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The Mittag-Leffler function appears as an analytical solution of some fractional differential equations. The behavior of the zeros of the Mittag-Leffler function, especially their asymptotic distribution, plays a fundamental role in the study of eigenvalue problems. The goal here is to derive a different form of the higher-order asymptotic distribution of the zeros of the \mbox{Mittag-Leffler} function.

Abstract 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.

In this paper, we introduce the concept of the weighted generalized Atangana-Baleanu fractional derivative. We prove the existence of the stability of solutions of non-local differential equations and non-local differential inclusions, in Banach spaces, with this new fractional derivative in the presence of instantaneous and non-instantaneous impulses. We considered the case in which the lower limit of the fractional derivative was kept at the initial point and where it was changed to the impulsive points. To prove our results, we established the relationship between solutions to each of the four studied problems and those of the corresponding fractional integral equation. There has been no previous study of the weighted generalized Atangana-Baleanu fractional derivative, and so, our findings are new and interesting. The technique we used based on the properties of this new fractional differential operator and suitable fixed point theorems for single valued and set valued functions. Examples are given to illustrate the theoretical results.

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.

This paper examines the existence, uniqueness, and Ulam-Hyers stability of solutions to nonlinear $\mho$-fractional differential equations with boundary conditions with a $\mho$-Caputo fractional derivative. The acquired results for the suggested problem are validated using a novel technique and minimum assumptions about the function $f$. The analysis reduces the problem to a similar integral equation and uses Banach and Sadovskii fixed point theorems to reach the desired findings. Finally, the inquiry is demonstrated by illustrative example to validate the theoretical findings.

In this article, we employ the Laplace transform (LT) method to study fractional differential equations with the problem of displacement of motion of mass for free oscillations, damped oscillations, damped forced oscillations, and forced oscillations (without damping). These problems are solved by using the Caputo and Atangana-Baleanu (AB) fractional derivatives, which are useful fractional derivative operators consist of a non-singular kernel and are efficient in solving non-local problems. The mathematical modelling for the displacement of motion of mass is presented in fractional form. Moreover, some examples are solved.

This paper explores a fractional integro-differential equation with boundary conditions that incorporate the Hilfer-Hadamard fractional derivative. We model the RLC circuit using fractional calculus and define weighted spaces of continuous functions. The existence and uniqueness of solutions are established, along with their Ulam-Hyers and Ulam-Hyers-Rassias stability. Our analysis employs Schaefer's fixed-point theorem and Banach's contraction principle. An illustrative example is presented to validate our findings.

This paper investigates nonlinear implicit fractional differential equations involving Hilfer-Katugampola fractional derivatives. Novel techniques are developed to establish the existence and uniqueness of solutions for the proposed problem using fixed-point theorems. Additionally, Hyers-Ulam stability and generalized Hyers-Ulam stability are analyzed, and an example is provided to validate the theoretical results.

This paper uses the fractional perturbed Gerdjikov–Ivanov (PGI) model, a basic mathematical framework in mathematical physics and nonlinear dynamics, to examine complex wave structures using the M-fractional operator and modified Extended Direct Algebraic Method (mEDAM). We find a wide variety of new optical wave solutions, such as kink-type, dark, brilliant, periodic, combo, exponential, trigonometric, and hyperbolic solutions. our examine the dynamic behavior and free parameters of these soliton solutions using contour plots and three-dimensional charts. The uniqueness of the study is shown by the noteworthy consistency and divergence of our results from earlier answers. This work makes a substantial contribution to the PGI model’s ability to extract many solitary wave solutions. The proposed suggested method shows dependability while assessing analytical solutions for fractional differential equations. This research intends to extend mathematical approaches for solving fractional differential equations, which will enable answers to a wide range of practical scientific and engineering problems, including implications for ultrafast pulse transmission in optical fibers.

The main motivation of this study is to extend the use of the operational matrices approach to solve fractional-order two-point boundary value problems (TPBVPs), a method often employed in the literature for solving fractional-order initial value problems. Our proposed approach employs innovative operational matrices, specifically the integral operational matrices based on Chelyshkov polynomials (CPs), a type of orthogonal polynomials. These operational matrices enable us to integrate monomial terms into the algorithm, effectively converting the problem into easily solvable Sylvester-type equations. We provide a comprehensive comparison to demonstrate the accuracy and computational advantages of our proposed approach against existing methods, including the exact solution, the Haar wavelet method (HWM), the Bessel collocation method (BCM), the Pseudo Spectral Method (PSM), the Generalized Adams–Bashforth–Moulton Method (GABMM) and the fractional central difference scheme (FCDS) through numerical examples. Additionally, our proposed approach is well-suited for solving problems with both polynomial and non-polynomial solutions.

This paper studies an Ebola epidemic model with an exponential nonlinear incidence function that considers the efficacy and the behaviour change. The current model also incorporates a new density-dependent treatment that catches the impact of the disease transmission on the treatment. Firstly, we provide a theoretical study of the nonlinear differential equations model obtained. More precisely, we derive the effective reproduction number and, under suitable conditions, prove the stability of equilibria. Afterwards, we show that the model exhibits the phenomenon of backward-bifurcation whenever the bifurcation parameter and the reproduction number are less than one. We find that the bi-stability and backward-bifurcation are not automatically connected in epidemic models. In fact, when a backward-bifurcation occurs, the disease-free equilibrium may be globally stable. Numerically, we use well-known standard tools to fit the model to the data reported for the 2018–2020 Kivu Ebola outbreak, and perform the sensitivity analysis. To control Ebola epidemics, our findings recommend a combination of a rapid behaviour change and the implementation of a proper treatment strategy with a high level of efficacy. Secondly, we propose and analyze a fractional-order Ebola epidemic model, which is an extension of the first model studied. We use the Caputo operator and construct the Grünwald-Letnikov nonstandard finite difference scheme, and show its advantages.

In this paper, we study the existence and uniqueness of solutions for a coupled system of Hilfer-Hadamard sequential fractional differential equations with multi-point Riemann-Liouville fractional integral boundary conditions via standard fixed point theorems. The existence of solutions is proved using Krasnoselskii's fixed point theorem, while the existence and uniqueness of solutions is established using the Banach fixed point theorem. We also discuss the stability of the problem in terms of Ulam-Hyers, Ulam-Hyers-Rassias, generalized Ulam-Hyers, and generalized Ulam-Hyers-Rassias stability. As an application, some examples are presented to illustrate our theoretical results.

This work aims to provide a new Galerkin algorithm for solving the fractional Rayleigh-Stokes equation (FRSE). We select the basis functions for the Galerkin technique to be appropriate orthogonal combinations of the second kind of Chebyshev polynomials (CPs). By implementing the Galerkin approach, the FRSE, with its governing conditions, is converted into a matrix system whose entries can be obtained explicitly. This system can be obtained by expressing the derivatives of the basis functions in terms of the second-kind CPs and after computing some definite integrals based on some properties of CPs of the second kind. A thorough investigation is carried out for the convergence analysis. We demonstrate that the approach is applicable and accurate by providing some numerical examples.

This study investigates the Atangana-Baleanu time-stochastic fractional neutral integro-differential equation, a complex mathematical model with broad applications in various scientific disciplines. Utilizing Banach's fixed point theory, we rigorously establish the existence and uniqueness of the mild solution to this equation. Our analysis centrally revolves around investigating the Mittag-Leffler non-singular and non-local kernel, emphasizing its crucial significance in elucidating the behavior of the equation. By integrating concepts from fractional differential equations and stochastic differential systems, we contribute to a deeper comprehension of these mathematical phenomena. Our findings not only contribute significantly to advancing theoretical understanding but also establish a solid groundwork for practical applications across various fields.

Fractional differential operators are inherently non-local, so global methods, such as spectral methods, are well suited for handling these non-local operators. Long-time integration of differential models such as chaotic dynamical systems poses specific challenges and considerations that make multi-domain numerical methods advantageous when dealing with such problems. This study proposes a novel multi-domain pseudospectral method based on the first kind of Chebyshev polynomials and the Gauss–Lobatto quadrature for fractional initial value problems.The proposed technique involves partitioning the problem’s domain into non-overlapping sub-domains, calculating the fractional differential operator in each sub-domain as the sum of the ‘local’ and ‘memory’ parts and deriving the corresponding differentiation matrices to develop the numerical schemes. The linear stability analysis indicates that the numerical scheme is absolutely stable for certain values of arbitrary non-integer order and conditionally stable for others. Numerical examples, ranging from single linear equations to systems of non-linear equations, demonstrate that the multi-domain approach is more appropriate, efficient and accurate than the single-domain scheme, particularly for problems with long-term dynamics.

In this study, we employ the Laplace Variational Iterational Method (LVIM) as a sophisticated mathematical tool to investigate into the complex dynamics of cancer cells under the influence of chemotherapy. The LVIM, a method combining Laplace transformations and variational iteration techniques, is specifically adapted to address a system of time fractional differential equations (FDEs) that characterizes the temporal behavior of cancer cells. To enhance the efficacy of our approach, we introduce a semi-analytic version of LVIM, which proves to be a powerful and versatile tool for solving mathematical problems involving fractional derivatives. The focus of our analysis centers on elucidating the impact of chemotherapy, with a particular emphasis on drug diffusion within cancer cells and the fractality of DNA walks. Through numerical exploration encompassing varying fractional order derivatives, our study exposes nonlinear behaviors that remain secret in systems featuring only integer order derivatives. Notably, the methodology we propose is not only applicable to the specific cases examined in this research but also exhibits broad versatility, making it suitable for exploring the effects of different drugs and types of cancers. This research contributes valuable insights into the dynamics of cancer cells, aiding in the understanding of the implications for therapeutic strategies in the context of cancer treatment.

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.

This study uses the Laplace transformation method to solve the fractional-order T. Regge problem. In this paper, we develop formulations for the fractional Laplace transform applied to fractional integrals and derivatives, and we use this method to solve the T. Regge problem. Moreover, several examples are presented to demonstrate the method's value and effectiveness. Examples prove that the Laplace transformation method significantly advances the fractional computation field and can potentially solve fractional differential equations (FDEs). On the other hand, the advantages and disadvantages of the method are provided.

In this article, the existence and uniqueness of solutions for non-linear fractional differential equation with Tempered Ψ−Caputo derivative with three-point boundary conditions were studied. The existence and uniqueness of the solution were proved by applying the Banach contraction mapping principle and Schaefer’s fixed point theorem.

In this work, we introduce a new definition of weighted derivatives along with corresponding integral operators, which aim to facilitate the solution of both linear and non-linear differential equations. A significant finding is that the fractional derivative of Caputo–Fabrizio type is a special case within this framework, allowing us to build upon existing research in this area. Additionally, we provide closed-form analytical solutions for autonomous and logistic equations using our newly defined derivatives and integrals. We thoroughly explore the properties associated with these weighted derivatives and integrals. To demonstrate the reliability and practical applicability of our results, we include several examples and applications that highlight the effectiveness of our approach.

In this paper, Lie symmetry analysis method is applied to time fractional Caudrey–Dodd–Gibbon equation. We obtain a symmetric group spanned by two generators for the governing equation. The obtained group generators are used to reduce the studied fractional partial differential equation to some fractional ordinary differential equations with Riemann–Liouville fractional derivative or Erdélyi-Kober fractional derivative, thereby getting one trivial solution and one convergent power series solution for the reduced equations. Then we present the dynamic behavior of the obtained analytical solutions graphically. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the equation studied.

This research delves into investigating the presence of solutions to fractional differential equations with an order σ∈(2,3]. These equations include the Caputo derivative and introduce innovative nonlocal antiperiodic boundary conditions. These boundary conditions, defined at a nonlocal intermediary point 0≤δ

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.

In this manuscript, we prove new extensions of Nashine, Wardowski, Feng-Liu, and Ćirić-type contractive inequalities using orbitally lower semi-continuous functions in an orbitally complete bb-metric space. We accomplish new multivalued common fixed point results for two families of dominated set-valued mappings in an ordered complete orbitally bb-metric space. Some new definitions and illustrative examples are given to validate our new results. To show the novelty of our results, applications are given to obtain the solution of nonlinear integral and fractional differential equations. Our results expand the hypothetical consequences of Nashine et al. (Feng–Liu-type fixed point result in orbital b-metric spaces and application to fractal integral equation, Nonlinear Anal. Model. Control. 26 (2021), no. 3, 522–533) and Rasham et al. (Common fixed point results for new Ciric-type rational multivalued-contraction with an application, J. Fixed Point Theory Appl. 20 (2018), no. 1, Paper No. 45).

Abstract 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.

In this paper, the Lie symmetry analysis was executed for the nonlinear fractional-order conduction-diffusion Buckmaster model (BM), which involves the Riemann-Liouville (R-L) derivative of fractional-order 'β'. In the study of groundwater flow and oil reservoir engineering where fluid flow through porous materials is crucial, BM played an important role. The Lie point infinitesimal generators and Lie algebra were constructed for the equation. The Lie symmetries were acquired for the ordinary fractional-order BM. The power series solution and its convergence were also analyzed with the application of the implicit theorem. Noether's theorem was employed to ensure the consistency of a system by deriving the conservation laws of its physical model.

The main aim of this paper is to study the Cauchy problem for nonlinear differential equations of fractional order containing the weighted Riemann-Liouville fractional derivative of a function with respect to another function. The equivalence of this problem and a nonlinear Volterra-type integral equation of the second kind have been presented. In addition, the existence and uniqueness of the solution to the considered Cauchy problem are proved using Banach's fixed point theorem and the method of successive approximations. Finally, we obtain a new estimate of the weighted Riemann-Liouville fractional derivative of a function with respect to functions at their extreme points. With the assistance of the estimate obtained, we develop the comparison theorems of fractional differential inequalities, strict as well as nonstrict, involving weighted Riemann-Liouville differential operators of a function with respect to functions of order $ \delta $, $ 0 < \delta < 1 $.

This paper proposes a numerical technique to solve the time-fractional generalized Kawahara differential equation (TFGKDE). Certain shifted Lucas polynomials are utilized as basis functions. We first establish some new formulas concerned with the introduced polynomials and then tackle the equation using a suitable collocation procedure. The integer and fractional derivatives of the shifted polynomials are used with the typical collocation method to convert the equation with its governing conditions into a system of algebraic equations. The convergence and error analysis of the proposed double expansion are rigorously investigated, demonstrating its accuracy and efficiency. Illustrative examples are provided to validate the effectiveness and applicability of the proposed algorithm.

In this paper, the use of a class of fractional-order dynamical systems with discontinuous right-hand side defined with Caputo’s derivative is considered. The existence of the solutions is analyzed. For this purpose, differential inclusions theory is used to transform, via the Filippov regularization, the discontinuous right-hand side into a set-valued function. Next, via Cellina’s Theorem, the obtained set-valued differential inclusion of fractional order can be restarted as a single-valued continuous differential equation of fractional order, to which the existing numerical schemes for fractional differential equations can be applied. In this way, the delicate problem of integrating discontinuous problems of fractional order, as well as integer order, is solved by transforming the discontinuous problem into a continuous one. Also, it is noted that even the numerical methods for fractional-order differential equations can be applied abruptly to the discontinuous problem, without considering the underlying discontinuity, so the results could be incorrect. The technical example of a single-degree-of-freedom preloaded compliance system of fractional order is presented.

In this work, a new relationship is established between the solutions of higher order fractional differential equations and a Wright-type transformation. Solutions could be interpreted as expected values of functions in a random time process. As applications, we solve the fractional beam equation, fractional electric circuits with special functions as external sources, derive d’Alembert’s formula and show the existence of explicit solutions for a general fractional wave equation with variable coefficients. Due to this relationship, we present two methods for simulating solutions of fractional differential equations. The two approaches use the interpretation of the Caputo derivative of a function as a Wright-type transformation of the higher derivative of the function. In the first approach, we use the Runge–Kutta method of hybrid orders 4 and 5 to solve ordinary differential equations combined with the Monte Carlo integration to conduct the Wright-type transformation. The second method uses a feedforward neural network to simulate the fractional differential equation.

In this paper we apply the Riemann–Liouville, Erdelyi–Kober and Caputo fractional operators to the modified beta, modified Gauss hypergeometric and modified confluent hypergeometric functions in which the generalized M-series are included in their kernels. Furthermore, as examples, we obtain solutions of some fractional differential equations involving the above modified special functions.

The Lotka–Volterra model arising in biology for simulating interactions between two species is considered in the paper with fractional order derivatives in Caputo sense. Two cases of the model are solved numerically using fractional reduced differential transform method (FRDTM). In both the cases, the predator and prey populations are computed for different fractional orders, over a range of time and absolute errors are studied for integer order. The results are presented in the form of tables and graphs depicting accuracy of the FRDTM and its agreement with the model’s dynamics.

Background: The dynamics of fractional oscillators are generally described by fractional differential equations, which include the fractional derivative of the Caputo or Riemann-Liouville type. These equations induce classical oscillator equations like the harmonic oscillator equation, to include fractional order derivatives. Solving fractional differential equations numerically can be challenging due to the non-local nature of fractional derivatives. Objective: In this paper, a recently developed integral Rohit transform is utilized for solving systems of undamped and damped fractional oscillators characterized by differential equations of fractional or non-integral order involving the Caputo-fractional derivative operator. The solutions of fractional systems which include undamped-simple fractional oscillators, undamped-driven fractional oscillators, damped-driven fractional oscillators, and damped-fractional oscillators are obtained. Methods: by applying the integral Rohit transform, also written as RT. Differential equations of fractional or non-integral order are generally solved by utilizing methods which include the fractional variational iteration approach, the homotopy-perturbation method, the equivalent linearized method, the Adomian decomposition method, etc. Results: This paper demonstrates the effectiveness, reliability, and efficiency of the integral Rohit transform in solving fractional systems, which include undamped-simple fractional oscillators, undamped-driven fractional oscillators, damped-driven fractional oscillators, and damped-fractional oscillators and are characterized by differential equations of fractional or non-integral order involving the Caputo-fractional derivative operator. Conclusions: The Rohit transform brought the progressive principles or methodologies that offer new insights or views on the problems examined in the paper, distinguishing itself from existing methods and doubtlessly beginning up new research instructions. It provided precise results for the specific problems discussed in the paper, surpassing the capabilities of other methods in terms of decision, constancy, or robustness to noise and disturbances.

The primary focus of this paper is to thoroughly examine and analyze a class of a fractional quadratic integral equation via generalized fractional integral. To achieve this, we introduce an operator that possesses fixed points corresponding to the solutions of the fractional quadratic integral equation, effectively transforming the given equation into an equivalent fixed-point problem. By applying the Banach fixed-point theorems, we prove the uniqueness of solutions to fractional quadratic integral equation. Additionally, The Adomian decomposition method is used, to solve the resulting fractional quadratic integral equation. This technique rapidly provides convergent successive approximations of the exact solution to the given fractional quadratic integral equation, therefore, we investigate the convergence of approximate solutions, using the Adomian decomposition method. Finally, we provide some examples, to demonstrate our results. Our findings contribute to the current understanding of fractional quadratic integral equation and their solutions and have the potential to inform future research in this area.

This paper discusses the existence of solutions for fractional differential equations with nonlocal boundary conditions (NFDEs) under essential assumptions. The boundary conditions incorporate a point $ 0\leq c < d $ and fixed points at the end of the interval $ [0, d] $. For $ i = 0, 1 $, the boundary conditions are as follows: $ a_{i}, b_{i} > 0 $, $ a_{0} p(c) = -b_{0} p(d), \ a_{1} p^{'}(c) = -b_{1} p^{'}(d) $. Furthermore, the research aims to expand the usability and comprehension of these results to encompass not just NFDEs but also classical fractional differential equations (FDEs) by using the Krasnoselskii fixed-point theorem and the contraction principle to improve the completeness and usefulness of the results in a wider context of fractional differential equations. We offer examples to demonstrate the results we have achieved.

In this paper, we expanded the concept of tempered fractional derivatives within both the Riemann-Liouville and Caputo frameworks, introducing a novel class of fractional operators. These operators are characterized by their dependence on a specific arbitrary smooth function. We then investigated the existence and uniqueness of solutions for a particular class of fractional differential equations, subject to specified initial conditions. To aid our analysis, we introduced and demonstrated the application of Picard's iteration method. Additionally, we utilized the Gronwall inequality to explore the stability of the system under examination. Finally, we studied the attractivity of the solutions, establishing the existence of at least one attractive solution for the system. Throughout the paper, we provide examples and remarks to support and reinforce our findings.

Abstract In this paper, we apply two methods for solving nonlinear system of fractional differential equations (FDEs); these two methods are Picard and Adomian decomposition methods (ADM). The type of fractional derivative in this system will be the Atangana–Baleanu derivative. The existence and uniqueness of the solution will be proved. In addition, the convergence of ADM series solution and the maximum expected error will be discussed. Some numerical examples will be solved by using these two method and a comparison between their solutions will be done. There exist an important application to these types of systems, this application is the fractional-order rabies model and it will be solved here. From the obtained results, it is noticed that the obtained results from using these two methods are coincide with each other, and also these results are coincide with the obtained results from the classical fractional derivatives such as Caputo sense.

In this article, we propose two numerical schemes for solving the time-fractional heat equation (TFHE). The proposed methods are based on applying the collocation and tau spectral methods. We introduce and employ a new set of basis functions: The unified Chebyshev polynomials (UCPs) of the first and second kinds. We establish some new theoretical results regarding the new UCPs. We employ these results to derive the proposed algorithms and analyze the convergence of the proposed double expansion. Furthermore, we compute specific integer and fractional derivatives of the UCPs in terms of their original UCPs. The derivation of these derivatives will be the fundamental key to deriving the proposed algorithms. We present some examples to verify the efficiency and applicability of the proposed algorithms.

In this work, we focus on investigating the existence of solutions to nonlinear fractional boundary value problems (FBVPs) with generalized nonlinear boundary conditions. By extending the framework of the technique based on well-ordered coupled lower and upper solutions, we guarantee the existence of solutions in a sector defined by these solutions. One notable aspect of our study is that the proposed approach unifies the existence results for the problems that have previously been discussed separately in the literature. To substantiate these findings, we have added three illustrative examples.

Abstract Several epidemiological models use the Caputo fractional-order differential operator without establishing its significance. This study verifies a Caputo operator-based fractional-order epidemiological model of the SAIVR type. COVID-19 kills. Infection weakens the immune system. The fractional Caputo operator describes COVID-19 immunization. Fundamental system characteristics are determined using fractional calculus. Our analysis included the fractional system’s Hyers–Ulam–Rassias stability and stable states. The uniqueness and existence of fractional Caputo system solutions are explored. The least-squares approach determines system parameters. The Caputo fractional-order α value is optimized to 6.757 e − 01 $6.757\text{e}{-}01$ , indicating that the system best fits real-life medical data for infection. Caputo and classical systems were compared for absolute mean errors. The Box-Whisker chart case summaries show the Caputo operator superiority. When α → 1 $\alpha \rightarrow 1$ , the memory traces and hereditary traits are also observed. Finally, the Caputo fractional framework simulates COVID-19 using strong numerical methods.

In this paper, we study nonlinear differential equations with Caputo fractional derivatives with respect to other functions and impulses. The main characteristic of the impulses is that the time between two consecutive impulsive moments is defined by random variables. These random variables are independent. As the distribution of these random variables is very important, we consider the Erlang distribution. It generalizes the exponential distribution, which is very appropriate for describing the time between the appearance of two consecutive events. We describe a detailed solution to the studied problem, which is a stochastic process. We define the p-exponential stability of the solutions and obtain sufficient conditions. The study is based on the application of appropriate Lyapunov functions. The obtained sufficient conditions depend not only on the nonlinear function and impulsive functions, but also on the function used in the fractional derivative. The obtained results are illustrated using some examples.

This study explores innovative insights into the realms of dynamic programming and fractional differential equations, situated explicitly within the framework of partial modular b-metric spaces enriched with a binary relation R, proposing a novel definition for a generalized ℷC-type Suzuki R-contraction specific to these spaces. By doing so, we pave the way for a range of relation-theoretical common fixed-point theorems, highlighting the versatility of our approach. To illustrate the practical relevance of our findings, we present a compelling example. Ultimately, this work aims to enrich the existing academic discourse and stimulate further research and practical applications within the field.

The paper considers the solution of a fractional differential boundary value problem, that is, a diffusion-type equation with arbitrary values of the derivative orders on an infinite axis. The difference between the obtained results and other authors’ ones is that these involve arbitrary values of the derivative orders. The solutions described in the literature, as a rule, are considered in the case when the fractional time derivative β lies in the range: 0 < β ≤ 1, and the fractional state derivative α (the variable describing the state of the process) is in the range: 1 < α ≤ 2. The solution presented in the article allows us to consider any ranges for α and β, if the inequality 0 < β/α ≤ 0.865 is satisfied in the range β/α. In order to solve the boundary value problem, the probability density function of the observed state x of a certain process (for example, the magnitude of the deviation of the levels of a time series) from time t (for example, the time interval for calculating the amplitudes of the deviation of the levels of a time series) can be captured.

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

In this paper, we propose a global numerical method for approximating Caputo fractional derivatives of order α(Dαf)(y)=1Γ(m−α)∫0y(y−x)m−α−1f(m)(x)dx,y>0, with m−1<α≤m,m∈N. The numerical procedure is based on approximating f(m) by the m-th derivative of a Lagrange polynomial, interpolating f at Jacobi zeros and some additional nodes suitably chosen to have corresponding logarithmically diverging Lebsegue constants. Error estimates in a uniform norm are provided, showing that the rate of convergence is related to the smoothness of the function f according to the best polynomial approximation error and depending on order α. As an application, we approximate the solution of a Volterra integral equation, which is equivalent in some sense to the Bagley–Torvik initial value problem, using a Nyström-type method. Finally, some numerical tests are presented to assess the performance of the proposed procedure.

This paper presents a study of the application of the finite element method for solving a fractional differential filtration problem in heterogeneous fractured porous media with variable orders of fractional derivatives. A numerical method for the initial-boundary value problem was constructed, and a theoretical study of the stability and convergence of the method was carried out using the method of a priori estimates. The results were confirmed through a comparative analysis of the empirical and theoretical orders of convergence based on computational experiments. Furthermore, we analyzed the effect of variable-order functions of fractional derivatives on the process of fluid flow in a heterogeneous medium, presenting new practical results in the field of modeling the fluid flow in complex media. This work is an important contribution to the numerical modeling of filtration in porous media with variable orders of fractional derivatives and may be useful for specialists in the field of hydrogeology, the oil and gas industry, and other related fields.