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Conditions for the existence and uniqueness of mild solutions for a system of semilinear impulsive differential equations with infinite fractional Brownian movements and the Wiener process are established. Our approach is based on a novel application of Burton and Kirk's fixed point theorem in extended Banach spaces. This paper aims to extend current results to a differential-inclusions scenario. The motivation of this paper for impulsive neutral differential equations is to investigate the existence of solutions for impulsive neutral differential equations with fractional Brownian motion and a Wiener process (topics that have not been considered and are the main focus of this paper). [ABSTRACT FROM AUTHOR]

In this paper, we present an efficient method for solving a class of higher order fractional differential equations with general boundary conditions. The convergence of the numerical method is proved and an error estimate is given. Finally, eight numerical examples, both linear and nonlinear, are presented to demonstrate the accuracy of our method. The proposed method introduces suitable base functions to calculate the approximate solutions and only requires us to deal with the linear or nonlinear systems. Thus, our method is convenient to implement. Furthermore, the numerical results show that the proposed method performs better compared to the existing ones. [ABSTRACT FROM AUTHOR]

This paper presents a groundbreaking numerical technique for solving nonlinear time fractional differential equations, combining the conformable continuity equation (CCE) with the Non-Polynomial Spline (NPS) interpolation to address complex mathematical challenges. By employing conformable descriptions of fractional derivatives within the CCE framework, our method ensures enhanced accuracy and robustness when dealing with fractional order equations. To validate our approach's applicability and effectiveness, we conduct a comprehensive set of numerical examples and assess stability using the Fourier method. The proposed technique demonstrates unconditional stability within specific parameter ranges, ensuring reliable performance across diverse scenarios. The convergence order analysis reveals its efficiency in handling complex mathematical models. Graphical comparisons with analytical solutions substantiate the accuracy and efficacy of our approach, establishing it as a powerful tool for solving nonlinear time-fractional differential equations. We further demonstrate its broad applicability by testing it on the Burgers–Fisher equations and comparing it with existing approaches, highlighting its superiority in biology, ecology, physics, and other fields. Moreover, meticulous evaluations of accuracy and efficiency using ( L 2 and L ∞ ) norm errors reinforce its robustness and suitability for real-world applications. In conclusion, this paper presents a novel numerical technique for nonlinear time fractional differential equations, with the CCE and NPS methods' unique combination driving its effectiveness and broad applicability in computational mathematics, scientific research, and engineering endeavors. [ABSTRACT FROM AUTHOR]

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 paper, a new class of coupled hybrid systems of proportional sequential ψ -Hilfer fractional differential equations, subjected to nonlocal boundary conditions were investigated. Based on a generalization of the Krasnosel'ski i ˘ 's fixed point theorem due to Burton, sufficient conditions were established for the existence of solutions. A numerical example was constructed illustrating the main theoretical result. For special cases of the parameters involved in the system many new results were covered. The obtained result is new and significantly contributes to existing results in the literature on coupled systems of proportional sequential ψ -Hilfer fractional differential equations. [ABSTRACT FROM AUTHOR]

This paper solves a generalized class of first-order fractional ordinary differential equations (1st-order FODEs) by means of Riemann–Liouville fractional derivative (RLFD). The principal incentive of this paper is to generalize some existing results in the literature. An effective approach is applied to solve non-homogeneous fractional differential systems containing 2 n periodic terms. The exact solutions are determined explicitly in a straightforward manner. The solutions are expressed in terms of entire functions with fractional order arguments. Features of the current solutions are discussed and analyzed. In addition, the existing solutions in the literature are recovered as special cases of our results. [ABSTRACT FROM AUTHOR]

The paper focuses on exploring the existence and uniqueness of a specific solution to a class of Caputo impulsive fractional differential equations with boundary value conditions on Banach space, referred to as (ω , ρ) -BVP solution. The proof of the main results of this study involves the application of the Banach contraction mapping principle and Schaefer's fixed point theorem. Furthermore, we provide the necessary conditions for the convexity of the set of solutions of the analyzed impulsive fractional differential boundary value problem. To enhance the comprehension and practical application of our findings, we conclude the paper by presenting two illustrative examples that demonstrate the applicability of the obtained results. [ABSTRACT FROM AUTHOR]

Fractional differential equations are used to construct mathematical models and can describe the characteristics of real systems. In this paper, the parameter estimation problem of a fractional Wiener system is studied by designing linear filters which can obtain smaller tunable parameters and maintain the stability of the parameters in any case. To improve the identification performance of the stochastic gradient algorithm, this paper derives two modified stochastic gradient algorithms for the fractional nonlinear Wiener systems with colored noise. By introducing the forgetting factor, a forgetting factor stochastic gradient algorithm is deduced to improve the convergence rate. To achieve more efficient and accurate algorithms, we propose a multi-innovation forgetting factor stochastic gradient algorithm by means of the multi-innovation theory, which expands the scalar innovation into the innovation vector. To test the developed algorithms, a fractional-order dynamic photovoltaic model is employed in the simulation, and the dynamic elements of this photovoltaic model are estimated using the modified algorithms. Concurrently, a numerical example is given, and the simulation results verify the feasibility and effectiveness of the proposed procedures. [ABSTRACT FROM AUTHOR]

In this paper, we present an application of the renormalization group method developed by Chen, Goldenfeld and Oono for a stochastic differential equation in a space of Hilbert space-valued generalized random variables with multiplicative noise. The driving process is a real-valued fractional white noise with a Hurst parameter greater than 1 / 2 . The stochastic integration is understood in the Wick–Itô–Skorohod sense. This article is a generalization of results of Glatt-Holtz and Ziane, which were for the systems with white noise. We firstly demonstrate the process of formulating the renormalization group equation and the asymptotic solution. Then, we give rigorous proof of the consistency of the approximate solution. In addition, some numerical comparisons are given to illustrate the validity of our results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence and multiplicity of monotone positive solutions for a class of nonlinear fractional differential equation with a disturbance parameter in the integral boundary conditions on the infinite interval. By using Guo–Krasnosel'skii fixed-point theorem and the analytic technique, we divide the range of parameter for the existence of at least two, one and no positive solutions for the problem. In the end, an example is given to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, we present a fractional version of the Sakiadis flow described by a nonlinear two-point fractional boundary value problem on a semi-infinite interval, in terms of the Caputo derivative. We derive the fractional Sakiadis model by substituting, in the classical Prandtl boundary layer equations, the second derivative with a fractional-order derivative by the Caputo operator. By using the Lie symmetry analysis, we reduce the fractional partial differential equations to a fractional ordinary differential equation, and, then, a finite difference method on quasi-uniform grids, with a suitable variation of the classical L1 approximation formula for the Caputo fractional derivative, is proposed. Finally, highly accurate numerical solutions are reported. [ABSTRACT FROM AUTHOR]

Fractional-order partial differential equations have gained significant attention due to their wide range of applications in various fields. This paper employed a novel technique for solving nonlinear multidimensional fractional differential equations by means of a modified version of the Bernstein polynomials called the Bhatti-fractional polynomials basis set. The method involved approximating the desired solution and treated the resulting equation as a matrix equation. All fractional derivatives are considered in the Caputo sense. The resulting operational matrix was inverted, and the desired solution was obtained. The effectiveness of the method was demonstrated by solving two specific types of nonlinear multidimensional fractional differential equations. The results showed higher accuracy, with absolute errors ranging from 10 − 12 to 10 − 6 when compared with exact solutions. The proposed technique offered computational efficiency that could be implemented in various programming languages. The examples of two partial fractional differential equations were solved using Mathematica symbolic programming language, and the method showed potential for efficient resolution of fractional differential equations. [ABSTRACT FROM AUTHOR]

The purpose of this research is to examine the uniqueness and existence of the (ω , ρ) -BVP solution for a particular solution to a class of Hadamard fractional differential equations with impulsive boundary value requirements on Banach spaces. The notion of Banach contraction and Schaefer's theorem are used to prove the study's key findings. In addition, we offer the prerequisites for the set of solutions to the investigated boundary value with impulsive fractional differential issue to be convex. To enhance the comprehension and practical application of our findings, we offer two illustrative examples at the end of the paper to show how the results can be applied. [ABSTRACT FROM AUTHOR]

This paper is devoted to the spectral analysis of one class of integral operators, associated with the boundary-value problems for differential equations of fractional order. In particular, we show the positive definiteness of studying operators, which makes it possible to select areas in the complex plane where there are no eigenvalues for these operators. [ABSTRACT FROM AUTHOR]

Differential equations containing fractional derivatives, for both time and spatial variables, have now begun to attract the attention of mathematicians and physicists; they are used in connection with these equations as mathematical models of various processes. The fractional derivative equation tool plays a crucial role in describing plenty of natural processes concerning physics, biology, geology, and so on. In this paper, we studied a loaded equation in relation to a spatial variable for a linear pseudoparabolic equation, with an initial and second boundary value condition (the Neumann condition), and a fractional Caputo derivative. A distinctive feature of the considered problem is that the load at the point is in the higher partial derivatives of the solution. The problem is reduced to a loaded equation with a nonlocal boundary value condition. A way to solve the considered problem is by using the method of energy inequalities, so that a priori estimates of solutions for non-local boundary value problems are obtained. To prove that this nonlocal problem is solvable, we used the method of continuation with parameters. The existence and uniqueness theorems for regular solutions are proven. [ABSTRACT FROM AUTHOR]

In recent years, various qualitative investigations of the properties of differential equations with different types of generalizations of Riemann–Liouville fractional derivatives were studied and stability properties were investigated, usually using Lyapunov functions. In the application of Lyapunov functions, we need appropriate inequalities for the fractional derivatives of these functions. In this paper, we consider several Riemann–Liouville types of fractional derivatives and prove inequalities for derivatives of convex Lyapunov functions. In particular, we consider the classical Riemann–Liouville fractional derivative, the Riemann–Liouville fractional derivative with respect to a function, the tempered Riemann–Liouville fractional derivative, and the tempered Riemann–Liouville fractional derivative with respect to a function. We discuss their relations and their basic properties, as well as the connection between them. We prove inequalities for Lyapunov functions from a special class, and this special class of functions is similar to the class of convex functions of many variables. Note that, in the literature, the most common Lyapunov functions are the quadratic ones and the absolute value ones, which are included in the studied class. As a result, special cases of our inequalities include Lyapunov functions given by absolute values, quadratic ones, and exponential ones with the above given four types of fractional derivatives. These results are useful in studying types of stability of the solutions of differential equations with the above-mentioned types of fractional derivatives. To illustrate the application of our inequalities, we define Mittag–Leffler stability in time on an interval excluding the initial time point. Several stability criteria are obtained. [ABSTRACT FROM AUTHOR]

Proposing a matrix transform method to solve a fractional partial differential equation is the main aim of this paper. The main model can be transferred to a partial-integro differential equation (PIDE) with a weakly singular kernel. The spatial direction is approximated by a fourth-order difference scheme. Also, the temporal derivative is discretized via a second-order numerical procedure. First, the spatial derivatives are approximated by a fourth-order operator to compute the second-order derivatives. This process produces a system of differential equations related to the time variable. Then, the Crank–Nicolson idea is utilized to achieve a full-discrete scheme. The kernel of the integral term is discretized by using the Lagrange polynomials to overcome its singularity. Subsequently, we prove the convergence and stability of the new difference scheme by utilizing the Rayleigh–Ritz theorem. Finally, some numerical examples in one-dimensional and two-dimensional cases are presented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, by using fixed-point theorems, the existence and uniqueness of positive solutions to a class of four-point impulsive fractional differential equations with p-Laplacian operators are studied. In addition, three examples are given to justify the conclusion. The interest of this paper is to study impulsive fractional differential equations with p-Laplacian operators. [ABSTRACT FROM AUTHOR]

This paper aims to investigate the Caputo fractional differential equation involving the ρ (τ) Laplacian operator and nonlocal multi-point of Riemann–Liouville's fractional integral. We also prove the uniqueness of the positive solutions for Boyd and Wong's nonlinear contraction via the Guo–Krasnoselskii fixed-point theorem in Banach spaces. Finally, we illustrate the theoretical results and show that by solving the nonlocal problems, it is possible to obtain accurate approximations of the solutions. An example is also provided to illustrate the applications of our theorem. [ABSTRACT FROM AUTHOR]

The numerical solution for fractional dynamics problems can create a high computational load, which makes it necessary to implement efficient algorithms for their solution. The main contribution to the computational load of such computations is created by heredity (memory), which is determined by the dependence of the current value of the solution function on previous values in the time interval. In terms of mathematics, the heredity here is described using a fractional differentiation operator in the Gerasimov–Caputo sense of variable order. As an example, we consider the Cauchy problem for the non-linear fractional Riccati equation with non-constant coefficients. An efficient parallel implementation algorithm has been proposed for the known sequential non-local explicit finite-difference numerical solution scheme. This implementation of the algorithm is a hybrid one, since it uses both GPU and CPU computational nodes. The program code of the parallel implementation of the algorithm is described in C and CUDA C languages, and is developed using OpenMP and CUDA hardware, as well as software architectures. This paper presents a study on the computational efficiency of the proposed parallel algorithm based on data from a series of computational experiments that were obtained using a computing server NVIDIA DGX STATION. The average computation time is analyzed in terms of the following: running time, acceleration, efficiency, and the cost of the algorithm. As a result, it is shown on test examples that the hybrid version of the numerical algorithm can give a significant performance increase of 3–5 times in comparison with both the sequential version of the algorithm and OpenMP implementation. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of bicomplex-valued controlled metric spaces and prove fixed point theorems. Our results mainly focus on generalizing and expanding some recently established results. Finally, we explain an application of our main result to a certain type of fractional differential equation. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce new forms of quasi-contractions in metric-like spaces and initiate more general conditions for the existence of invariant points for such operators. The proposed notions are then applied to study novel existence criteria for the existence of solutions to two-point boundary value problems in the domains of integer and fractional orders. To attract further research in this direction, important consequences are deduced and discussed to indicate the novelty and generality of our proposed concepts. [ABSTRACT FROM AUTHOR]

Within the last few years, many of the efforts of the fractional calculus (FC) community have been directed towards clarifying the nature and essential properties of the operators known as fractional integrals and derivatives. In [[3]], some integral and integro-differential equations with Prabhakar fractional integrals and derivatives are solved in explicit form and their asymptotic behavior is discussed. One of the most interesting and promising recent approaches to the time-fractional integrals and derivatives is that based on the integral and integro-differential operators with Sonin kernels. [Extracted from the article]

In economics, depreciation functions (operator kernels) are certain decreasing functions, which are assumed to be equal to unity at zero. Usually, an exponential function is used as a depreciation function. However, exponential functions in operator kernels do not allow simultaneous consideration of memory effects and depreciation effects. In this paper, it is proposed to consider depreciation of a non-exponential type, and simultaneously take into account memory effects by using the Prabhakar fractional derivatives and integrals. Integro-differential operators with the Prabhakar (generalized Mittag-Leffler) function in the kernels are considered. The important distinguishing features of the Prabhakar function in operator kernels, which allow us to take into account non-exponential depreciation and fading memory in economics, are described. In this paper, equations with the following operators are considered: (a) the Prabhakar fractional integral, which contains the Prabhakar function as the kernels; (b) the Prabhakar fractional derivative of Riemann–Liouville type proposed by Kilbas, Saigo, and Saxena in 2004, which is left inverse for the Prabhakar fractional integral; and (c) the Prabhakar operator of Caputo type proposed by D'Ovidio and Polito, which is also called the regularized Prabhakar fractional derivative. The solutions of fractional differential equations with the Prabhakar operator and its special cases are suggested. The asymptotic behavior of these solutions is discussed. [ABSTRACT FROM AUTHOR]

The price V of a contingent claim in finance, insurance and economics is defined as an expectation of a stochastic expression. If the underlying uncertainty is modeled as a strong Markov process X, the Feynman–Kac theorem suggests that V is the unique solution of a boundary problem for a parabolic equation. In the case of PDO with constant symbols, simple probabilistic tools explained in this paper can be used to explicitly calculate expectations under very weak conditions on the process and study the regularity of the solution. Assuming that the Feynman–Kac theorem holds, and a more general boundary problem can be localized, the local results can be used to study the existence and regularity of solutions, and derive efficient numerical methods. In the paper, difficulties for the realization of this program are analyzed, several outstanding problems are listed, and several closely efficient methods are outlined. [ABSTRACT FROM AUTHOR]

In this paper, we first consider the general fractional derivatives of arbitrary order defined in the Riemann–Liouville sense. In particular, we deduce an explicit form of their null space and prove the second fundamental theorem of fractional calculus that leads to a closed form formula for their projector operator. These results allow us to formulate the natural initial conditions for the fractional differential equations with the general fractional derivatives of arbitrary order in the Riemann–Liouville sense. In the second part of the paper, we develop an operational calculus of the Mikusiński type for the general fractional derivatives of arbitrary order in the Riemann–Liouville sense and apply it for derivation of an explicit form of solutions to the Cauchy problems for the single- and multi-term linear fractional differential equations with these derivatives. The solutions are provided in form of the convolution series generated by the kernels of the corresponding general fractional integrals. [ABSTRACT FROM AUTHOR]

The growth of colorectal cancer tumors and their reactions to chemo-immunotherapeutic treatment with monoclonal antibodies (mAb) are discussed in this paper using a system of fractional order differential equations (FDEs). mAb medications are still at the research stage; however, this research takes into account the mAbs that are already in use. The major goal is to demonstrate the effectiveness of the mAb medication Cetuximab and the significance of IL-2 levels in immune system support. The created model is broken down into four sub-systems: cell populations, irinotecan (CPT11) concentration for treatment, IL-2 concentration for immune system support, and monoclonal antibody Cetuximab. We show the existence and uniqueness of the initial value problem (IVP). After that, we analyze the stability of the equilibrium points (disease-free and co-existing) using the Routh–Hurwitz criteria. In addition, in applying the discretization process, we demonstrate the global stability of the constructed system around the equilibrium points based on specific conditions. In the end, simulation results were carried out to support the theory of the manuscript. [ABSTRACT FROM AUTHOR]

This paper introduces the generalized fractional differential quadrature method, which is based on the generalized Caputo type and is used for the first time to solve nonlinear fractional differential equations. One of the effective shape functions of this method is the Cardinal Sine shape function, which is used in combination with the fractional operator of the generalized Caputo kind to convert nonlinear fractional differential equations into a nonlinear algebraic system. The nonlinearity problem is then solved using an iterative approach. Numerical results for a variety of chaotic systems are introduced using the MATLAB program and compared with previous theoretical and numerical results to ensure their reliability, convergence, accuracy, and efficiency. The fractional parameters play an effective role in studying the proposed problems. The achieved solutions prove the viability of the presented method and demonstrate that this method is easy to implement, effective, highly accurate, and appropriate for studying fractional differential equations emerging in fields related to chaotic systems and generalized Caputo-type fractional problems in the future. [ABSTRACT FROM AUTHOR]

Despite recent advances in the mathematical modeling of biological processes and real-world situations raised in the day-to-day life phase, some phenomena such as immune cell populations remain poorly understood. The mathematical modeling of complex phenomena such as immune cell populations using nonlinear differential equations seems to be a quite promising and appropriate tool to model such complex and nonlinear phenomena. Fractional differential equations have recently gained a significant deal of attention and demonstrated their relevance in modeling real phenomena rather than their counterpart, classical (integer) derivative differential equations. We report in this paper a mathematical approach susceptible to answering some relevant questions regarding the side effects of ionizing radiation (IR) on DNA with a particular focus on double-strand breaks (DSBs), leading to the destruction of the cell population. A theoretical elucidation of the population memory was carried out within the framework of fractional differential equations (FODEs). Using FODEs, the mathematical approach presented herein ensures connections between fractional calculus and the nonlocal feature of the fractional order of immune cell populations by taking into account the memory trace and genetic qualities that are capable of integrating all previous actions and considering the system's long-term history. An illustration of both fractional modeling, which provides an excellent framework for the description of memory and hereditary properties of immune cell populations, is elucidated. The mathematics presented in this research hold promise for modeling real-life phenomena and paves the way for obtaining accurate model parameters resulting from the mathematical modeling. Finally, the numerical simulations are conducted for the analytical approach presented herein to elucidate the effect of various parameters that govern the influence of ionizing irradiation on DNA in immune cell populations as well as the evolution of cell population dynamics, and the results are presented using plots and contrasted with previous theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, the problem of the existence and uniqueness of solutions for a nonlinear fractional-order coupled delayed system with a new kind of boundary condition is studied. For this reason, we transform the above problem into an equivalent fixed point problem using the integral operator. Moreover, by applying fixed point theorems, a novel set of sufficient conditions that guarantee the existence and uniqueness of solutions of the coupled system is derived. Eventually, an example is presented to illustrate the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

Recently, researchers have been interested in studying fractional differential equations and their solutions due to the wide range of their applications in many scientific fields. In this paper, a new approach called the Hussein–Jassim (HJ) method is presented for solving nonlinear fractional ordinary differential equations. The new method is based on a power series of fractional order. The proposed approach is employed to obtain an approximate solution for the fractional differential equations. The results of this study show that the solutions obtained from solving the fractional differential equations are highly consistent with those obtained by exact solutions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a new class of nonlinear fractional integrodifferential systems that includes the Ψ -Riemann–Liouville fractional integral term. Using the technique of upper and lower solutions, the solvability of the system is examined. We add two examples to demonstrate and validate the main result. The main results highlight crucial contributions to the general theory of fractional differential equations. [ABSTRACT FROM AUTHOR]

The fractional Fourier transform is a natural generalization of the Fourier transform. In this work, we recall the definition of the fractional Fourier transform and its relation to the conventional Fourier transform. We exhibit that this relation permits one to obtain easily the main properties of the fractional Fourier transform. We investigate the sharp Hausdorff-Young inequality for the fractional Fourier transform and utilize it to build Matolcsi-Szücs inequality related to this transform. The other versions of the inequalities concerning the fractional Fourier transform is also discussed in detail. The results obtained in this paper are very significant, especially in the field of fractional differential equations. [ABSTRACT FROM AUTHOR]

This paper aims to propose a generalized fractional Fokker–Planck equation based on a stable Lévy stochastic process. To develop the general fractional equation, we will use the Lévy process rather than the Brownian motion. Due to the Lévy process, this fractional equation can provide a better description of heavy tails and skewness. The analytical solution is chosen to solve the fractional equation and is expressed using the H-function to demonstrate the indicator entropy production rate. We model market data using a stable distribution to demonstrate the relationships between the tails and the new fractional Fokker–Planck model, as well as develop an R code that can be used to draw figures from real data. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the controllability of the system with non-local conditions. The existence of a mild solution is established. We obtain the results by using resolvent operators functions, the Hausdorff measure of non-compactness, and Monch's fixed point theorem. We also present an example, in order to elucidate one of the results discussed. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a new class of boundary value problems, consisting of a mixed-type ψ 1 -Hilfer and ψ 2 -Caputo fractional order differential equation supplemented with integro-differential nonlocal boundary conditions. The uniqueness of solutions is achieved via the Banach contraction principle, while the existence of results is established by using the Leray–Schauder nonlinear alternative. Numerical examples are constructed illustrating the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and Ulam–Hyers–Rassias stability results for a class of boundary value problems for implicit ψ -Caputo fractional differential equations with non-instantaneous impulses involving both retarded and advanced arguments. The results are based on the Banach contraction principle and Krasnoselskii's fixed point theorem. In addition, the Ulam–Hyers–Rassias stability result is proved using the nonlinear functional analysis technique. Finally, illustrative examples are given to validate our main results. [ABSTRACT FROM AUTHOR]

This paper considers a class of non-homogeneous fractional systems with harmonic terms by means of the Riemann–Liouville definition. Two different approaches are applied to obtain the dual solution of the studied class. The first approach uses the Laplace transform (LT) and the solution is given in terms of the Mittag-Leffler functions. The second approach avoids the LT and expresses the solution in terms of exponential and periodic functions which is analytic in the whole domain. The current methods determine the solution directly and efficiently. The results are applicable for other problems of higher order. [ABSTRACT FROM AUTHOR]

As is known to all, Lipschitz condition, which is very important to guarantee existence and uniqueness of solution for differential equations, is not frequently satisfied in real-world problems. In this paper, without the Lipschitz condition, we intend to explore a kind of novel coupled systems of fuzzy Caputo Generalized Hukuhara type (in short, gH-type) fractional partial differential equations. First and foremost, based on a series of notions of relative compactness in fuzzy number spaces, and using Schauder fixed point theorem in Banach semilinear spaces, it is naturally to prove existence of two classes of gH-weak solutions for the coupled systems of fuzzy fractional partial differential equations. We then give an example to illustrate our main conclusions vividly and intuitively. As applications, combining with the relevant definitions of fuzzy projection operators, and under some suitable conditions, existence results of two categories of gH-weak solutions for a class of fire-new fuzzy fractional partial differential coupled projection neural network systems are also proposed, which are different from those already published work. Finally, we present some work for future research. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence and stability results for the (ρ k , ϕ k) -Hilfer fractional integro-differential equations under instantaneous impulse with non-local multi-point fractional integral boundary conditions. We achieve the formulation of the solution to the (ρ k , ϕ k) -Hilfer fractional differential equation with constant coefficients in term of the Mittag–Leffler kernel. The uniqueness result is proved by applying Banach's fixed point theory with the Mittag–Leffler properties, and the existence result is derived by using a fixed point theorem due to O'Regan. Furthermore, Ulam–Hyers stability and Ulam–Hyers–Rassias stability results are demonstrated via the non-linear functional analysis method. In addition, numerical examples are designed to demonstrate the application of the main results. [ABSTRACT FROM AUTHOR]

In this paper, we show several connections between special functions arising from generalized Conway-Maxwell-Poisson (COM-Poisson) type statistical distributions and integro-differential equations with varying coefficients involving Hadamard-type operators. New analytical results are obtained, showing the particular role of Hadamard-type derivatives in connection with a recently introduced generalization of the Le Roy function. We are also able to prove a general connection between fractional hyper-Bessel-type equations involving Hadamard operators and Le Roy functions. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce a new family of functions like an implicit function called Γ -functions. Furthermore, we introduce a new concept of α - Γ F -fuzzy contractive mappings, which is weaker than the class of fuzzy F-contractive mappings. Then, the existence and uniqueness of the fixed point are established for a new type of fuzzy contractive mappings in the setting of fuzzy metric spaces. Moreover, some examples and an application to nonlinear fractional differential equation are given, and these show the importance of the introduced theorems in fuzzy settings. [ABSTRACT FROM AUTHOR]

In this paper, we establish existence and uniqueness results for single-valued as well as multi-valued (k , ψ) -Hilfer boundary value problems of order in (1 , 2 ] , subject to nonlocal integro-multi-point boundary conditions. In the single-valued case, we use Banach and Krasnosel'skiĭ fixed point theorems as well as a Leray–Schauder nonlinear alternative to derive the existence and uniqueness results. For the multi-valued problem, we prove two existence results for the convex and non-convex nature of the multi-valued map involved in a problem by applying a Leray–Schauder nonlinear alternative for multi-valued maps, and a Covitz–Nadler fixed point theorem for multi-valued contractions, respectively. Numerical examples are presented for illustration of all the obtained results. [ABSTRACT FROM AUTHOR]

The generalized proportional Caputo fractional derivative is a comparatively new type of derivative that is a generalization of the classical Caputo fractional derivative, and it gives more opportunities to adequately model complex phenomena in physics, chemistry, biology, etc. In this paper, the presence of noninstantaneous impulses in differential equations with generalized proportional Caputo fractional derivatives is discussed. Generalized proportional Caputo fractional derivatives with fixed lower limits at the initial time as well as generalized proportional Caputo fractional derivatives with changeable lower limits at each impulsive time are considered. The statements of the problems in both cases are set up and the integral representation of the solution of the defined problem in each case is presented. Ulam-type stability is also investigated and some examples are given illustrating these concepts. [ABSTRACT FROM AUTHOR]

We consider the following stochastic fractional differential equation C D 0 + α , ρ φ (t) = κ ϑ (t , φ (t)) w ˙ (t) , 0 < t ≤ T , where φ (0) = φ 0 is the initial function, C D 0 + α , ρ is the Caputo–Katugampola fractional differential operator of orders 0 < α ≤ 1 , ρ > 0 , the function ϑ : [ 0 , T ] × R → R is Lipschitz continuous on the second variable, w ˙ (t) denotes the generalized derivative of the Wiener process w (t) and κ > 0 represents the noise level. The main result of the paper focuses on the energy growth bound and the asymptotic behaviour of the random solution. Furthermore, we employ Banach fixed point theorem to establish the existence and uniqueness result of the mild solution. [ABSTRACT FROM AUTHOR]

Practical stability properties of generalized proportional Caputo fractional differential equations with bounded delay are studied in this paper. Two types of stability, practical stability and exponential practical stability, are defined and considered, and also some sufficient conditions to guarantee stability are presented. The study is based on the application of Lyapunov like functions and their generalized proportional Caputo fractional derivatives among solutions of the studied system where appropriate Razumikhin like conditions are applied (appropriately modified in connection with the fractional derivative considered). The theory is illustrated with several nonlinear examples. [ABSTRACT FROM AUTHOR]

This paper is devoted to investigating one type of nonlinear two-term fractional order delayed differential equations involving Caputo fractional derivatives. The Leray–Schauder alternative fixed-point theorem and Banach contraction principle are applied to analyze the existence and uniqueness of solutions to the problem with infinite delay. Additionally, the Hyers–Ulam stability of fractional differential equations is considered for the delay conditions. [ABSTRACT FROM AUTHOR]

In this paper, we present a qualitative study of an implicit fractional differential equation involving Riemann–Liouville fractional derivative with delay and its corresponding integral equation. Under some sufficient conditions, we establish the global and local existence results for that problem by applying some fixed point theorems. In addition, we have investigated the continuous and integrable solutions for that problem. Moreover, we discuss the continuous dependence of the solution on the delay function and on some data. Finally, further results and particular cases are presented. [ABSTRACT FROM AUTHOR]

For the first time, a self-consistent mathematical approach to describe economic processes with a general form of a memory function is proposed. In this approach, power-type memory is a special case of such general memory. The memory is described by pairs of memory functions that satisfy the Sonin and Luchko conditions. We propose using general fractional calculus (GFC) as a mathematical language that allows us to describe a general form of memory in economic processes. The existence of memory (non-locality in time) means that the process depends on the history of changes to this process in the past. Using GFC, exactly solvable economic models of natural growth with a general form of memory are proposed. Equations of natural growth with general memory are equations with general fractional derivatives and general fractional integrals for which the fundamental theorems of GFC are satisfied. Exact solutions for these equations of models of natural growth with general memory are derived. The properties of dynamic maps with a general form of memory are described in the general form and do not depend on the choice of specific types of memory functions. Examples of these solutions for various types of memory functions are suggested. [ABSTRACT FROM AUTHOR]

In this work, we study the equation of the catenary curve in the context of the Caputo derivative. We solve this equation and compare the solution with real physical models. From the experiments, we find that the best approximation is achieved in the classical case. Therefore, introducing a fractional parameter arbitrarily can be detrimental. However, we observe that, when adding a certain weight to the chain, fractional calculus produces better results than classical calculus for modeling the minimum height. [ABSTRACT FROM AUTHOR]