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A numerical method based on fractional B-spline is developed and examined to approximate the solution of fractional multiterm pantograph equations. The method is applied to a uniform partition, and the midpoints of the partition are used as extra points to construct a uniquely solvable system. Convergence analysis of the method is discussed via Green's function approach, and an error bound dependent on the regularity of the exact solution is obtained. Additionally, to demonstrate the efficiency and accuracy of the proposed method, several examples are solved. The results indicate that the practical orders of convergence obtained by our proposed method are in good agreement with the theoretical results. [ABSTRACT FROM AUTHOR]

This work uses the modified Extended Direct Algebraic Method (mEDAM) with conformable derivatives to obtain accurate solutions for the diffusion-reaction equation with cubic nonlinearity and the nonlinear fractional generalised density-independent DR problem. We use fractional derivatives in the conformable sense to achieve precise polynomial-form solutions by converting the equations into autonomous two-dimensional plane systems. The technique solves nonlinear ordinary differential equations (ODEs) produced by a fractional transformation to produce travelling wave solutions. Through the visualisation of various solution profiles, as periodic, shock, kink, anti-kink, and soliton waves, numerical simulations offer important insights into the dynamics of nonlinear fractional partial differential equations. This work builds on earlier research and broadens our understanding of complex events. [ABSTRACT FROM AUTHOR]

This study delves into the existence, uniqueness, and stability of solutions for a nonlinear coupled system incorporating mixed generalized fractional derivatives. The system is characterized by ψ-Caputo and ϕ-Riemann-Liouville fractional derivatives with mixed boundary conditions. We provide essential preliminaries and definitions, followed by a detailed analysis using fixed point theory to establish the main results. Furthermore, we discuss the Hyers-Ulam stability of the proposed system and illustrate the theoretical findings with several examples. This study extends and generalizes various results in the literature and provides new insights into the qualitative behavior of fractional differential systems. [ABSTRACT FROM AUTHOR]

In this manuscript, we investigated the existence, uniqueness, and Ulam–Hyers stability results of solutions to implicit Caputo–Hadamard fractional differential equations with noninstantaneous impulses and δ − d e r i v a t i v e initial conditions. By employing Banach's and Schaefer's fixed-point theorems, we attain the required results. Finally, we provide examples that support the main findings we arrived at. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of solutions of Hilfer-Katugampola fractional differential equations with boundary conditions. We first establish existence theory of solutions for the mentioned problem by the fixed point theory and the measure of noncompactness. Then we investigate the ε-approximate solution to our concerned problem via a generalized Gronwall inequality. Finally, as the application of abstract results, we give an example to illustrate our main results. [ABSTRACT FROM AUTHOR]

Abstract This paper investigates a hybrid Langevin inclusion involving the generalized Caputo fractional derivative with a variable coefficient. The proposed Langevin inclusion is based on the ρ-Caputo fractional derivative, which extends the well-known fractional derivatives by incorporating a variable coefficient that depends on different values of the function ρ. By applying Dhage’s fixed point theorem, we establish the needed conditions for the existence of a solution. Based on this theorem, we investigate the problem by focusing on some cases for variable coefficients. To validate our theoretical findings, we provide a detailed example that highlights the practical significance and applicability of the proposed approach.

The nonlinear fractional Langevin equation is an important equation used in various areas of physics and mathematics. In this article, we focus on the singular-hybrid case of the generalized Langevin equations and prove the existence of a solution using fixed point theory. We also provide numerical and graphical simulations to demonstrate the convergence and existence of a solution for this singular problem. Finally, we present numerical examples in the form of tables and graphs to examine the accuracy of the method. [ABSTRACT FROM AUTHOR]

This paper presents a novel spectral algorithm for the numerical solution of multi-dimensional fractional-order telegraph equations, a critical model used to capture the combined effects of diffusion and wave propagation. The core innovation of this work is the application of Jacobi-Romanovski polynomials as the basis functions for spectral discretization. These polynomials offer unique advantages, including the ability to handle nonstandard domains and boundary conditions, making them particularly suitable for partial differential equation (PDE) applications. A comprehensive error analysis is conducted, providing deep insights into the convergence rates and factors affecting the accuracy of the numerical solutions. Extensive numerical experiments further demonstrate the superior performance of the proposed spectral algorithm in solving a wide range of multi-dimensional fractional-order telegraph equation models. The results show a significant improvement in accuracy and computational efficiency compared to traditional numerical methods, such as finite difference or finite element techniques. This research advances the field of computational science by offering a robust, efficient, and versatile numerical framework for the precise solution of complex multi-dimensional PDEs. [ABSTRACT FROM AUTHOR]

Abstract In this work, we introduce a novel idea of generalized ( k , ψ ) $({{k}},\psi )$ -Hilfer proportional fractional operators. The proposed operator combines the ( k , ψ ) $({{k}},\psi )$ -Riemann–Liouville and ( k , ψ ) $({{k}},\psi )$ -Caputo proportional fractional operators. Some properties and auxiliary results of the proposed operators are investigated. The ψ-Laplace transform and its properties of the proposed operators are established and utilized to solve Cauchy-type problems. Furthermore, the uniqueness result for a higher-order initial value problem under ( k , ψ ) $({{k}},\psi )$ -Hilfer proportional fractional operators is proved by using Picard’s iterative technique. At the end, examples are provided to present the theoretical results. This new type of proposed operator can help other researchers who are still working on real-world problems.

In this paper, we focus on studying a fractional Schrödinger equation of the form { (− Δ) s u + V (x) u = f (x , u) in Ω , u > 0 in Ω , u = 0 in R n ∖ Ω , where 0 < s < 1 , n > 2 s , Ω is a smooth bounded domain in R n , (− Δ) s denotes the fractional Laplacian of order s, f (x , t) is a function in C (Ω ‾ × R) , and f (x , t) / t is nondecreasing in t and converges uniformly to an L ∞ function q (x) as t approaches infinity. The potential energy V satisfies appropriate assumptions. In the first part of our study, we analyze the asymptotic linearity of the nonlinearity and investigate the occurrence of the bifurcation phenomenon. We employ variational techniques and a "mountain pass" approach in our proof, notable for not assuming the Ambrosetti–Rabinowitz condition or any replacement condition on the nonlinearity. Additionally, we extend our methods to handle cases where the function f (x , t) exhibits superlinearity in t at infinity, represented by q (x) ≡ + ∞ . [ABSTRACT FROM AUTHOR]

In this study, an ordinary-deterministic equation system modeling the spread dynamics of malware under mutation is analyzed with fractional derivatives and random variables. The original model is transformed into a system of fractional-random differential equations (FRDEs) using Caputo fractional derivatives. Normally distributed random variables are defined for the parameters of the original system that are related to the mutations and infections of the nodes in the network. The resulting system of FRDEs is simulated using the predictor-corrector method based fde12 algorithm and the forward fractional Euler method (ffEm) for various values of the model components such as the standard deviations, orders of derivation, and repetition numbers. Additionally, the sensitivity analysis of the original model is investigated in relation to the random nature of the components and the basic reproduction number ( R 0 ) to underline the correspondence of random dynamics and sensitivity indices. Both the normalized forward sensitivity indices (NFSI) and the standard deviation of R 0 with random components give matching results for analyzing the changes in the spread rate. Theoretical results are backed by the simulation outputs on the numerical characteristics of the fractional-random model for the expected number of infections and mutations, expected timing of the removal of mutations from the network, and measurement of the variability in the results such as the coefficients of variation. Comparison of the results from the original model and the fractional-random model shows that the fractional-random analysis provides a more generalized perspective while facilitating a versatile investigation with ease and can be used on different models as well. [ABSTRACT FROM AUTHOR]

This article is devoted to proving the uniqueness of positive solutions for p-Laplacian equations with Caputo and Riemann-Liouville fractional derivative. The uniqueness result and the dependence of the solution on a parameter are established based on the fixed point point theorem of mixed monotone operators. In the end, a numerical simulation is given to verify the main results. [ABSTRACT FROM AUTHOR]

In this paper, the coupled system of sequential ψ-Hilfer fractional boundary value problems with non-instantaneous impulses is investigated. The existence results of the system are proved by means of topological degree theory. An example is constructed to demonstrate our results. Additionally, a graphical analysis is performed to verify our results. [ABSTRACT FROM AUTHOR]

In this work, we introduce a novel idea of generalized (k , ψ) -Hilfer proportional fractional operators. The proposed operator combines the (k , ψ) -Riemann–Liouville and (k , ψ) -Caputo proportional fractional operators. Some properties and auxiliary results of the proposed operators are investigated. The ψ-Laplace transform and its properties of the proposed operators are established and utilized to solve Cauchy-type problems. Furthermore, the uniqueness result for a higher-order initial value problem under (k , ψ) -Hilfer proportional fractional operators is proved by using Picard's iterative technique. At the end, examples are provided to present the theoretical results. This new type of proposed operator can help other researchers who are still working on real-world problems. [ABSTRACT FROM AUTHOR]

Abstract This paper focuses on exploring the existence of solutions for a specific class of FDEs by leveraging fixed point theorem. The equation in question features the Caputo fractional derivative of order 3 < u ˆ ≤ 4 $3

Abstract We provide here a novel approach for solving IVPs in ODEs and MTFDEs numerically by means of a class of MSJPs. Using the SCM, we build OMs for RIs and RLFI for MSJPs as part of our process. These architectures guarantee accurate and efficient numerical computations. We provide theoretical assurances for the efficacy of an algorithm by establishing its convergence and error analysis features. We offer five numerical examples to prove that our method is accurate and applicable. Through these examples, we demonstrate the greater accuracy and efficiency of our approach by comparing our results with previously published findings. Tables and graphs show that the method produces exact and approximate solutions that agree quite well with each other.

This research article introduces a novel approach based on the fuzzy Adomian decomposition method (FADM) to solve specific time fuzzy fractional partial differential equations with initial and boundary conditions (IBCs). The proposed approach addresses the challenge of incorporating both initial and boundary conditions into the FADM framework by employing a modified approach. This approach iteratively generates a new initial solution using the decomposition method. The method presented here offers a significant contribution to solving fuzzy fractional partial differential equations (FFPDEs) with fuzzy IBCs, a topic that has received limited attention in the literature. Furthermore, it satisfies a high convergence rate with minimal computational complexity, establishing a novel aspect of this research. By providing a series solution with a small number of recursive formulas, this method enhances accuracy and emerges as a preferred choice for tackling FFPDEs with mixed initial and boundary conditions. The effectiveness of the proposed technique is further supported by the inclusion of several illustrative examples. [ABSTRACT FROM AUTHOR]

The present research work investigates some new results for a fractional generalized Sturm–Liouville–Langevin (FGSLL) equation involving the Ψ-Caputo fractional derivative with a modified argument. We prove the uniqueness of the solution using the Banach contraction principle endowed with a norm of the Ψ-Bielecki-type. Meanwhile, the fixed-point theorems of the Leray–Schauder and Krasnoselskii type associated with the Ψ-Bielecki-type norm are used to derive the existence properties by removing some strong conditions. We use the generalized Gronwall-type inequality to discuss Ulam–Hyers (UH), generalized Ulam–Hyers (GUH), Ulam–Hyers–Rassias (UHR), and generalized Ulam–Hyers–Rassias (GUHR) stability of these solutions. Lastly, three examples are provided to show the effectiveness of our main results for different cases of (FGSLL)-problem such as Caputo-type Sturm–Liouville, Caputo-type Langevin, Caputo–Erdélyi–Kober-type Langevin problems. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study a generalized Riemann–Liouville fractional differential equation and system with nonlocal boundary conditions. Firstly, some properties of the Green function are presented and then Lyapunov-type inequalities for a sequential ψ-Riemann–Liouville fractional boundary value problem are established. Also, the existence and uniqueness of solutions are proved by using Banach and Schauder fixed-point theorems. Furthermore, the existence and uniqueness of solutions to a sequential nonlinear differential system is established by means of Schauder's and Perov's fixed-point theorems. Examples are given to validate the theoretical results. [ABSTRACT FROM AUTHOR]

This work is devoted to using topological degree theory to establish a mathematical analysis for a class of fractional-order evolution hybrid differential equations using a modified Mittag–Leffler-type derivative. In addition, two kinds of Ulam–Hyers (U–H) stability results are deduced for the mentioned problem. A pertinent example is given to verify the results. [ABSTRACT FROM AUTHOR]

We provide here a novel approach for solving IVPs in ODEs and MTFDEs numerically by means of a class of MSJPs. Using the SCM, we build OMs for RIs and RLFI for MSJPs as part of our process. These architectures guarantee accurate and efficient numerical computations. We provide theoretical assurances for the efficacy of an algorithm by establishing its convergence and error analysis features. We offer five numerical examples to prove that our method is accurate and applicable. Through these examples, we demonstrate the greater accuracy and efficiency of our approach by comparing our results with previously published findings. Tables and graphs show that the method produces exact and approximate solutions that agree quite well with each other. [ABSTRACT FROM AUTHOR]

This paper focuses on exploring the existence of solutions for a specific class of FDEs by leveraging fixed point theorem. The equation in question features the Caputo fractional derivative of order 3 < u ˆ ≤ 4 and includes a term Θ (β , Z (β)) alongside boundary conditions. Through the application of a fixed point theorem in appropriate function spaces, we consider nonlocal conditions along with necessary assumptions under which solutions to the given FDE exist. Furthermore, we offer an example to illustrate the results. [ABSTRACT FROM AUTHOR]

This article investigates sufficient conditions for the existence and uniqueness of solutions to the ψ-Hilfer sequential type pantograph fractional boundary value problem. Considering the system depends on a lower-order fractional derivative of an unknown function, the study is carried out in a special working space. Standard fixed point theorems such as the Banach contraction principle and Krasnosel'skii's fixed point theorem are applied to prove the uniqueness and the existence of a solution, respectively. Finally, an example demonstrating our results with numerical simulations is presented. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and uniqueness of positive solutions for a class of a fractional differential equation system of Riemann–Liouville type on infinite intervals with infinite-point boundary conditions. First, the higher-order equation is reduced to the lower-order equation, and then it is transformed into the equivalent integral equation. Secondly, we obtain the existence and uniqueness of positive solutions for each fixed parameter λ > 0 by using the mixed monotone operators fixed-point theorem. The results obtained in this paper show that the unique positive solution has good properties: continuity, monotonicity, iteration, and approximation. Finally, an example is given to demonstrate the application of our main results. [ABSTRACT FROM AUTHOR]

This paper solves fractional differential equations using the Shehu transform in combination with the q-homotopy analysis transform method (q-HATM). As the Shehu transform is only applicable to linear equations, q-HATM is an efficient technique for approximating solutions to nonlinear differential equations. In nonlinear systems that explain the emergence of stripes in 2D systems, the Newell–Whitehead–Segel equation plays a significant role. The findings indicate that the outcomes derived from the tables yield superior results compared to the existing LTDM in the literature. Maple is utilized to depict three-dimensional surfaces and find numerical values that are displayed in a table. [ABSTRACT FROM AUTHOR]

In this paper, we study fractional p 1 (x , ⋅) & p 2 (x , ⋅) -Laplacian Schrödinger-type equations for Robin boundary conditions. Under some suitable assumptions, we show that two solutions exist using the mountain pass lemma and Ekeland's variational principle. Then, the existence of infinitely many solutions is derived by applying the fountain theorem and the Krasnoselskii genus theory, respectively. Different from previous results, the topic of this paper is the Robin boundary conditions in R N ∖ Ω ‾ for fractional order p 1 (x , ⋅) & p 2 (x , ⋅) -Laplacian Schrödinger-type equations, including concave-convex nonlinearities, which has not been studied before. In addition, two examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

Over the years, the research of backward stochastic differential equations (BSDEs) has come a long way. As a extension of the BSDEs, the BSDEs with time delay have played a major role in the stochastic optimal control, financial risk, insurance management, pricing, and hedging. In this paper, we study a class of BSDEs with time-delay generators driven by Caputo fractional derivatives. In contrast to conventional BSDEs, in this class of equations, the generator is also affected by the past values of solutions. Under the Lipschitz condition and some new assumptions, we present a theorem on the existence and uniqueness of solutions. [ABSTRACT FROM AUTHOR]

In this work, we analyze a q-fractional jerk problem having anti-periodic boundary conditions. The focus is on investigating whether a unique solution exists and remains stable under specific conditions. To prove the uniqueness of the solution, we employ a Banach fixed point theorem and a mathematical tool for establishing the presence of distinct fixed points. To demonstrate the availability of a solution, we utilize Leray–Schauder's alternative, a method commonly employed in mathematical analysis. Furthermore, we examine and introduce different kinds of stability concepts for the given problem. In conclusion, we present several examples to illustrate and validate the outcomes of our study. [ABSTRACT FROM AUTHOR]

In this research, we investigate the existence and multiplicity of solutions for fractional differential equations on infinite intervals. By using monotone iteration, we identify two solutions, and the multiplicity of solutions is demonstrated by the Leggett–Williams fixed point theorem. [ABSTRACT FROM AUTHOR]

In this paper, we propose the notion of extended neutrosophic rectangular metric space and prove some fixed point results under contraction mapping. Finally, as an application of the obtained results, we prove the existence and uniqueness of the Caputo fractional differential equation. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence and uniqueness of solutions for the following nonlinear multi-order fractional differential equation with integral boundary conditions { (C D 0 + α u) (t) + ∑ i = 1 m λ i (t) (C D 0 + α i u) (t) + ∑ j = 1 n μ j (t) (C D 0 + β j u) (t) + ∑ k = 1 p ξ k (t) (C D 0 + γ k u) (t) + ∑ l = 1 q ω l (t) (C D 0 + δ l u) (t) + σ (t) u (t) + f (t , u (t)) = 0 , t ∈ [ 0 , 1 ] , u ″ (0) = u ‴ (0) = 0 , u ′ (0) = η 1 ∫ 0 1 u (s) d s , u (1) = η 2 ∫ 0 1 u (s) d s , where 0 < δ 1 < δ 2 < ⋯ < δ q < 1 < γ 1 < γ 2 < ⋯ < γ p < 2 < β 1 < β 2 < ⋯ < β n < 3 < α 1 < α 2 < ⋯ < α m < α < 4 and η 1 + 2 (1 − η 2) ≠ 0 . Using a fixed point theorem and Banach contractive mapping principle, we obtain some existence and uniqueness results. [ABSTRACT FROM AUTHOR]

Abstract We introduce a notion of nonlinear cyclic orbital ( ξ − F ) $(\xi -\mathscr{F})$ -contraction and prove related results. With these results, we address the existence and uniqueness results with periodic/anti-periodic boundary conditions for: 1. The nonlinear multi-order fractional differential equation L ( D ) θ ( ς ) = σ ( ς , θ ( ς ) ) , ς ∈ J = [ 0 , A ] , A > 0 , $$ \mathcal{L}(\mathcal{D})\theta (\varsigma )=\sigma \bigl(\varsigma , \theta ( \varsigma ) \bigr), \quad \varsigma \in \mathscr{J}=[0,\mathscr{A}], \mathscr{A}>0, $$ where L ( D ) = γ w c D δ w + γ w − 1 c D δ w − 1 + ⋯ + γ 1 c D δ 1 + γ 0 c D δ 0 , γ ♭ ∈ R ( ♭ = 0 , 1 , 2 , 3 , … , w ) , γ w ≠ 0 , 0 ≤ δ 0 < δ 1 < δ 2 < ⋯ < δ w − 1 < δ w < 1 ; $$\begin{aligned} &\mathcal{L}(\mathcal{D})=\gamma _{w} \,{}^{c} \mathcal{D}^{\delta _{w}}+ \gamma _{w-1} \,{}^{c} \mathcal{D}^{\delta _{w-1}}+\cdots+\gamma _{1} \,{}^{c} \mathcal{D}^{\delta _{1}}+\gamma _{0} \,{}^{c} \mathcal{D}^{\delta _{0}},\\ &\gamma _{\flat}\in \mathbb{R}\quad (\flat =0,1,2,3,\ldots,w), \qquad \gamma _{w} \neq 0, \\ &0\leq \delta _{0}< \delta _{1}< \delta _{2}< \cdots< \delta _{w-1}< \delta _{w}< 1; \end{aligned}$$ 2. The nonlinear multi-term fractional delay differential equation L ( D ) θ ( ς ) = σ ( ς , θ ( ς ) , θ ( ς − τ ) ) , ς ∈ J = [ 0 , A ] , A > 0 ; θ ( ς ) = σ ¯ ( ς ) , ς ∈ [ − τ , 0 ] , $$\begin{aligned} &\mathcal{L}(\mathcal{D})\theta (\varsigma ) =\sigma \bigl(\varsigma , \theta ( \varsigma ),\theta (\varsigma -\tau ) \bigr), \quad \varsigma \in \mathscr{J}=[0, \mathscr{A}], \mathscr{A}>0; \\ &\theta (\varsigma ) =\bar{\sigma}(\varsigma ),\quad \varsigma \in [-\tau ,0], \end{aligned}$$ where L ( D ) = γ w c D δ w + γ w − 1 c D δ w − 1 + ⋯ + γ 1 c D δ 1 + γ 0 c D δ 0 , γ ♭ ∈ R ( ♭ = 0 , 1 , 2 , 3 , … , w ) , γ w ≠ 0 , 0 ≤ δ 0 < δ 1 < δ 2 < ⋯ < δ w − 1 < δ w < 1 ; $$\begin{aligned} &\mathcal{L}(\mathcal{D})=\gamma _{w} \,{}^{c} \mathcal{D}^{\delta _{w}}+ \gamma _{w-1} \,{}^{c} \mathcal{D}^{\delta _{w-1}}+\cdots+\gamma _{1} \,{}^{c} \mathcal{D}^{\delta _{1}}+\gamma _{0} \,{}^{c} \mathcal{D}^{\delta _{0}},\\ &\gamma _{\flat}\in \mathbb{R}\quad (\flat =0,1,2,3,\ldots,w), \qquad \gamma _{w} \neq 0, \\ &0\leq \delta _{0}< \delta _{1}< \delta _{2}< \cdots< \delta _{w-1}< \delta _{w}< 1; \end{aligned}$$ moreover, here D δ c ${}^{c}\mathcal{D}^{\delta}$ is predominantly called Caputo fractional derivative of order δ.

The purpose of this study is to establish fixed-point results for new interpolative contraction mappings in the setting of Busemann space involving a convex hull. To illustrate our findings, we also offer helpful and compelling examples. Finally, the theoretical results are applied to study the existence of solutions to fractional Navier–Stokes and fractional-functional differential equations as applications. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of solutions for a boundary value problem of singular two-term fractional differential equation with delay and sign-changing nonlinearity. By means of the Guo–Krasnosel'skii fixed point theorem and the Leray–Schauder nonlinear alternative theorem, we obtain some results on the existence and multiplicity of solutions, respectively. [ABSTRACT FROM AUTHOR]

This paper presents a new way to solve numerically multiterm variable-order fractional differential equations (MTVOFDEs) with initial conditions by using a class of modified shifted Jacobi polynomials (MSJPs). As their defining feature, MSJPs satisfy the given initial conditions. A key aspect of our methodology involves the construction of operational matrices (OMs) for ordinary derivatives (ODs) and variable-order fractional derivatives (VOFDs) of MSJPs and the application of the spectral collocation method (SCM). These constructions enable efficient and accurate numerical computation. We establish the error analysis and the convergence of the proposed algorithm, providing theoretical guarantees for its effectiveness. To demonstrate the applicability and accuracy of our method, we present five numerical examples. Through these examples, we compare the results obtained with other published results, confirming the superiority of our method in terms of accuracy and efficiency. The suggested algorithm yields very accurate agreement between the approximate and exact solutions, which are shown in tables and graphs. [ABSTRACT FROM AUTHOR]

Abstract This paper aims to extend the Caputo–Atangana–Baleanu ( A B C $ABC$ ) and Riemann–Atangana–Baleanu ( A B R $ABR$ ) fractional derivatives with respect to another function, from fractional order μ ∈ ( 0 , 1 ] $\mu \in (0,1]$ to an arbitrary order μ ∈ ( n , n + 1 ] $\mu \in (n,n+1]$ , n = 0 , 1 , 2 , … $n=0,1,2,\dots $ . Also, their corresponding Atangana–Baleanu (AB) fractional integral is extended. Additionally, several properties of such definitions are proved. Moreover, the generalization of Gronwall’s inequality in the framework of the AB fractional integral with respect to another function is introduced. Furthermore, Picard’s iterative method is employed to discuss the existence and uniqueness of the solution for a higher-order initial fractional differential equation involving an A B C $ABC$ operator with respect to another function. Finally, examples are given to illustrate the effectiveness of the main findings. The idea of this work may attract many researchers in the future to study some inequalities and fractional differential equations that are related to AB fractional calculus with respect to another function.

We introduce a notion of nonlinear cyclic orbital (ξ − F) -contraction and prove related results. With these results, we address the existence and uniqueness results with periodic/anti-periodic boundary conditions for: 1. The nonlinear multi-order fractional differential equation L (D) θ (ς) = σ (ς , θ (ς)) , ς ∈ J = [ 0 , A ] , A > 0 , where L (D) = γ w c D δ w + γ w − 1 c D δ w − 1 + ⋯ + γ 1 c D δ 1 + γ 0 c D δ 0 , γ ♭ ∈ R (♭ = 0 , 1 , 2 , 3 , ... , w) , γ w ≠ 0 , 0 ≤ δ 0 < δ 1 < δ 2 < ⋯ < δ w − 1 < δ w < 1 ; 2. The nonlinear multi-term fractional delay differential equation L (D) θ (ς) = σ (ς , θ (ς) , θ (ς − τ)) , ς ∈ J = [ 0 , A ] , A > 0 ; θ (ς) = σ ¯ (ς) , ς ∈ [ − τ , 0 ] , where L (D) = γ w c D δ w + γ w − 1 c D δ w − 1 + ⋯ + γ 1 c D δ 1 + γ 0 c D δ 0 , γ ♭ ∈ R (♭ = 0 , 1 , 2 , 3 , ... , w) , γ w ≠ 0 , 0 ≤ δ 0 < δ 1 < δ 2 < ⋯ < δ w − 1 < δ w < 1 ; moreover, here D δ c is predominantly called Caputo fractional derivative of order δ. [ABSTRACT FROM AUTHOR]

This paper investigates the existence of positive solutions for a class of fractional boundary value problems involving an improper integral and the infinite-point on the half-line by making use of properties of the Green function and Avery–Peterson fixed point theorem. In addition, an example is presented to illustrate the applicability of our main result. [ABSTRACT FROM AUTHOR]

In this paper, the existence of the multiple positive solutions for a class of higher-order fractional differential equations on infinite intervals with infinite-point boundary value conditions is mainly studied. First, we construct the Green function and analyze its properties, and then by using the Leggett–Williams fixed point theorem, some new results on the existence of positive solutions for the boundary value problem are obtained. Finally, we illustrate the application of our conclusion by an example. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new coupled system of sequential fractional differential equations with coupled boundary conditions. We establish existence and uniqueness results using the Leray–Schauder alternative and Banach contraction principle. We examine the stability of the solutions involved in the Hyers–Ulam type. As an application, we present a few examples to illustrate the main results. [ABSTRACT FROM AUTHOR]

Variable order differential equations are the natural extension of the said area. In many situations, such problems have the ability to describe real-world problems more concisely. Therefore, keeping this validity in mind, we have considered a class of boundary value problems (BVPs) under the variable order differentiation. For the suggested problems, we have developed the existence and uniqueness (EU) by using some fixed point results due to Banach and Schauder. Sufficient adequate results have been established for the required need. Some stability results have also been elaborated based on the concepts of Ulam, Hyers, and Rassias. Proper examples have also been provided with detailed analysis to verify our results. [ABSTRACT FROM AUTHOR]

This paper aims to extend the Caputo–Atangana–Baleanu (A B C ) and Riemann–Atangana–Baleanu (A B R ) fractional derivatives with respect to another function, from fractional order μ ∈ (0 , 1 ] to an arbitrary order μ ∈ (n , n + 1 ] , n = 0 , 1 , 2 , ... . Also, their corresponding Atangana–Baleanu (AB) fractional integral is extended. Additionally, several properties of such definitions are proved. Moreover, the generalization of Gronwall's inequality in the framework of the AB fractional integral with respect to another function is introduced. Furthermore, Picard's iterative method is employed to discuss the existence and uniqueness of the solution for a higher-order initial fractional differential equation involving an A B C operator with respect to another function. Finally, examples are given to illustrate the effectiveness of the main findings. The idea of this work may attract many researchers in the future to study some inequalities and fractional differential equations that are related to AB fractional calculus with respect to another function. [ABSTRACT FROM AUTHOR]

In this paper, we consider a mathematical model of a coronavirus disease involving the Caputo–Fabrizio fractional derivative by dividing the total population into the susceptible population S (t) , the vaccinated population V (t) , the infected population I (t) , the recovered population R (t) , and the death class D (t) . A core goal of this study is the analysis of the solution of a proposed mathematical model involving nonlinear systems of Caputo–Fabrizio fractional differential equations. With the help of Lipschitz hypotheses, we have built sufficient conditions and inequalities to analyze the solutions to the model. Eventually, we analyze the solution for the formed mathematical model by employing Krasnoselskii's fixed point theorem, Schauder's fixed point theorem, the Banach contraction principle, and Ulam–Hyers stability theorem. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the existence and multiplicity of ρ-concave positive solutions for a p-Laplacian boundary value problem of two-sided fractional differential equations involving generalized-Caputo fractional derivatives. Using Guo–Krasnoselskii fixed point theorem and under some additional assumptions, we prove some important results and obtain the existence of at least three solutions. To establish the results, Green functions are used to transform the considered two-sided generalized Katugampola and Caputo fractional derivatives. Finally, applications with illustrative examples are presented to show the validity and correctness of the obtained results.

Abstract This work deals with the large deviation principle which studies the decay of probabilities of certain kind of extremely rare events. We consider stochastic neutral fractional functional differential equation with multiplicative noise and show large deviation principle for its solution processes in a suitable Polish space. The existence and uniqueness results are presented using the Picard iterative method, which is indeed essential for further analysis. The establishment of Freidlin–Wentzell type large deviation principle is solely based on the variational representation developed by Budhiraja and Dupuis in which the weak convergence technique is used to show the sufficient condition. Examples are provided to emphasize the theory.

This research inscription gets to grips with two novel varieties of boundary value problems. One of them is a hybrid Langevin fractional differential equation, whilst the other is a coupled system of hybrid Langevin differential equation encapsuling a collective fractional derivative known as the ψ-Caputo fractional operator. Such operators are generated by iterating a local integral of a function with respect to another increasing positive function Ψ. The existence of the solutions of the aforehand equations is tackled by using the Dhage fixed point theorem, whereas their uniqueness is handled using the Banach fixed point theorem. On the top of this, the stability within the scope of Ulam–Hyers of solutions to these systems are also considered. Two pertinent examples are presented to corroborate the reported results. [ABSTRACT FROM AUTHOR]

In this paper, we consider fractional neutral differential equations with multipoint boundary value conditions involving Hadamard derivatives and integrals. We obtain the existence and uniqueness of the solution of the equation by using several fixed point theorems, and we also consider the Ulam–Hyers stability of the solution. In addition, we study the differential inclusion problem with multipoint boundary value conditions and prove the existence of the solution of the boundary value problem when the multivalued map has convex values. We also give several examples to illustrate the feasibility of the results. [ABSTRACT FROM AUTHOR]

We prove the existence of the solutions for the new mixed differential equations, which is characteristic of the right-sided Caputo and the left-sided Riemann–Liouville fractional derivatives. There are four major ingredients. The first is composed of some basic definitions and lemmas. The second is the Green's function of the new mixed fractional differential equations. We calculate the corresponding Green's functions as well as their properties. The third, which is the main new ingredient of this paper, is demonstration of the existence of the solutions for fractional equations by the fixed-point theorem in cone expansion and compression of norm type. The fourth, as applications, is the example provided to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and uniqueness of fractional differential equations (FDEs) by using the fixed-point theory (FPT). We discuss also the Ulam–Hyers–Rassias (UHR) stability of some generalized FDEs according to some classical mathematical techniques and the FPT. Finally, two illustrative examples are presented to show the validity of our results. [ABSTRACT FROM AUTHOR]

In this article, we present an iterative transformation method for solving fractional partial differential equations that combines the Elzaki transform and iterative methods. By this iterative transformation method, numerical solutions in the form of series are obtained. When we apply this method to the fractional linear Klein–Gordon equation, we find that it yields the same results, just like the Homotopy perturbation method. The procedures and results of this method for solving the new generalized fractional Hirota–Satsuma coupled KdV equation are given in the paper. [ABSTRACT FROM AUTHOR]