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In this paper, we consider a Cauchy problem for a Caputo‐type time delay linear system of fractional differential equations with permutable matrices. First, we provide a new representation of solutions to linear homogeneous fractional differential equations using the Laplace integral transform and variation of constants formula via a newly defined delayed Mittag‐Leffler type matrix function introduced through a three‐parameter Mittag‐Leffler function. Second, with the help of a delayed perturbation of a Mittag‐Leffler type matrix function, we attain an explicit formula for solutions to a linear nonhomogeneous time delay fractional order system using the superposition principle. Furthermore, we prove the existence and uniqueness of solutions to nonlinear fractional delay differential equations using the contraction mapping principle. Finally, we present an example to illustrate the applicability of our results. [ABSTRACT FROM AUTHOR]

In this paper, we provide an extension result for the existence and uniqueness of solutions for a coupled system of fractional Langevin differential equations with antiperiodic boundary conditions. The system involves two different Caputo fractional derivatives defined on different intervals and associated with boundary conditions described by sequential fractional derivatives. As a conclusion for our main result, we deduce the results of H. Fazli, J.J. Nieto, Fractional Langevin equation with anti‐periodic boundary conditions, Chaos Soliton Fract, 114 (2018),332–337 under less restrictive conditions. The consistency of the main results is demonstrated by two numerical examples. For the sake of completeness, we end the paper by a concluding discussion. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to obtain the existence of at least one solution to the following coupled system of nonlinear fractional order differential equations under the integral type boundary conditions by using topological degree theory 0.1Dδμ(ℓ)=F1(ℓ,μ(λℓ),ν(λℓ)),ℓ∈[0,1],Dϱν(ℓ)=F2(ℓ,μ(λℓ),ν(λℓ)),ℓ∈[0,1],μ(0)=r(μ),μ(1)=1Γ(δ)∫01(1−η)δ−1φ(η,μ(η))dη,ν(0)=h(ν),ν(1)=1Γ(ϱ)∫01(1−η)ϱ−1ψ(η,ν(η))dη, where δ,ϱ∈(1,2], 0<λ<1, D denotes the standard Caputo fractional derivative, F1,F2:[0,1]×ℜ×ℜ→ℜ, φ,ψ:[0,1]×ℜ→ℜ and r,h:[0,1]→ℜ are continuous functions. For this intention, some results for the existence of at least one solution are constructed. For the validity of our results, an appropriate example is presented. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of fractional equations with nonlinear boundary condition. We establish a new comparison principle related to linear fractional equation and show the existence of extreme solution by using monotone iterative method and lower and upper solutions method. [ABSTRACT FROM AUTHOR]

In this paper, the implementation of the Fractional Variational iteration method (FVIM) is used to approximate the generalized fractional perturbed Zakharove-Kuznetsov equation. The obtained results show that the suggested approach is very efficient for dealing with various forms of differential equations of fractional order. The numerical comparison for the attained resolution is made through the q-homotopy analysis transmute manner (q-HATM) which confirms the convergence of the devised method. The applicability of the method is observed by applying the proposed technique to two numerical examples. Furthermore, the solutions obtained by the FVIM are completely well-matched with the solutions presented in the literature. [ABSTRACT FROM AUTHOR]

We derive the prolongation formula of the one-parameter Lie point transformations to the Hilfer fractional derivative and show that the existing prolongation formula for the Riemann Liouville and Caputo fractional derivatives are special cases of the proposed formula, corresponding to the type parameter γ = 0 and γ = 1 , respectively. The applicability of the proposed formula is demonstrated by deriving the Lie point symmetries of the time-fractional heat equation, the fractional Burgers equation, and the fractional KdV equation in Hilfer's sense. We use the obtained Lie point symmetries to find the similarity variables and transformations. Using the similarity transformations, we show that each is converted into a nonlinear fractional ordinary differential equation with a new independent variable. The fractional derivative in the reduced equation can be either the Hilfer-type modification of the Erdélyi Kober fractional derivative or the Hilfer fractional derivative itself. We demonstrate that the exact solution of the time-fractional differential equation in the Hilfer sense can be reduced to the exact solutions of the corresponding time-fractional differential equations in the Riemann–Liouville and Caputo senses by setting the type parameter to γ = 0 and γ = 1 , respectively. [ABSTRACT FROM AUTHOR]

Fractional derivatives are suitable for describing several physical phenomena. The construction of efficient analytical and numerical methods for the solutions of ordinary and partial fractional differential equations is an active research area and it is of great interest to the researchers. The Caputo fractional derivative is of great use in the modelling and simulation of phenomena where consideration is given to the interactions within the past and problems with nonlocal properties. This study considers the use of a hybrid of the Sumudu Transform method for constructing the solution of nonlinear equations that describe the processes in the functional and structural materials. This study considers the models that are given by the integer-order derivatives, Caputo derivatives of fractional variable orders and Caputo derivatives of fractional variable orders that are associated with delays. The study applies a hybrid of Sumudu Transform to present solutions for each considered model and makes use of graphs to show the correlation among the models. The study is of great importance in the numerical and experimental characterization of the decay properties of functional and structural materials. [ABSTRACT FROM AUTHOR]

For solving the semilinear fractional differential equations with the nonsmooth force term, we construct a class of Runge–Kutta-based convolution quadrature. Moreover, we analyse the convergence of the proposed scheme. In addition, we employ the fast Runge–Kutta approximation to reduce the calculation cost. Finally, we give some numerical experiments to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this study, using Caputo and ABC derivatives, we present a mathematical analysis of two fractional models for poliomyelitis, considering the presence of vaccination (V) and a post‐paralytic class (A). The existence and uniqueness of solutions are proved. The basic reproduction number R0$$ {\mathcal{R}}_0 $$ is computed. Local and global stability of the disease‐free stationary state, depending on the threshold R0$$ {\mathcal{R}}_0 $$, is provided, along with conditions for the existence of an endemic stationary state. Moreover, we performed a sensitivity analysis to study the influence of all biological parameters on R0$$ {\mathcal{R}}_0 $$. We concluded our study with numerical simulations to illustrate the models' dynamics and to compare the trajectories of Caputo and ABC solutions. We found that the Caputo and ABC operators are both convenient for the modelization of the poliomyelitis disease. However, the ABC operator not only refined the Caputo operator by removing singularity from the kernel expression but also brought out heredity and memory in the model's characteristics. [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]

This interdisciplinary study critically analyzes current research, establishing a profound connection between sea water, sea ice, sea temperature, and surface temperature through a 4D hyperchaotic Caputo fractional differential equation. Emphasizing the collective impact on climate, focusing on challenges from anthropogenic global warming, the study scrutinizes theoretical aspects, including existence and uniqueness. Two sliding mode controllers manage chaos in this 4D fractional system, assessed amid uncertainties and disruptions. The global stability of these controlled systems is also confirmed, considering both commensurate and non-commensurate 4D fractional order. To demonstrate the intricate chaotic motion within the system, we employ the Lyapunov exponent and Poincare sections. Numerical simulations are conducted by using the predictor-corrector method. The effects of surface temperature on chaotic dynamics are discussed. The crucial role of sea ice reflection in climate stability is highlighted in two scenarios. Correlation graphs, comparing model and observational data using the predictor-corrector method, enhance the proposed 4D hyperchaotic model's credibility. Subsequently, numerical simulations validate theoretical assertions about the controllers' influence. These controllers indicate which variable significantly contributes to controlling the chaos. [ABSTRACT FROM AUTHOR]

In the context of forward uncertainty quantification, we investigate the convergence of the Galerkin projections for random fractional differential equations. The governing system is formed by a finite set of independent input random parameters (a germ) and by a fractional derivative in the Caputo sense. Input uncertainty arises from biased measurements, and a fractional derivative, defined by a convolution, takes past history into account. While numerical experiments on the gPC-based Galerkin method are already available in the literature for random ordinary, partial and fractional differential equations, a theoretical analysis of mean-square convergence is still lacking for the fractional case. The aim of this contribution is to fill this gap, by establishing new inequalities and results and by raising new open problems. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the tempered subdiffusive Black–Scholes model. The main part of our work consists of the finite difference method as a numerical approach to option pricing in the considered model. We derive the governing fractional differential equation and the related weighted numerical scheme. The proposed method has an accuracy order 2 - α with respect to time, where α ∈ (0 , 1) is the subdiffusion parameter and 2 with respect to space. Furthermore, we provide stability and convergence analysis. Finally, we present some numerical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of finding the order 0<α<1$$ 0<\alpha <1 $$ in the fractional discrete system: C∇αun=Aun,n≥1,u0=x0,$$ \left\{\begin{array}{cll}{\kern0.1em }_C{\nabla}^{\alpha }{u}^n& =& A{u}^n,\kern0.30em n\ge 1,\\ {}{u}^0& =& {x}_0,\end{array}\right. $$where A$$ A $$ is a closed linear operator defined in a Banach space X,x0∈X$$ X,{x}_0\in X $$, and C∇αun$$ {\kern0.1em }_C{\nabla}^{\alpha }{u}^n $$ is the discrete Caputo fractional derivative of a given vector‐valued sequence (un)n∈ℕ0$$ {\left({u}^n\right)}_{n\in {\mathrm{\mathbb{N}}}_0} $$. [ABSTRACT FROM AUTHOR]

The symmetry features of fractional differential equations allow effective explanation of physical and biological phenomena in nature. The generalized form of the fractional differential equations is the variable-order fractional differential equations that describe the physical and biological applications. This paper discusses the closed-form traveling wave solutions for the nonlinear space-time variable-order fractional modified Kawahara and (2 + 1)-dimensional Burger hierarchy equations. The variable-order fractional differential equation has a derivative operator in the Caputo sense that is converted into the integer-order ordinary differential equation (ODE) by fractional transformation. The obtained ODE is solved by the exponential rational function method, and as a result, new exact solutions are constructed. Two problems are proposed to confirm the solutions of the space-time variable-order fractional differential equations. [ABSTRACT FROM AUTHOR]

Fractional pantograph stochastic differential equations (FPSDEs) combine elements of fractional calculus, pantograph equations, and stochastic processes to model complex systems with memory effects, time delays, and random fluctuations. Ensuring the well-posedness of these equations is crucial as it guarantees meaningful, reliable, and applicable solutions across various disciplines. In differential equations, regularity refers to the smoothness of solution behavior. The averaging principle offers an approximation that balances complexity and simplicity. Our research contributes to establishing the well-posedness, regularity, and averaging principle of FPSDE solutions in L p spaces with p ≥ 2 under Caputo derivatives. The main ingredients in the proof include the use of Hölder, Burkholder–Davis–Gundy, Jensen, and Grönwall–Bellman inequalities, along with the interval translation approach. To understand the theoretical results, we provide numerical examples at the end. [ABSTRACT FROM AUTHOR]

In this paper, the Caputo fractional derivative is assumed to be the prey–predator model. In order to create Caputo fractional differential equations for the prey–predator model, a discretization process is first used. The fixed points of the model are categorized topologically. We identify requirements for the fixed points of the suggested prey–predator model's local asymptotic stability. We demonstrate analytically that, under specific parametric conditions, a fractional order prey–predator model supports both a Neimark–Sacker (NS) bifurcation and a Flip bifurcation. We present evidence for NS and Flip bifurcations using central manifold and bifurcation theory. The parameter values and the initial conditions have been found to have a profound impact on the dynamical behavior of the fractional order prey–predator model. As the bifurcation parameter is increased, the system displays chaotic behavior. Numerical simulations are shown to demonstrate chaotic behaviors like bifurcations, phase portraits, invariant closed cycles, and attractive chaotic sets in addition to validating analytical conclusions. The suggested prey–predator dynamical system's chaotic behavior will be controlled by the OGY and hybrid control methodology, which will also visualize the chaotic state for various biological parameters. [ABSTRACT FROM AUTHOR]

The partial neutral functional fractional differential equation described by the fractional operator is considered in the present investigation. The used fractional operator is the Caputo derivative. In the present paper, the fractional resolvent operators have been defined and used to prove the existence of the unique solution of the fractional neutral differential equations. The fixed point theorem has been used in existence investigations. For an illustration of our results in this paper, an example has been provided as well. [ABSTRACT FROM AUTHOR]

The novel coronavirus SARS-Cov-2 is a pandemic condition and poses a massive menace to health. The governments of different countries and their various prohibitory steps to restrict the virus’s expanse have changed individuals’ communication processes. Due to physical and financial factors, the population’s density is more likely to interact and spread the virus. We establish a mathematical model to present the spread of the COVID-19 in worldwide. In this article, we propose a novel mathematical model (‘SLIIqIhRP’) to assess the impact of using hospitalization, quarantine measures, and pathogen quantity in controlling the COVID-19 pandemic. We analyse the boundedness of the model’s solution by employing the Laplace transform approach to solve the fractional Gronwall’s inequality. To ensure the uniqueness and existence of the solution, we rely on the Picard-Lindelof theorem. The model’s basic reproduction number, a crucial indicator of epidemic potential, is determined based on the greatest eigenvalue of the next-generation matrix. We then employ stability theory of fractional differential equations to qualitatively examine the model. Our findings reveal that both locally and globally, the endemic equilibrium and disease-free solutions demonstrate symptomatic stability. These results shed light on the effectiveness of the proposed interventions in managing and containing the COVID-19 outbreak. [ABSTRACT FROM AUTHOR]

In this study, we investigate novel fractional tuberculosis model with Caputo fractional derivative. The computational solution of the fractional tuberculosis disease is obtained with the help of the generalized Euler’s method (GEM). This article considers two treatment strategies: protective treatment for latent populations and main treatment for infected ones. The compartmental structure of the six-dimensional model includes the susceptible, latent, infected, recovered, and treatment classes. To examine the nature of differential equations of fractional order that arise in biological illness, the findings produced utilizing the method under consideration are more accurate and straightforward. Additionally, the stability of the equilibrium point is also investigated. The fractional model has been graphically simulated using MATLAB22. [ABSTRACT FROM AUTHOR]

In this paper, we examine the time-dependent two-dimensional cable equation of fractional order in terms of the Caputo fractional derivative. This cable equation plays a vital role in diverse areas of electrophysiology and modeling neuronal dynamics. This paper conveys a precise semi-analytical method called the q-homotopy analysis transform method to solve the fractional cable equation. The proposed method is based on the conjunction of the q-homotopy analysis method and Laplace transform. We explained the uniqueness of the solution produced by the suggested method with the help of Banach's fixed-point theory. The results obtained through the considered method are in the form of a series solution, and they converge rapidly. The obtained outcomes were in good agreement with the exact solution and are discussed through the 3D plots and graphs that express the physical representation of the considered equation. It shows that the proposed technique used here is reliable, well-organized and effective in analyzing the considered non-homogeneous fractional differential equations arising in various branches of science and engineering. [ABSTRACT FROM AUTHOR]

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In response to the challenges posed by complex boundary conditions and singularities in molecular systems and quantum chemistry, accurately determining energy levels (eigenvalues) and corresponding wavefunctions (eigenfunctions) is crucial for understanding molecular behavior and interactions. Mathematically, eigenvalues and normalized eigenfunctions play crucial role in proving the existence and uniqueness of solutions for nonlinear boundary value problems (BVPs). In this paper, we present an iterative procedure for computing the eigenvalues (μ ) and normalized eigenfunctions of novel fractional singular eigenvalue problems, D 2 α y (t) + k t α D α y (t) + μ y (t) = 0 , 0 < t < 1 , 0 < α ≤ 1 , with boundary condition, y ′ (0) = 0 , y (1) = 0 , where D α , D 2 α represents the Caputo fractional derivative, k ≥ 1 . We propose a novel method for computing Lagrange multipliers, which enhances the variational iteration method to yield convergent solutions. Numerical findings suggest that this strategy is simple yet powerful and effective. [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]

This study presents an enhanced fractional-order mathematical model for analyzing the dynamics of Klebsiella pneumonia infections and antibiotic resistance over time. The model incorporates fractional Caputo derivative operators and kernel, to provide a more comprehensive understanding of the complex temporal dynamics. The model consists of three groups: Susceptible (S), Infected (I), and Resistant (R) individuals, each controlled by a fractional differential equation. The model represents the interaction between infection, recovery from infection, and the possible development of antibiotic resistance in susceptible individuals. The existence, uniqueness, stability, and alignment of the model's prediction to the observed data were analyzed and buttressed with numerical simulations. The results show that imipenem has the highest efficacy compared with ertapenem and meropenem category drugs. The estimated reproduction number and reproduction coefficient illustrate the potential impact of this model in improving treatment strategies, while the memory effects highlight the advantages of fractional differentiation. The model predicts an increased possibility of antibiotic resistance despite effective treatment, suggesting a new treatment approach. [ABSTRACT FROM AUTHOR]

Around the world, the people are simultaneously susceptible to or infected with several infections. This work aims at the analysis of the dynamics of transmission of two deadly viruses, COVID-19 and Influenza, using a co-infection epidemiological model by applying the Caputo fractional derivative. Fractional differential equations are currently used worldwide to model physical and biological phenomena. Our comprehension of complicated phenomena is improved when fractional-order derivatives are used to model systems with memory effects and long-range interactions. Mathematical depictions of infectious disease dynamics and dissemination across communities are provided by epidemiological models, which are essential resources for understanding and controlling infectious diseases. These models support informed decision making to prevent outbreaks, evaluate intervention measures, and help researchers and policymakers understand how diseases spread. A subclass of epidemiological models called co-infection models focuses on studying the dynamics of several infectious illnesses that occur in the same population at the same time. They are especially useful in situations where people are simultaneously susceptible to or infected with several infections. Co-infection models provide information on the development of effective control techniques, the progression of disease, and the interactions between several pathogens. The qualitative study via stability analysis is discussed at equilibrium, the reproduction number R 0 is computed, and the results are simulated using the Laplace Adomian Decomposition Method (LADM) for Fractional Differential Equations. We employ MATLAB R2023a for graphical presentations and numerical simulations. [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]

We propose a novel dynamical model for blood alcohol concentration that incorporates ψ$$ \psi $$‐Caputo fractional derivatives. Using the generalized Laplace transform technique, we successfully derive an analytic solution for both the alcohol concentration in the stomach and the alcohol concentration in the blood of an individual. These analytical formulas provide us a straightforward numerical scheme, which demonstrates the efficacy of the ψ$$ \psi $$‐Caputo derivative operator in achieving a better fit to real experimental data on blood alcohol levels available in the literature. In comparison with existing classical and fractional models found in the literature, our model outperforms them significantly. Indeed, by employing a simple yet nonstandard kernel function ψ(t)$$ \psi (t) $$, we are able to reduce the error by more than half, resulting in an impressive gain improvement of 59%. [ABSTRACT FROM AUTHOR]

We consider the l-th order linear fractional differential equations with constant coefficients. Here l ∈ ℕ is the ceiling for the highest derivative of order α, l - 1 < α ≤ l . If β i < α are the other derivatives, the existing theory requires α - max ⁡ { β i } ≥ l - 1 for the existence of l linearly independent solutions. Thus, at most one derivative may have order greater than one, but all other derivatives must be between zero and one. We remove this essential restriction and construct l linearly independent solutions. With this aim, we remodel the series approaches and elaborate the multi-sum fractional series method in order to obtain the existence and linear independence results. We consider both Riemann–Liouville or Caputo fractional derivatives. [ABSTRACT FROM AUTHOR]

The present work introduces a novel approach, the Adomian Decomposition Formable Transform Method (ADFTM), and its application to solve the fractional order Sharma-Tasso-Olver problem. The method's distinctive outcomes are highlighted through a comparative analysis with established non-local Caputo fractional derivatives and the non-singular Atangana–Baleanu (ABC) fractional derivatives. To provide a comprehensive understanding, the proposed ADFTM's approximate solution is compared with the homotopy perturbation method (HPM) and residual power series method (RPSM). Further, numerical and graphical results demonstrate the reliability and accuracy of the ADFTM approach. The novel outcomes presented in this work emphasize its capability to address complex engineering problems effectively. By demonstrating its efficacy in solving the fractional order problems, the new ADFTM proves to be a valuable tool in solving scientific problems. [ABSTRACT FROM AUTHOR]

This work explores the existence and uniqueness criteria for the solution of hybrid Caputo–Hadamard fractional sequential differential equations (HCHFSDEs) by employing Darbo's fixed-point theorem. Fractional differential equations play a pivotal role in modeling complex phenomena in various areas of science and engineering. The hybrid approach considered in this work combines the advantages of both the Caputo and Hadamard fractional derivatives, leading to a more comprehensive and versatile model for describing sequential processes. To address the problem of the existence and uniqueness of solutions for such hybrid fractional sequential differential equations, we turn to Darbo's fixed-point theorem, a powerful mathematical tool that establishes the existence of fixed points for certain types of mappings. By appropriately transforming the differential equation into an equivalent fixed-point formulation, we can exploit the properties of Darbo's theorem to analyze the solutions' existence and uniqueness. The outcomes of this research expand the understanding of HCHFSDEs and contribute to the growing body of knowledge in fractional calculus and fixed-point theory. These findings are expected to have significant implications in various scientific and engineering applications, where sequential processes are prevalent, such as in physics, biology, finance, and control theory. [ABSTRACT FROM AUTHOR]

Inequalities serve as fundamental tools for analyzing various important concepts in stochastic differential problems. In this study, we present results on the existence, uniqueness, and averaging principle for fractional neutral stochastic differential equations. We utilize Jensen, Burkholder–Davis–Gundy, Grönwall–Bellman, Hölder, and Chebyshev–Markov inequalities. We generalize results in two ways: first, by extending the existing result for p = 2 to results in the L p space; second, by incorporating the Caputo–Katugampola fractional derivatives, we extend the results established with Caputo fractional derivatives. Additionally, we provide examples to enhance the understanding of the theoretical results we establish. [ABSTRACT FROM AUTHOR]

Ulam type stability is an important property studied for different types of differential equations. When this type of stability is applied to boundary value problems, there are some misunderstandings in the literature. In connection with this, initially, we give a brief overview of the basic ideas of the application of Ulam type stability to initial value problems. We provide several examples with simulations to illustrate the main points in the application. Then, we focus on some misunderstandings in the application of Ulam stability to boundary value problems. We suggest a new way to avoid these misunderstandings and how to keep the main idea of Ulam type stability when it is applied to boundary value problems of differential equations. We present one possible way to connect both the solutions of the given problem and the solutions of the corresponding inequality. In addition, we provide several examples with simulations to illustrate the ideas for boundary value problems and we also show the necessity of the new way of applying the Ulam type stability. To illustrate the theoretical application of the suggested idea to Ulam type stability, we consider a linear boundary value problem for nonlinear impulsive fractional differential equations with the Caputo fractional derivative with respect to another function and piecewise-constant variable order. We define the Ulam–Hyers stability and obtain sufficient conditions on a finite interval. As partial cases, integral presentations of the solutions of boundary value problems for various types of fractional differential equations are obtained and their Ulam type stability is studied. [ABSTRACT FROM AUTHOR]

In this article, we consider the nonlinear stochastic partial differential equation of fractional order in both space and time variables with constant initial condition: \begin{equation*} \left (\partial ^{\beta }_t+\dfrac {\nu }{2}\left (-\Delta \right)^{\alpha / 2}\right) u(t, x) = \: I_{t}^{\gamma }\left [\lambda u(t, x) \dot {W}(t, x)\right ] \quad t>0,\: x\in \mathbb {R}^d, \end{equation*} with constants \lambda \ne 0 and \nu >0, where \partial ^{\beta }_t is the Caputo fractional derivative of order \beta \in (0,2], I_{t}^{\gamma } refers to the Riemann-Liouville integral of order \gamma \ge 0, and \left (-\Delta \right)^{\alpha /2} is the standard fractional/power of Laplacian with \alpha >0. We concentrate on the scenario where the noise \dot {W} is the space-time white noise. The existence and uniqueness of solution in the Itô-Skorohod sense is obtained under Dalang's condition. We obtain explicit formulas for both the second moment and the second moment Lyapunov exponent. We derive the p-th moment upper bounds and find the matching lower bounds. Our results solve a large class of conjectures regarding the order of the p-th moment Lyapunov exponents. In particular, by letting \beta = 2, \alpha = 2, \gamma = 0, and d = 1, we confirm the following standing conjecture for the stochastic wave equation: \begin{align*} \frac {1}{t}\log \mathbb {E}[|u(t,x)|^p ] \asymp p^{3/2}, \quad \text {for p\ge 2 as t\to \infty.} \end{align*} The method for the lower bounds is inspired by a recent work of Hu and Wang, where the authors focus on the space-time colored Gaussian noise case. [ABSTRACT FROM AUTHOR]

Linear correlated fractional fuzzy differential equations (LCFFDEs) are one of the best tools for dealing with physical problems with uncertainty. The LCFFDEs mostly do not have unique solutions, especially if the basic fuzzy number is symmetric. The LCFFDEs of symmetric basic fuzzy numbers extend to the new system by extension and produce many solutions. The existing literature does not have any criteria to ensure the existence of unique solutions to LCFFDEs. In this study, we will explore the main causes of the extension and the unavailability of unique solutions. Next, we will discuss the existence and uniqueness conditions of LCFFDEs by using the concept of metric fixed point theory. For the useability of established results, we will also provide numerical examples and discuss their unique solutions. To show the authenticity of the solutions, we will also provide 2D and 3D plots of the solutions. [ABSTRACT FROM AUTHOR]

In the present study, established fixed-point theories are utilized to explore the requisite conditions for the existence and uniqueness of solutions within the realm of sequential fractional differential equations, incorporating both Caputo fractional operators and nonlocal boundary conditions. Subsequently, the stability of these solutions is assessed through the Ulam-Hyers stability method. The research findings are validated with a practical example that corroborate and reinforce the theoretical results. [ABSTRACT FROM AUTHOR]

This article dealt with a class of coupled hybrid fractional differential system. It consisted of a mixed type of Caputo and Hilfer fractional derivatives with respect to two different kernel functions, ψ1 and ψ2, respectively, in addition to coupled boundary conditions. The existence of the solution of the system was investigated using the Dhage fixed point theorem. Finally, an illustration was presented to validate our findings. [ABSTRACT FROM AUTHOR]

We present the application of Lie group theory analysis with ζ -Riemann–Liouville fractional derivative (ζ -RLFD, for short) detailing the construction of infinitesimal prolongation to obtain Lie symmetries. In addition, it addresses the invariance condition without necessarily imposing that the lower limit of the fractional integral is fixed. We find an expression that expands the knowledge regarding the study of exact solutions for fractional differential equations. We apply the Leibniz-type rule for the derivative operator in question to build the prolongation. At last, we calculate the Lie symmetries of the generalized Burgers equation and fractional porous medium equation. [ABSTRACT FROM AUTHOR]

New explicit criteria for the oscillation of all solutions of first and second-order delay differential equations including the Caputo fractional derivative are presented. Examples illustrating the importance and novelty of the main results are included. [ABSTRACT FROM AUTHOR]

All biological forms of life rely on plants for many of their basic needs. Nonetheless, viruses can often infect plants. The ecosystem that depends on it could be destroyed as a result. An insect vector may be responsible for the virus's spread from plant to plant. A fractional order epidemic model is developed since it gives the more realistic solutions as the epidemics always persists in some or the other region and never becomes zero. The dynamics of the plant virus propagation model with effect from seasonality (PVP-S) and intraspecific competition among predators are investigated in terms of existence of solutions, boundedness, and uniqueness. Analysis of plant virus propagation with effect from seasonality and intraspecific competition among predators in terms of fractional order is the novelty of this model. Here, we observe that the infection almost becomes zero in case of integral value, whereas it always persists as the fractional order is introduced, which is more realistic in nature. The occurrence of transcritical bifurcation for the model has been investigated. The proposed nonlinear model is numerically studied by the Adams-Bashforth-Moulton method. This study reveals the effectiveness of the numerical technique as well as the effect of the fractional order derivative on PVP-S dynamics. [ABSTRACT FROM AUTHOR]

In recent years, various complex systems and real-world phenomena have been shown to include memory and hereditary properties that change with respect to time, space, or other variables. Consequently, fractional partial differential equations containing variable-order fractional operators have been extensively resorted for modeling such phenomena accurately. In this paper, we consider the two-dimensional fractional cable equation with the Caputo variable-order fractional derivative in the time direction, which is preferable for describing neuronal dynamics in biological systems. A point-wise scheme, namely, the Crank–Nicolson finite difference method, along with a group-wise scheme referred to as the explicit decoupled group method are proposed to solve the problem under consideration. The stability and convergence analyses of the numerical schemes are provided with complete details. To demonstrate the validity of the proposed methods, numerical simulations with results represented in tabular and graphical forms are given. A quantitative analysis based on the CPU timing, iteration counting, and maximum absolute error indicates that the explicit decoupled group method is more efficient than the Crank–Nicolson finite difference scheme for solving the variable-order fractional equation. [ABSTRACT FROM AUTHOR]

Purpose: Fractional Fokker-Planck equation (FFPE) and time fractional coupled Boussinesq-Burger equations (TFCBBEs) play important roles in the fields of solute transport, fluid dynamics, respectively. Although there are many methods for solving the approximate solution, simple and effective methods are more preferred. This paper aims to utilize Laplace Adomian decomposition method (LADM) to construct approximate solutions for these two types of equations and gives some examples of numerical calculations, which can prove the validity of LADM by comparing the error between the calculated results and the exact solution. Design/methodology/approach: This paper analyzes and investigates the time-space fractional partial differential equations based on the LADM method in the sense of Caputo fractional derivative, which is a combination of the Laplace transform and the Adomian decomposition method. LADM method was first proposed by Khuri in 2001. Many partial differential equations which can describe the physical phenomena are solved by applying LADM and it has been used extensively to solve approximate solutions of partial differential and fractional partial differential equations. Findings: This paper obtained an approximate solution to the FFPE and TFCBBEs by using the LADM. A number of numerical examples and graphs are used to compare the errors between the results and the exact solutions. The results show that LADM is a simple and effective mathematical technique to construct the approximate solutions of nonlinear time-space fractional equations in this work. Originality/value: This paper verifies the effectiveness of this method by using the LADM to solve the FFPE and TFCBBEs. In addition, these two equations are very meaningful, and this paper will be helpful in the study of atmospheric diffusion, shallow water waves and other areas. And this paper also generalizes the drift and diffusion terms of the FFPE equation to the general form, which provides a great convenience for our future studies. [ABSTRACT FROM AUTHOR]

The author considers a nonlinear Caputo fractional-order delay differential equation (CFrDDE) with multiple variable delays. First, we study the existence and uniqueness of the solutions of the CFrDDE with multiple variable delays. Second, we obtain two new results on the Ulam–Hyers–Mittag–Leffler (UHML) stability of the same equation in a closed interval using the Picard operator, Chebyshev norm, Bielecki norm and the Banach contraction principle. Finally, we present three examples to show the applications of our results. Although there is an extensive literature on the Lyapunov, Ulam and Mittag–Leffler stability of fractional differential equations (FrDEs) with and without delays, to the best of our knowledge, there are very few works on the UHML stability of FrDEs containing a delay. Thereby, considering a CFrDDE containing multiple variable delays and obtaining new results on the existence and uniqueness of the solutions and UHML stability of this kind of CFrDDE are the important aims of this work. [ABSTRACT FROM AUTHOR]

In this paper, we present the fractional backward differentiation formulas for the numerical solutions of two-term fractional differential Sylvester matrix equations in the Caputo derivative sense, which includes the celebrated two-term fractional differential Lyapunov matrix equations. We give two applications in a two-term time-fractional telegraph equation with illustrative examples. In addition, we also consider two examples to illustrate the effectiveness of the proposed approaches. [ABSTRACT FROM AUTHOR]

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This paper aims to establish the existence and uniqueness of mild solutions to abstract sequential fractional differential equations. The approach employed involves the utilization of resolvent operators and the fixed-point theorem. Additionally, we investigate a specific example concerning a partial differential equation incorporating the Caputo fractional derivative. [ABSTRACT FROM AUTHOR]

In this study, the initial-boundary value problems to semilinear integro-differential equations with multi-term fractional Caputo derivatives are analyzed. A particular case of these equations models oxygen diffusion through capillaries. Under proper requirements on the given data in the model, the classical and strong solvability of these problems for any finite time interval [0, T] are proved via so-called continuation method. The key point in this approach is finding suitable a priori estimates of a solution in the fractional Hölder and Sobolev spaces. [ABSTRACT FROM AUTHOR]

In this paper, we focus on solving nonlinear fractional logistic differential equation within the Caputo–Fabrizio fractional derivative operator. First, we present a new numerical scheme called new decomposition transform method (NDTM) for solving general nonlinear fractional differential equations. The proposed scheme is obtained by combining the Jafari transform method and a new decomposition method. Furthermore, we establish the uniqueness theorem of the solution using Banach's fixed point theorem. Finally, after the convergence analysis of the NDTM, we provide two numerical examples of nonlinear fractional logistic differential equations involving the Caputo–Fabrizio fractional derivative operator to validate the results. The obtained results have shown that the new method for analytical solutions is simple to implement and very effective for analyzing the complex fractional problems that arise in related fields of science and engineering. [ABSTRACT FROM AUTHOR]

This paper focuses on revisiting and improving the results regarding the existence of a solution to the implicit fractional differential equations (FDEs) given in the following form: DCρω(ϰ)=fϰ,ω(ϰ),DCρω(ϰ)$$ {\mathcal{D}}_C&#x0005E;{\rho}\omega \left(\mathrm{\varkappa}\right)&#x0003D;f\left(\mathrm{\varkappa},\omega \left(\mathrm{\varkappa}\right),{\mathcal{D}}_C&#x0005E;{\rho}\omega \left(\mathrm{\varkappa}\right)\right) $$for all ϰ∈[0,T]$$ \mathrm{\varkappa}\in \left[0,T\right] $$ with the initial condition ω(0)=ω0$$ \omega (0)&#x0003D;{\omega}_0 $$, where 0<ρ<1,DCρ$$ 0<\rho <1,{\mathcal{D}}_C&#x0005E;{\rho } $$ is the Caputo fractional derivative (CFD). Mainly, we aim to weaken the necessary conditions often found in the literature for guaranteeing the existence of solution. [ABSTRACT FROM AUTHOR]