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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]

Within the context of fractional calculus, we investigate novel mathematical possibilities. In this context, using the fractional dispersion relations for the fractional wave equation, we explore a class of the generalized fractional wave equation numerically. Some important classes of differential equations in the theory of wave studies are Drinfeld–Sokolov–Wilson and Shallow Water equations. In this effort, the natural transform decomposition technique has been implemented to investigate the explicit result of fractional-order coupled schemes of Drinfeld–Sokolov–Wilson and Shallow Water coupled systems. The proposed method is obtained by coupling the Natural transform with the Adomian decomposition process. The current technique significantly works to find the approximate solution without any discretization or constraining parameter assumptions. The obtained numerical and graphical outcomes by the devised technique are compared with the available exact result to verify the convergence of the method. For mathematical calculations, the Mathematica software package is used. [ABSTRACT FROM AUTHOR]

In this study, we employ the effective iterative method to address the fractional Wu-Zhang Equation within the framework of the Caputo Derivative. The effective iterative method offers a practical approach to obtaining approximate solutions for fractional differential equations. We seek to provide insights into its solution and behavior by applying this method to the Wu-Zhang Equation. Through numerical analysis and the presentation of relevant tables and Figures, we demonstrate the accuracy and efficiency of this method in solving the fractional Wu-Zhang Equation. This research contributes to the understanding and solution of fractional-order differential equations and their applications in various scientific and engineering domains. [ABSTRACT FROM AUTHOR]

This work discusses the beginning of fractional calculus and how the Sumudu and Elzaki transforms are applied to fractional derivatives. This approach combines a double Sumudu-Elzaki transform strategy to discover analytic solutions to space-time fractional partial differential equations in Mittag- Leffler functions subject to initial and boundary conditions. Where this method gets closer and closer to the correct answer, and the technique's efficacy is demonstrated using numerical examples performed with Matlab R2015a. [ABSTRACT FROM AUTHOR]

In this paper, we study new classes of differential equations of fractional order. The first considered problem involves the derivative of Caputo, while the second one involves conformable Khalil derivative. For the first class, we prove an existence and uniqueness result, then, we discuss an example to show the applicability of the result. For the second one, we apply the exp-function method to find new traveling wave solutions for a generalised conformable fractional partial differential equation, then some examples on Ostrovsky and KdV equations are given to illustrate the efficiency and accuracy of the method. Some graphs are plotted and discussed to show more the importance of the obtained results. [ABSTRACT FROM AUTHOR]

We investigate the existence and uniqueness of the solution and also the rate of convergence of a numerical method for a fractional differential equation in both q-calculus and (p; q)-calculus versions. We use the Banach and Schauder fixed point theorems in this study. We provide two examples, one by definition of the q-derivative and the other by (p; q)-derivative. We compare the rate of convergence of the numerical method. We like to clear some facts on (p; q)-calculus. The data from our numerical calculations show well that q-calculus works better than (p; q)-calculus in each case. [ABSTRACT FROM AUTHOR]

In this work, the Optimal Homotopy Asymptotic Method (OHAM) is prolifically implemented to find the optimal solutions of fractional order of fuzzy differential equations. We inspect the competence of the method by examining fractional order time-dependent first order fuzzy initial value problems in the Caputo derivative sense. The concepts of fuzzy sets theory and fractional calculus were used to present a new general fuzzification formulation of the OHAM method followed by fuzzy analysis for the proposed fractional problems are presented in detail. OHAM is recognized as a feasible and convergent method for solving linear and nonlinear models, and it provides a convenient technique for controlling the convergence of approximate solutions optimally. The method's capability is demonstrated by obtaining approximate solutions to linear and nonlinear fuzzy initial value problems in various orders of fractional derivative. The numerical results acknowledge that OHAM is a feasible and transformation mechanism for solving linear and nonlinear fuzzy fractional initial value problems. Also, the results ensure that the method used was succinct, efficient, and simple to use, to deal with a more comprehensive fuzzy fractional differential equation. [ABSTRACT FROM AUTHOR]

In this paper, a reliable numerical scheme is developed and reviewed in order to obtain approximate solution of time fractional parabolic partial differential equations. The introduced scheme is based on Legendre tau spectral approximation and the time fractional derivative is employed in the Caputo sense. The L² convergence analysis of the numerical method is analysed. Numerical results for different examples are examined to verify the accuracy of spectral method and justification the theoretical analysis, and to compare with other existing methods in the literatures. [ABSTRACT FROM AUTHOR]

In this paper, we study a nonlinear differential equation of Lane Emden type. The considered problem, with its nonlocal conditions, involves Caputo fractional derivative and Riemann-Liouville integral in its nonlinearities. We investigate the existence and uniqueness of solution for our problem. Then, we study the Ulam-Hyers stability of its solutions. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence and uniqueness of a mild solution to a class of semilinear fractional differential equation in an infinite Banach space with Caputo derivative order 0 < α ≤ 1. Furthermore, we establish the stability conditions and then prove that the considered initial value problem is exponentially stabilizable when the stabilizer acts linearly on the control system. [ABSTRACT FROM AUTHOR]

Purpose The purpose of this paper was to verify the possibility of applying differential calculus of incomplete order to describe relaxation of the material obtained using selective laser sintering (SLS) technology.Design/methodology/approach The samples were made using the incremental technology for three print directions. Relaxation tests were conducted. The theoretical curves, which are the solution of the equation describing the five-parameter Maxwell-Wiechert model for derivatives in relation to the total time of complete order and fractional order, were adjusted to the obtained experimental curves.Findings The SLS technology creates new possibilities regarding modelling polymeric elements which might be applied as functional models (products). Therefore, it is necessary to conduct an in-depth study of their properties, including relaxation properties, which is associated with the necessity to use proper mathematical tools to describe those properties. The differential calculus of incomplete order was applied herein to describe the anisotropy of relaxation properties because of the print direction in relation to the relaxation curves adjusted with the five-parameter Maxwell-Wiechert model.Research limitations/implications As a result of the conducted considerations, the authors obtained the dependencies describing the anisotropy of relaxation properties with the use of coefficients alpha and beta, which stand for the derivative order of the differential equation, whereas coefficient kappa stands for the translation coefficient which is an innovative application of this type of mathematical apparatus.Practical implications The developed method might be applied to describe the anisotropy of a broader group of materials manufactured with the use of incremental technologies.Originality/value The application of the differential calculus of incomplete order to describe the anisotropy of the materials manufactured from polyamide powder using the SLS technology is a distinctive feature of this paper. A crucial cognitive element of the conducted research is the fact which confirms that the dynamic viscosity coefficients have the greatest impact on the anisotropy of material properties depending on the print directions. [ABSTRACT FROM AUTHOR]

In this paper, a Bernoulli pseudo-spectral method for solving nonlinear fractional Volterra integro-differential equations is considered. First existence of a unique solution for the problem under study is proved. Then the Caputo fractional derivative and Riemman-Liouville fractional integral properties are employed to derive a new approximate formula for unknown function of the problem. The suggested technique transforms this type of equations to the solution of a system of algebraic equations. In the next step, the error analysis of the proposed method is investigated. Finally, the technique is applied to some problems to show its validity and applicability. [ABSTRACT FROM AUTHOR]

In this work, an efficient generalized differential transform method (GDTM) is proposed for solving the twodimensional Volterra-Integro differential equation (2-DVIDE) of fractional order. The results of the proposed method are compared with exact solution, a numerical example is considered for testing the accuracy and validity of this method. [ABSTRACT FROM AUTHOR]

We propose a generalized definition of Caputo derivatives from t = 0 of order γ∈in (0, 1) using a convolution group, and we build a convenient framework for studying initial value problems of general nonlinear time fractional differential equations. Our strategy is to define a modified Riemann-Liouville fractional calculus which agrees with the traditional Riemann-Liouville definition for t > 0 but includes some singularities at t = 0 so that the group property holds. Then, making use of this fractional calculus, we introduce the generalized definition of Caputo derivatives. The new definition is consistent with various definitions in the literature while revealing the underlying group structure. The underlying group property makes many properties of Caputo derivatives natural. In particular, it allows us to deconvolve the fractional differential equations to integral equations with completely monotone kernels, which then enables us to prove the general comparison principle with the most general conditions. This then allows for a priori energy estimates of fractional PDEs. Since the new definition is valid for locally integrable functions that can blow up in finite time, it provides a framework for solutions to fractional ODEs and fractional PDEs. Many fundamental results for fractional ODEs are revisited within this framework under very weak conditions. [ABSTRACT FROM AUTHOR]

This paper studies the existence and uniqueness of polynomial solutions for a class of fractional differential equations and systems involving Caputo derivatives. Several illustrative examples with explicit solutions are also presented. [ABSTRACT FROM AUTHOR]

This paper presents a new computational technique for solving fractional pantograph differential equations. The fractional derivative is described in the Caputo sense. The main idea is to use Müntz-Legendre wavelet and its operational matrix of fractional-order integration. First, the Müntz-Legendre wavelet is presented. Then a family of piecewise functions is proposed, based on which the fractional order integration of the Müntz-Legendre wavelets are easy to calculate. The proposed approach is used this operational matrix with the collocation points to reduce the under study problem to a system of algebraic equations. An estimation of the error is given in the sense of Sobolev norms. The efficiency and accuracy of the proposed method are illustrated by several numerical examples. [ABSTRACT FROM AUTHOR]

This work studies the approximate controllability of a class of impulsive neutral stochastic differential equations with infinite delay and Poisson jumps involving generalized Caputo fractional derivative under the condition that the corresponding linear system is approximately controllable. Utilizing the fixed point theory and sectorial operator theory, the existence of the mild solution of the impulsive neutral stochastic equation is established imposing weaker regularity on nonlinear terms. A set of sufficient conditions establishing controllability results is derived with the help of stochastic analysis and fractional calculus. Finally, an example is provided to illustrate the obtained abstract result. [ABSTRACT FROM AUTHOR]

According to semigroup theories, fractional calculus, Banach contraction principle and Schaefer's fixed point theorem, this paper is fundamentally involved with the existence of mild solutions for an impulsive fractional neutral differential systems (abbreviated, IFNDS) with nonlocal conditions (abbreviated, NLC) in Banach space X. At last, an illustration is also presented to exhibit the use of our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we first introduce fractional orthogonal Jacobi functions then we obtain a new fractional derivative operational matrix for these orthogonal functions. It is based on the relationship between the coefficients of the fractional Taylor series and fractional Jacobi function expansions. We also apply this new operational matrix to the collocation method for solving general multi-order fractional differential equations (FDEs) and nonlinear fractional integro–differential equations (FIDEs). We also present several test problems. The numerical results show that our new scheme is very effective and convenient for solving FDEs and FIDEs. [ABSTRACT FROM PUBLISHER]

Abstract: In this paper, we establish a local stable manifold theorem near a hyperbolic equilibrium point for planar fractional differential equations. The construction of this stable manifold is based on the associated Lyapunov–Perron operator. An example is provided to illustrate the result. [Copyright &y& Elsevier]

Abstract: The flow through porous media can be better described by fractional models than the classical ones since they include inherently memory effects caused by obstacles in the structures. The variational iteration method was extended to find approximate solutions of fractional differential equations with the Caputo derivatives, but the Lagrange multipliers of the method were not identified explicitly. In this paper, the Lagrange multiplier is determined in a more accurate way and some new variational iteration formulae are presented. [Copyright &y& Elsevier]

Purpose – The purpose of this paper is to show how an application of fractional two dimensional differential transformation method (DTM) obtained approximate analytical solution of time-fraction modified equal width wave (MEW) equation. Design/methodology/approach – The fractional derivative is described in the Caputo sense. Findings – It is indicated that the solutions obtained by the two dimensional DTM are reliable and that this is an effective method for strongly nonlinear partial equations. Originality/value – The paper shows that exact solutions can also be obtained from the known forms of the series solutions. [ABSTRACT FROM AUTHOR]

The diffusion equations with the local and the nonlocal fractional derivatives have been used to describe the flow through disorder media. Recently, the variational iteration method is successfully developed to find approximate solutions of the two kinds of fractional differential equations. This study reveals the new development of the method and compares the applications in two types of fractional diffusions. [ABSTRACT FROM AUTHOR]

Abstract: We consider a class of boundary value problems for linear multi-term fractional differential equations which involve Caputo-type fractional derivatives. Using an integral equation reformulation of the boundary value problem, some regularity properties of the exact solution are derived. Based on these properties, the numerical solution of boundary value problems by piecewise polynomial collocation methods is discussed. In particular, we study the attainable order of convergence of proposed algorithms and show how the convergence rate depends on the choice of the grid and collocation points. Theoretical results are verified by two numerical examples. [Copyright &y& Elsevier]

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The definition, properties and applications of the Sumudu transform to ordinary differential equations are described in [1-3]. In this manuscript we derive the formulae for the Sumudu and double Sumudu transforms of ordinary and partial fractional derivatives and apply them in solving Caputo-fractional differential equations. Our purpose here is to show the applicability of this new transform and its efficiency in solving such problems. [ABSTRACT FROM AUTHOR]

Abstract: Enlightened by the Caputo type of fractional derivative, here we bring forth a concept of “memory-dependent derivative”, which is simply defined in an integral form of a common derivative with a kernel function on a slipping interval. In case the time delay tends to zero it tends to the common derivative. High order derivatives also accord with the first order one. Comparatively, the form of kernel function for the fractional type is fixed, yet that of the memory-dependent type can be chosen freely according to the necessity of applications. So this kind of definition is better than the fractional one for reflecting the memory effect (instantaneous change rate depends on the past state). Its definition is more intuitionistic for understanding the physical meaning and the corresponding memory-dependent differential equation has more expressive force. [Copyright &y& Elsevier]

Abstract: Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. For that reason we need a reliable and efficient technique for the solution of fractional differential equations. This paper deals with the numerical solution of a class of fractional differential equations. The fractional derivatives are described in the Caputo sense. Our main aim is to generalize the Legendre operational matrix to the fractional calculus. In this approach, a truncated Legendre series together with the Legendre operational matrix of fractional derivatives are used for numerical integration of fractional differential equations. The main characteristic behind the approach using this technique is that it reduces such problems to those of solving a system of algebraic equations thus greatly simplifying the problem. The method is applied to solve two types of fractional differential equations, linear and nonlinear. Illustrative examples are included to demonstrate the validity and applicability of the presented technique. [Copyright &y& Elsevier]

Abstract: Present paper deals with fractional version of a dynamical system introduced by C. Liu, L. Liu and T. Liu [C. Liu, L. Liu, T. Liu, A novel three-dimensional autonomous chaos system, Chaos Solitons Fractals 39 (4) (2009) 1950–1958]. Numerical investigations on the dynamics of this system have been carried out. Properties of the system have been analyzed by means of Lyapunov exponents. Furthermore the minimum effective dimensions have been identified for chaos to exist in commensurate and incommensurate orders. It is noteworthy that the results obtained are consistent with the analytical conditions given in the literature. [Copyright &y& Elsevier]

In this study, exact and approximate solutions of higher-dimensional time-fractional diffusion equations were obtained using a relatively new method, the fractional reduced differential transform method (FRDTM). The exact solutions can be found with the benefit of a special function, and we applied Caputo fractional derivatives in this method. The numerical results and graphical representations specified that the proposed method is very effective for solving fractional diffusion equations in higher dimensions. [ABSTRACT FROM AUTHOR]

In this article we consider a class of singular linear systems of first order, and introduce a generalized fractional order feedback controller of Caputo type. The closed loop system in question is a singular system of differential equations having both first, and fractional order derivatives. We provide a comprehensive theory for the existence and uniqueness of solutions, as well as for the stability of the system with inclusion of the fractional order controller. An example of a singular system with a fractional order proportional integral controller, as well as an example on a 3-bus power system with inclusion of a fractional order damping controller, is given to illustrate our theory. [ABSTRACT FROM AUTHOR]

In this paper, we derive some sufficient conditions which ensure the existence and uniqueness of a solution for a class of nonlinear three point boundary value problems of fractional order implicit differential equations (FOIDEs) with some boundary and impulsive conditions. Also we investigate various types of Hyers-Ulam stability (HUS) for our concerned problem. Using classical fixed point theory and nonlinear functional analysis, we obtain the required conditions. In the last section we give an example to show the applicability of our obtained results. [ABSTRACT FROM AUTHOR]