{\Gamma (\eta )}\Big ( {\tilde{f}}(t,\xi )+\int _{0}^t \Upsilon _1 (t,\varrho ) \Lambda _{1} (\tilde{\vartheta }(\varrho ,\xi ))d\varrho \\ &+\int _{0}^T \Upsilon _{2} (t,\varrho ) \Lambda _{2} (\tilde{\vartheta }(\varrho ,\xi ))d\varrho \Big )dt \\ &- {\frac{1}{p_{1}+p_{2}}}\Big [\int _{0}^T\frac{p_{2}(T-t)^{\eta -1}}{\Gamma (\eta )}\Big ({\tilde{f}}(t,\xi )+\int _{0}^t \Upsilon _1 (t,\varrho ) \Lambda _{1}(\tilde{\vartheta }(\varrho ,\xi ))d\varrho \\ &+\int _{0}^T \Upsilon _{2} (t,\varrho ) \Lambda _{2} (\tilde{\vartheta }(\varrho ,\xi ))d\varrho \Big )dt-\tilde{p_3} \Big ],\ for\ y\in [0,1]. \end{aligned}$$

Let’s define the operator $\Upsilon$ according to the Theorem (1). If $\tilde{\vartheta } \in C\left( X, {\mathcal {R}}_{f}\right)$ is a fixed point of $\Upsilon$, then $\tilde{\vartheta }$ is the solution of the Eq. (1). Let $\tilde{\vartheta }_{1}, \tilde{\vartheta }_{2} \in C\left( X, {\mathcal {R}}_{f}\right)$, then

$$\begin{aligned} |\Upsilon {\tilde{\vartheta _1}(y,\xi )} - \Upsilon {\tilde{\vartheta _2}(y,\xi )}|&= \frac{1}{\Gamma (\eta )}\int _{0}^y (y-t)^{\eta -1}\Big (\int _{0}^{t}|\Upsilon _1(t,\varrho )||\Lambda _{1}(\tilde{\vartheta _1}(\varrho ,\xi ))-\Lambda _{1}(\tilde{\vartheta _2}(\varrho ,\xi ))|d\varrho \\ &\quad + \int _{0}^{T}|\Upsilon _{2}(t,\varrho )||\Lambda _{2}(\tilde{\vartheta _1}(\varrho ,\xi ))-\Lambda _{2}(\tilde{\vartheta _2}(\varrho ,\xi ))|d\varrho \Big )dt \\&\quad + \frac{|p_{2}|}{|p_{1}+p_{2}|\Gamma (\eta )} \int _{0}^T (T-t)^{\eta -1}\Big (\int _{0}^{t}|\Upsilon _1(t,\varrho )||\Lambda _{1}(\tilde{\vartheta _1}(\varrho ,\xi )) \\&\quad -\Lambda _{1}(\tilde{\vartheta _2}(\varrho ,\xi ))|d\varrho + \int _{0}^{T}|\Upsilon _{2}(t,\varrho )||\Lambda _{2}(\tilde{\vartheta _1}(\varrho ,\xi ))-\Lambda _{2}(\tilde{\vartheta _2}(\varrho ,\xi ))|d\varrho \Big ) dt \end{aligned}$$

Taking supremum on both sides, we can get,

$$\begin{aligned} \Vert \Upsilon {\tilde{\vartheta _1}(y,\xi )} - \Upsilon {\tilde{\vartheta _2} (y,\xi )}\Vert _{\infty }&\le \frac{1}{\Gamma (\eta )}\Big (\Upsilon ^{*}_{1}(c_1)+\Upsilon ^{*}_{2}(c_2) \Big )\left\| \tilde{\vartheta }_{1}-\tilde{\vartheta }_{2}\right\| _{\infty } \int _{0}^y (y-t)^{\eta -1}dt \\ &\quad + \frac{|p_{2}|}{|p_{1}+p_{2}|\Gamma (\eta )} \Big (\Upsilon ^{*}_{1}(c_1)+\Upsilon ^{*}_{2}(c_2)\Big )\left\| \tilde{\vartheta }_{1}-\tilde{\vartheta }_{2}\right\| _{\infty } \int _{0}^T (T-t)^{\eta -1}dt \\ \Vert \Upsilon {\tilde{\vartheta _1}(y,\xi )} - \Upsilon {\tilde{\vartheta _2} (y,\xi )}\Vert _{\infty }&\le \frac{\Big ( \Upsilon ^{*}_{1}c_1+\Upsilon ^{*}_{2}c_2\Big )}{\Gamma (\eta +1)}\left( 1+\frac{|{p_{2}}|}{|p_{1}+p_{2}|} \right) \left\| \tilde{\vartheta }_{1}-\tilde{\vartheta }_{2}\right\| _{\infty }< 1 \\ \Vert \Upsilon {\tilde{\vartheta _1}(y,\xi )} - \Upsilon {\tilde{\vartheta _2} (y,\xi )}\Vert _{\infty }&<\left\| \tilde{\vartheta }_{1}-\tilde{\vartheta }_{2}\right\| _{\infty }. \end{aligned}$$

Hence, $\Upsilon$ is a contraction. Thus, by the Banach contraction principle, using the theorem (2.12) in²⁰, $\Upsilon$ has a unique fixed point $\tilde{\vartheta } \in C\left( [0,1], {\mathcal {R}}_{f}\right)$. $\square$

Description of ADM

The Adomian decomposition method, abbreviated ADM, is the subject of this essay. ADM is a kind of method that uses a decomposition approach to generate approximative solutions and even exact solutions for nonlinear systems when the proper initial data. Numerous benefits of this approach include. The diagrammatic representation of the proposed method is shown in Fig. 1.

Advantages of the adomian decomposition method

• Numerous types of nonlinear systems, including algebraic equations, differential equations, integral equations, integro-differential equations, and so on, can be solved using this method, and it is relatively simple to use.

• It avoids the time-consuming Picard method integrations.

• It can solve certain nonlinear problems that conventional numerical approaches cannot address.

• The number of variables has no impact on applications since it need not directly affect the solutions series.

Fig. 1

Diagrammatic representation of proposed method.

Before using ADM to construct approximations of solutions for nonlinear equations with boundary conditions.

Consider this equation: When the operator $I^{\eta }$ is applied to both sides of Eq. (1), we obtain

$$\begin{aligned} \begin{aligned} D^{\eta }{\tilde{\vartheta } (y,\xi )} = {\tilde{h}}(y,\xi )-&{\frac{1}{p_{1}+p_{2}}} \int _{0}^T\frac{p_{2}(T-t)^{\eta -1}}{\Gamma (\eta )} \Big ( \int _{0}^t \Upsilon _1 (t,\varrho ) \Lambda _{1} (\tilde{\vartheta }(\varrho ,\xi ))d\varrho \\+&\int _{0}^T \Upsilon _{2} (t,\varrho )\Lambda _{2}(\tilde{\vartheta }(\varrho ,\xi ))d\varrho \Big )dt \\+&\int _{0}^y \frac{(y-t)^{\eta -1}}{\Gamma (\eta )} \Big (\int _{0}^t \Upsilon _1 (t,\varrho ) \Lambda _{1} (\tilde{\vartheta }(\varrho ,\xi ))d\varrho \\+&\int _{0}^T \Upsilon _{2} (t,\varrho ) \Lambda _{2} (\tilde{\vartheta }(\varrho ,\xi ))d\varrho \Big )dt. \end{aligned} \end{aligned}$$

In the form of a series, Adomian’s method provides the solution $\tilde{\vartheta }(y,\xi )$,

$$\tilde{\vartheta }=\sum _{n=0}^{\infty } \tilde{\vartheta }_{n}$$

and the decomposition of the nonlinear $\Lambda _{1}$ and $\Lambda _{2}$ is

$$\Lambda _{1}=\sum _{n=0}^{\infty } M_{n},~~ \Lambda _{2}=\sum _{n=0}^{\infty } N_{n}.$$

Where $M_{n}$ and $N_{n}$ are Adomian polynomials and the expression for each of them is:

$$\begin{aligned}&M_{n}=\frac{1}{n !} \frac{d^{n}}{d y^{n}}\left[ \Lambda _{1}\left( \sum _{l=0}^{\infty } U^{l} \tilde{\vartheta }_{l}\right) \right] _{U=0} \\&N_{n}=\frac{1}{n !} \frac{d^{n}}{d y^{n}}\left[ \Lambda _{2}\left( \sum _{l=0}^{\infty } U^{l} \tilde{\vartheta }_{l}\right) \right] _{U=0}. \end{aligned}$$

Therefore,

$$\begin{aligned}&M_{0}=\Lambda _{1}\left( \tilde{\vartheta }_{0}\right) , \\&M_{1}=\tilde{\vartheta }_{1} \Lambda _{1}^{\prime }\left( \tilde{\vartheta }_{0}\right) , \\&M_{2}=\tilde{\vartheta }_{2} \Lambda _{1}^{\prime }\left( \tilde{\vartheta }_{0}\right) +\frac{1}{2}\tilde{\vartheta }_{1}^{2} \Lambda _{1}^{\prime \prime }\left( \tilde{\vartheta }_{0}\right) , \\&M_{3}=\tilde{\vartheta }_{3} \Lambda _{1}^{\prime }\left( \tilde{\vartheta }_{0}\right) +\tilde{\vartheta }_{1} \tilde{\vartheta }_{2} \Lambda _{1}^{\prime \prime }\left( \tilde{\vartheta }_{0}\right) +\frac{1}{3}\tilde{\vartheta }_{1}^{3} \Lambda _{1}^{\prime \prime \prime }\left( \tilde{\vartheta }_{0}\right) ,\\ \vdots \end{aligned}$$

and

$$\begin{aligned}&N_{0}=\Lambda _{2}\left( \tilde{\vartheta }_{0}\right) , \\&N_{1}=\tilde{\vartheta }_{1} \Lambda _{2}^{\prime }\left( \tilde{\vartheta }_{0}\right) , \\&N_{2}=\tilde{\vartheta }_{2} \Lambda _{2}^{\prime }\left( \tilde{\vartheta }_{0}\right) +\frac{1}{2} \tilde{\vartheta }_{1}^{2} \Lambda _{2}^{\prime \prime }\left( \tilde{\vartheta }_{0}\right) , \\&N_{3}=\tilde{\vartheta }_{3} \Lambda _{2}^{\prime }\left( \tilde{\vartheta }_{0}\right) +\tilde{\vartheta }_{1} \tilde{\vartheta }_{2} \Lambda _{2}^{\prime \prime }\left( \tilde{\vartheta }_{0}\right) +\frac{1}{3} \tilde{\vartheta }_{1}^{3} \Lambda _{2}^{\prime \prime \prime }\left( \tilde{\vartheta }_{0}\right) ,\\ \vdots \end{aligned}$$

The components $\tilde{\vartheta }_{0}, \tilde{\vartheta }_{1}, \tilde{\vartheta }_{2}, \ldots$ are determined in a recursive manner by

$$\begin{aligned} \tilde{\vartheta }_{0}(y,\xi )=&{\tilde{h}}(y,\xi ) \\ \tilde{\vartheta }_{1}(y,\xi )=&-{\frac{1}{p_{1}+p_{2}}} \int _{0}^T\frac{p_{2}(T-t)^{\eta -1}}{\Gamma (\eta )} \Big ( \int _{0}^t \Upsilon _1 (t,\varrho ) M_0 d\varrho +\int _{0}^T \Upsilon _{2} (t,\varrho ) N_0 d\varrho \Big )dt \\ &\quad + \int _{0}^y \frac{(y-t)^{\eta -1}}{\Gamma (\eta )} \Big (\int _{0}^t \Upsilon _1 (t,\varrho ) M_0 d\varrho +\int _{0}^T \Upsilon _{2} (t,\varrho ) N_0 d\varrho \Big )dt\\ \tilde{\vartheta }_{k+1}(y,\xi )=&-{\frac{1}{p_{1}+p_{2}}} \int _{0}^T\frac{p_{2}(T-t)^{\eta -1}}{\Gamma (\eta )} \Big ( \int _{0}^t \Upsilon _1 (t,\varrho ) M_k d\varrho +\int _{0}^T \Upsilon _{2} (t,\varrho ) N_k d\varrho \Big )dt \\&\quad + \int _{0}^y \frac{(y-t)^{\eta -1}}{\Gamma (\eta )} {\Big (\int _{0}^t \Upsilon _1 (t,\varrho ) M_k d\varrho +\int _{0}^T \Upsilon _{2} (t,\varrho ) N_k dr\varrho \Big )dt,~~ k\ge 1.} \end{aligned}$$

We approximate the BVP solution by solving the relation using boundary conditions. The recurrence relations are easily considered, in contrast to the boundary conditions for (3), which are important in the development of the solution.

Remark 1

Note that the notation $\Lambda _{1}(\tilde{\vartheta }_{0}(y,\xi ))$ always must be replaced by $\Lambda _{1}(\tilde{\vartheta }_{0})$.

Limitations of the adomian decomposition method

• The method’s accuracy relies heavily on the choice of the initial fuzzy approximation, which can affect the convergence and stability of the solution.

• ADM does not provide a straightforward way to estimate approximation errors in fuzzy fractional contexts.

• For FFVIDEs with strong nonlinear kernels or long-memory fractional terms, ADM may require many iterations to achieve acceptable accuracy.

Numerical experiments

The examples in this section demonstrate how precisely the methodology yields the FFVIDEs solution. The existence and uniqueness requirement in (1-2) is required for BVPs to solve the problem.

Example 1

Consider the Caputo-type Volterra-Fredholm integro-differential equation with fuzzy boundary conditions.

$$\begin{aligned} {\left\{ \begin{array}{ll} & D^{\frac{1}{2}}\tilde{\vartheta }(y,\xi )=(2+\frac{{y^2}}{3})[\xi -1,1-\xi ]+\frac{1}{20}\int \limits _{0}^{y}(3y+4\varrho )\tilde{\vartheta }(\varrho ,\xi )d\varrho +\frac{1}{8}\int \limits _{0}^{1}(3y-1)\tilde{\vartheta }(\varrho ,\xi )d\varrho \\ & 2\tilde{\vartheta }(0,\xi )+\tilde{\vartheta }(1,\xi )=[\xi -1, 1-\xi ], \quad y\in (0,1] \\ \end{array}\right. } \end{aligned}$$

$$p_{1},p_{2}\in R, ~ {\tilde{p}}_3\in {\mathcal {R}}_f \quad p_{1}=2,p_{2}=1,{\tilde{p}}_3=[{\underline{p}}_3,{\overline{p}}_3]=[\xi -1, 1-\xi ].$$

The equivalent form of the above equation for type(i)-differentiability is shown below. ( using the definition 2.9 in³⁸)

$$\begin{aligned} {\left\{ \begin{array}{ll} & D^{\frac{1}{2}}\underline{\vartheta }(y,\xi )=(2+\frac{{y^2}}{3})(\xi -1)+\frac{1}{20}\int \limits _{0}^{x}(3y+4\varrho )\underline{\vartheta }(\varrho ,\xi )d\varrho +\frac{1}{8}\int \limits _{0}^{1}(3y-1)\underline{\vartheta }(\varrho ,\xi )d\varrho ,\\ & D^{\frac{1}{2}}\overline{\vartheta }(y,\xi )=(2+\frac{{y^2}}{3})(1-\xi )+\frac{1}{20}\int \limits _{0}^{x}(3y+4\varrho )\overline{\vartheta }(\varrho ,\xi )d\varrho +\frac{1}{8}\int \limits _{0}^{1}(3y-1)\overline{\vartheta }(\varrho ,\xi )d\varrho . \end{array}\right. } \end{aligned}$$

let’s construct $\underline{\vartheta }(y,\xi )$ and apply the operator $I^{\frac{1}{2}}$ on both sides

$$\begin{aligned} {\left\{ \begin{array}{ll} \underline{\vartheta }(y,\xi )= & {\underline{h}}(y,\xi )-\frac{p_{2}}{p_{1}+p_{2}}\frac{1}{\Gamma (1/2)}\int \limits _{0}^{1}(1-t)^{-1/2}\left[ \int \limits _{0}^{y}\frac{(3y+4\varrho )}{20}\underline{\vartheta }(\varrho ,\xi )d\varrho +\int \limits _{0}^{1}\frac{(3y-1)}{8} \underline{\vartheta }(\varrho ,\xi )d\varrho \right] dt\\ & +\frac{1}{(1/2)}\int \limits _{0}^{y}(y-t)^{-1/2}\left[ \int \limits _{0}^{x}\frac{(3y+4\varrho )}{20} \underline{\vartheta }(\varrho ,\xi )d\varrho +\int \limits _{0}^{1}\frac{(3y-1)}{8} \underline{\vartheta }(\varrho ,\xi )d\varrho \right] dt, \\ \overline{\vartheta }(y,\xi )= & {\overline{h}}(y,\xi )-\frac{p_{2}}{p_{1}+p_{2}}\frac{1}{\Gamma (1/2)}\int \limits _{0}^{1}(1-t)^{-1/2}\left[ \int \limits _{0}^{y}\frac{(3y+4\varrho )}{20}\overline{\vartheta }(\varrho ,\xi )d\varrho +\int \limits _{0}^{1}\frac{(3y-1)}{8} \overline{\vartheta }(\varrho ,\xi )d\varrho \right] dt\\ & +\frac{1}{(1/2)}\int \limits _{0}^{y}(y-t)^{-1/4}\left[ \int \limits _{0}^{y}\frac{(3y+4\varrho )}{20}\overline{\vartheta }(\varrho ,\xi )d\varrho +\int \limits _{0}^{1}\frac{(3y-1)}{8} \overline{\vartheta }(\varrho ,\xi )d\varrho \right] dt. \end{array}\right. } \end{aligned}$$

Fig. 2

Visualizes the approximated fuzzy results of $\underline{\vartheta }(y,\xi ), \overline{\vartheta }(y,\xi )$ at different values of uncertainty $\xi$= 0.2,0.5,0.7 and Caputo derivative $\eta =1/5, 3/4$ for Example 1.

Fig. 3

Visualizes the fuzzy results of $\underline{\vartheta }(y,\xi ), \overline{\vartheta }(y,\xi )$ at different values of $\eta =1/5, 3/4$ and space variable $y= 0.2, 0.5, 0.7$ for Example 1.

Fig. 4

Visualizes the fuzzy results of $\underline{\vartheta }(y,\xi ), \overline{\vartheta }(y,\xi )$ at different values of fractional order $\eta$, uncertainty $\xi$ and space variable y for Example 1.

The next step is to determine the solution of $\underline{\vartheta }(y,\xi )$ by using the ADM.

$$\begin{aligned} \underline{\vartheta }_0(y,\xi )=&{\underline{h}}(y,\xi ) \\ \underline{\vartheta }_0(y,\xi )=&\underline{h}(y,\xi )= {\frac{{\underline{p}}_3}{p_{1}+p_{2}}} - {\frac{1}{p_{1}+p_{2}}} \int _{0}^1\frac{p_{2}(1-t)^{\eta -1}}{\Gamma (\eta )} [f(t,\xi )]dt+\int _{0}^y \frac{(y-t)^{\eta -1}}{\Gamma (\eta )} [f(t,\xi )]dt \\ \underline{\vartheta }_0(y,\xi )=&\underline{h}(y,\xi )= {\frac{(\xi -1)}{3}} - {\frac{(\xi -1)}{3\Gamma (1/2)}}\int _{0}^1{(1-t)^{-1/2}} [t^3+3{t^2}+1]dt\\ &+\frac{(\xi -1)}{\Gamma (1/2)}\int _{0}^y {(y-t)^{ -1/2}} [t^3+3{t^2}+1]dt \\ \underline{\vartheta }_1(y,\xi ) =&-\frac{p_{2}}{p_{1}+p_{2}}\frac{1}{\Gamma (1/2)}\int \limits _{0}^{1}(1-t)^{-1/2}\left[ \frac{1}{20}\int \limits _{0}^{y}(3y+4\varrho ) M_0 d\varrho +\frac{1}{8}\int \limits _{0}^{1}(3y-1) N_0 d\varrho \right] dt\\&+\frac{1}{\Gamma (1/2)}\int \limits _{0}^{y}(y-t)^{-1/2}\left[ \frac{1}{20}\int \limits _{0}^{y}(3y+4\varrho )\ M_0 d\varrho +\frac{1}{8}\int \limits _{0}^{1}(3y-1) N_0 d\varrho \right] dt \end{aligned}$$

In a similar manner, we can determine the following terms: $\underline{\vartheta }_2(y,\xi ),\underline{\vartheta }_3(y,\xi ),...$ and obtain the result.

$$\begin{aligned} \underline{\vartheta }(y,\xi )=\sum _{n=0}^{\infty } \underline{\vartheta }_{n}(y,\xi )=\frac{(\xi -1)}{3}-\frac{2(\xi -1)}{3\Gamma {(3/2)}}-\frac{2(\xi -1)}{9\Gamma {(7/2)}}+\frac{2(\xi -1)y^{1/2}}{\Gamma {(3/2)}}+\frac{2(\xi -1)y^{5/2}}{3\Gamma {(7/2)}}+... \end{aligned}$$

In the same way, we can find type(ii) differentiable,

$$\begin{aligned} \overline{\vartheta }(y,\xi )=\sum _{n=0}^{\infty } \overline{\vartheta }_{n}(y,\xi )=\frac{(1-\xi )}{3}-\frac{2(1-\xi )}{3\Gamma {(3/2)}}-\frac{2(1-\xi )}{9\Gamma {(7/2)}}+\frac{2(1-\xi )y^{1/2}}{\Gamma {(3/2)}}+\frac{2(1-\xi )y^{5/2}}{3\Gamma {(7/2)}}+... \end{aligned}$$

Using MATLAB, we obtain 2D and 3D graphs of the simulation result of approximated fuzzy solutions displayed in the figure and table values for different levels of uncertainty $\xi =0.2$(red), $\xi =0.5$(green), $\xi =0.7$(blue) shown in Fig. 2a, 2b and Table. 1, 2, 3 where us plots of $\xi$-cut representation of the approximated solution for Caputo derivative $\eta =1/5$, $\eta =3/4$ respectively. Different values of space variable $y=0.2, 0.5, 0.7$ are shown in Fig. 3a,3b, and both uncertainty $\xi$ and space variable y with the fuzzy solutions are shown in Fig. 4a, 4b the surface plot of the fuzzy solutions are presented respectively (green and blue are the upper and lower solutions respectively). The fuzzy solution and the solution representation changed when we changed the $\eta$ order. While this modification is tiny, it may make a significant effect when applied to real-world issues.

Note: Here, The 1st non linear term $\underline{\vartheta }_0(y,\xi ) = M_{0 }$,

and 2nd non linear term $\underline{\vartheta }_0(y,\xi ) = N_{0}$.

Table 1 Numerical results for Example 1, $\underline{\vartheta }(y,\xi )$ & $\overline{\vartheta }(y,\xi )$ at different values of uncertainty $\xi$ and fractional value $\eta =1/5.$.

Table 2 Numerical results for Example 1, $\underline{\vartheta }(y,\xi )$ & $\overline{\vartheta }(y,\xi )$ at different values of uncertainty $\xi$ and fractional value $\eta =3/4$.

Table 3 Numerical results for Example 1, where $p_{1}=2, p_{2}=1$ and boundary points $\underline{\vartheta }_{0}(y,\xi ),\underline{\vartheta }_{1}(y,\xi ),\overline{\vartheta }_{0}(y,\xi ),\overline{\vartheta }_{1}(y,\xi )$ at different values of uncertainty $\xi$ and fractional values $\eta =1/5$, $\eta =3/4$.

Example 2

Consider the Caputo-type Volterra-Fredholm integro-differential equation with fuzzy boundary conditions.

$$\begin{aligned} {\left\{ \begin{array}{ll} & D^{\frac{3}{4}}\tilde{\vartheta }(y,\xi )=(y+1)[\xi -1,1-\xi ]+\frac{1}{10}\int \limits _{0}^{y}(\varrho ^{2}e^{y})\tilde{\vartheta }(\varrho ,\xi )d\varrho +\int \limits _{0}^{1}{y^2}\tilde{\vartheta }(\varrho ,\xi )d\varrho \\ & 2\tilde{\vartheta }(0,\xi )+\tilde{\vartheta }(1,\xi )=[\xi -1, 1-\xi ], \quad y\in (0,1] \\ \end{array}\right. } \end{aligned}$$

$$p_{1},p_{2}\in R, ~ {\tilde{p}}_3\in {\mathcal {R}}_f \quad p_{1}=2,p_{2}=1,{\tilde{p}}_3=[{\underline{p}}_3,{\overline{p}}_3]=[\xi -1, 1-\xi ].$$

The equivalent form of the above equation for type(i)-differentiability is shown below.

$$\begin{aligned} {\left\{ \begin{array}{ll} & D^{\frac{3}{4}}\underline{\vartheta }(y,\xi )=(y+1)(\xi -1)+\frac{1}{10}\int \limits _{0}^{x}(\varrho ^{2}e^{y})\underline{\vartheta }(\varrho ,\xi )d\varrho +\int \limits _{0}^{1}{y^2}\underline{\vartheta }(\varrho ,\xi )d\varrho \\ & D^{\frac{3}{4}}\overline{\vartheta }(y,\xi )=(y+1)(1-\xi )+\frac{1}{10}\int \limits _{0}^{y}(\varrho ^{2}e^{y})\overline{\vartheta }(\varrho ,\xi )d\varrho +\int \limits _{0}^{1}{y^2} \overline{\vartheta }(\varrho ,\xi )d\varrho \end{array}\right. } \end{aligned}$$

let’s construct $\underline{\vartheta }(y,\xi )$ and apply the operator $I^{\frac{3}{4}}$ on both sides

$$\begin{aligned} {\left\{ \begin{array}{ll} \underline{\vartheta }(y,\xi )= & {\underline{h}}(y,\xi )-\frac{p_{2}}{p_{1}+p_{2}}\frac{1}{\Gamma (\eta )}\int \limits _{0}^{1}(1-t)^{-1/4}\left[ \frac{1}{10}\int \limits _{0}^{y}(\varrho ^{2}e^{y})\underline{\vartheta }(\varrho ,\xi )d\varrho +\int \limits _{0}^{1}{t^2} \underline{\vartheta }(\varrho ,\xi )d\varrho \right] dt\\ & +\frac{1}{(\eta )}\int \limits _{0}^{y}(y-t)^{-1/4}\left[ \frac{1}{10}\int \limits _{0}^{y}(\varrho ^{2}e^{y})\ \underline{\vartheta }(\varrho ,\xi )d\varrho +\int \limits _{0}^{1}{t^2} \underline{\vartheta }(\varrho ,\xi )d\varrho \right] dt \\ \overline{\vartheta }(y,\xi )= & {\overline{h}}(t,\xi )-\frac{p_{2}}{p_{1}+p_{2}}\frac{1}{\Gamma (\eta )}\int \limits _{0}^{1}(1-t)^{-1/4}\left[ \frac{1}{10}\int \limits _{0}^{y}(\varrho ^{2}e^{y})\overline{\vartheta }(\varrho ,\xi )d\varrho +\int \limits _{0}^{1}{t^2} \overline{\vartheta }(\varrho ,\xi )d\varrho \right] dt\\ & +\frac{1}{(\eta )}\int \limits _{0}^{y}(y-t)^{-1/4}\left[ \frac{1}{10}\int \limits _{0}^{y}(\varrho ^{2}e^{y})\overline{\vartheta }(\varrho ,\xi )d\varrho +\int \limits _{0}^{1}{t^2} \overline{\vartheta }(\varrho ,\xi )d\varrho \right] dt. \end{array}\right. } \end{aligned}$$

The next step is to determine the solution of $\underline{\vartheta }(y,\xi )$ by using the ADM.

$$\begin{aligned} \underline{\vartheta }_0(y,\xi )=&{\underline{h}}(y,\xi ) \\ \underline{h}(y,\xi )=&{\frac{{\underline{p}}_3}{p_{1}+p_{2}}} - {\frac{1}{p_{1}+p_{2}}} \int _{0}^1\frac{p_{2}(1-t)^{\eta -1}}{\Gamma (\eta )} [f(t,\xi )]dt+\int _{0}^y \frac{(y-t)^{\eta -1}}{\Gamma (\eta )} [f(t,\xi )]dt \\ \underline{\vartheta }_0(y,\xi )=&\underline{h}(y,\xi )= {\frac{(\xi -1)}{2+1}} - {\frac{1}{2+1}} \int _{0}^1\frac{(1-t)^{-1/4}}{\Gamma (3/4)} [(t+1)(\xi -1)]dt\\ &+\int _{0}^y \frac{(y-t)^{ -1/4}}{\Gamma (3/4)} [(t+1)(\xi -1)]dt \\ \underline{\vartheta }_1(y,\xi ) =&-\frac{p_{2}}{p_{1}+p_{2}}\frac{1}{\Gamma (\eta )}\int \limits _{0}^{1}(1-t)^{-1/4}\left[ \frac{1}{10}\int \limits _{0}^{y}(\varrho ^{2}e^{y}) M_0 d\varrho +\int \limits _{0}^{1}{t^2} N_0 d\varrho \right] dt\\&+\frac{1}{\Gamma (\eta )}\int \limits _{0}^{y}(y-t)^{-1/4}\left[ \frac{1}{10}\int \limits _{0}^{y}(\varrho ^{2}e^{y})\ M_0 d\varrho +\int \limits _{0}^{1}{t^2} N_0 d\varrho \right] dt \end{aligned}$$

Fig. 5

Visualizes the fuzzy results of $\underline{\vartheta }(y,\xi ), \overline{\vartheta }(y,\xi )$ at different values of uncertainty $\xi = 0.2, 0.5, 0.7$ and fractional order $\eta =1/5, 3/4$ for Example 2.

Fig. 6

Visualizes the fuzzy results of $\underline{\vartheta }(y,\xi ), \overline{\vartheta }(y,\xi )$ at different values of $\eta =1/5, 3/4$ and space variable $y= 0.2, 0.5, 0.7$ for Example 2.

Fig. 7

Visualizes the fuzzy results of $\underline{\vartheta }(y,\xi ), \overline{\vartheta }(y,\xi )$ at different values of fractional order $\eta$, uncertainty $\xi$ and space variable y of Example 2.

In a similar manner, we can determine the following terms: $\underline{\vartheta }_2(y,\xi ),\underline{\vartheta }_3(y,\xi )$ and obtain the result.

$$\begin{aligned} \underline{\vartheta }(y,\xi )=\sum _{n=0}^{\infty } \underline{\vartheta }(y,\xi )_{n}=\frac{(\xi -1)}{3}-\frac{(\xi -1)}{3\Gamma {(11/4)}}-\frac{(\xi -1)}{3\Gamma {(7/4)}}+\frac{(\xi -1)y^{7/4}}{\Gamma {(11/4)}}+\frac{(\xi -1)y^{3/4}}{\Gamma {(7/4)}}+... \end{aligned}$$

In the same way, we can find type(ii) differentiable,

$$\begin{aligned} \overline{\vartheta }(y,\xi )=\sum _{n=0}^{\infty } \overline{\vartheta }(y,\xi )_{n}=\frac{(1-\xi )}{3}-\frac{(1-\xi )}{3\Gamma {(11/4)}}-\frac{(1-\xi )}{3\Gamma {(7/4)}}+\frac{(1-\xi )y^{7/4}}{\Gamma {(11/4)}}+\frac{(1-\xi )y^{3/4}}{\Gamma {(7/4)}}+... \end{aligned}$$

Using MATLAB, we obtain 2D and 3D graphs of the simulation result of approximated fuzzy solutions displayed in the figure and table values for different levels of uncertainty $\xi =0.2$(red), $\xi =0.5$(green), $\xi =0.7$(blue) shown in Fig. 5a, 5b and Table. 4, 5, 6 where us plots of $\xi$-cut representation of the approximated solution for Caputo derivative $\eta =1/5$, $\eta =3/4$ respectively. Different values of space variable $y=0.2, 0.5, 0.7$ are shown in Fig. 6a, 6b, and both uncertainty $\xi$ and space variable y with the fuzzy solutions are shown in Fig. 7a, 7b the surface plot of the fuzzy solutions are presented respectively(green and blue are the upper and lower solutions respectively.)

Table 4 Numerical results for Example 2, $\underline{\vartheta }(y,\xi )$ & $\overline{\vartheta }(y,\xi )$ at different values of uncertainty $\xi$ and fractional value $\eta =1/5$.

Table 5 Numerical results for Example 2, $\underline{\vartheta }(y,\xi )$ & $\overline{\vartheta }(y,\xi )$ at different values of uncertainty $\xi$ and fractional value $\eta =3/4$.

Table 6 Numerical results for Example 2, where $p_{1}=2, p_{2}=1$ and boundary points $\underline{\vartheta }_{0}(y,\xi ),\underline{\vartheta }_{1}(y,\xi ),\overline{\vartheta }_{0}(y,\xi ),\overline{\vartheta }_{1}(y,\xi )$ at different values of uncertainty $\xi$ and fractional values $\eta =1/5$, $\eta =3/4$.

Conclusions

In this study, we examined a class of fuzzy fractional Volterra integro-differential equations (FFVIDEs) using both theoretical and numerical approaches based on the Adomian decomposition method. The existence and uniqueness of the solution were established through the contraction mapping principle under the given fuzzy boundary conditions. The proposed approach, derived from the decomposition rule, effectively generates approximate solutions whose behavior was analyzed both mathematically and graphically. The simulation results indicate that even a slight variation in the $\xi$-cut value can significantly affect the solution, confirming the sensitivity and accuracy of the numerical approach. Overall, the findings demonstrate that the proposed method provides reliable and precise fuzzy fractional solutions.

For future research, we intend to extend this work by incorporating the concept of time delay, as time-delay systems play a vital role in modeling real-world processes in engineering, biology, and control theory. The study will further explore variable-order fuzzy fractional Volterra integro-differential equations (VOFFVIDEs) using alternative and more advanced numerical techniques.