Search

We have studied initial value problems for Caputo fractional differential equations with singular nonlinearities involving the p-Laplacian operator. We have given a precise mathematical analysis of the equivalence of the fractional differential equations and Volterra integral equations studied in this paper. A theorem for the global existence of the solution was proven. In addition, an example was given at the end of the article as an application of the results found in this paper. [ABSTRACT FROM AUTHOR]

This paper presents numerical methods for solving the one-dimensional fractional reaction-diffusion equation with the fractional Caputo derivative. The proposed methods are based on transformation of the fractional differential equation to an equivalent form of a integro-differential equation. The paper proposes an improvement of the existing implicit method, and a new explicit method. Stability and convergence tests of the methods were also conducted. [ABSTRACT FROM AUTHOR]

This paper introduces a novel approach to obtaining analytical solutions for fuzzy fractional differential equations in the context of weighted Caputo-type generalized Hukuhara derivatives. The paper proposes the use of non-singular kernels to improve the accuracy of fractional calculus in fuzzy space and establishes the uniqueness of solutions for fuzzy fractional differential equations. The paper also introduces the concept of fuzzy Laplace transforms to facilitate the solution of these equations. Practical examples, such as the fuzzy fractional Newton’s law of heating and cooling, are provided to demonstrate the effectiveness of the proposed method. Overall, this paper contributes to the development of practical solutions for real-world problems in fuzzy space and enhances the accuracy of fractional calculus in this context. [ABSTRACT FROM AUTHOR]

In this paper, we primarily focused on the existence and uniqueness of the initial value problem for fractional order fuzzy ordinary differential equations in a fuzzy metric space. First, definitions and relevant properties of the Gamma function and Beta function within a fuzzy metric space were provided. Second, by employing the principle of fuzzy compression mapping and Choquet integral of fuzzy numerical functions, we established the existence and uniqueness of solutions to initial value problems for fuzzy ordinary differential equations. Finally, several examples were presented to demonstrate the validity of our obtained results. [ABSTRACT FROM AUTHOR]

This paper aims to investigate the existence, uniqueness, and stability properties for a class of fractional weighted Cauchy-type problem in the variable exponent Lebesgue space Lp(.). The obtained results are set up by employing generalized intervals and piece-wise constant functions so that the Lp(.) is transformed into the classical Lebesgue spaces. Moreover, the usual Banach Contraction Principle is utilized, and the Ulam-Hyers (UH) stability is studied. At the final stage, we provide an example to support the accuracy of the obtained results. [ABSTRACT FROM AUTHOR]

Many techniques have been recently employed by researchers to address the challenges posed by fractional differential equations. In this paper, we investigate the concept of Ulam-Hyers stability for a class of neutral fractional stochastic differential equations by using the Banach fixed point theorem and the stochastic analysis techniques. An example is presented at the end of the paper to show the interest and the applicability of the results. [ABSTRACT FROM AUTHOR]

This document discusses recent trends in mathematical models and solitons used to analyze real-world problems. The focus is on the application of fractional calculus in modeling phenomena such as anomalous diffusion, non-Markovian processes, and random walk. The document provides summaries of 15 papers accepted for publication in a special issue, covering topics such as digestive system modeling, COVID-19 pandemic modeling, control systems for motors, symmetry in pine wilt disease, soliton solutions for various equations, and epidemic modeling. The papers employ various mathematical techniques and provide insights into different aspects of real-world problems. [Extracted from the article]

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

Addressing the boundary value problems of fractional-order differential equations hold significant importance due to their applications in various fields. The aim of this paper was to approximate solutions for a class of boundary value problems involving Caputo fractional-order differential equations employing the AA-iterative scheme. Moreover, the stability and data dependence results of the iterative scheme were given for a certain class of mappings. Finally, a numerical experiment was illustrated to support the results presented herein. The results presented in this paper extend and unify some well-known comparable results in the existing literature. [ABSTRACT FROM AUTHOR]

A new class of nonlocal boundary value problems consisting of multi-term delay fractional differential equations and multipoint-integral boundary conditions is studied in this paper. We derive a more general form of the solution for the given problem by applying a fractional integral operator of an arbitrary order βξ instead of β1; for details, see Lemma 2.2. The given problem is converted into an equivalent fixed-point problem to apply the tools of fixed-point theory. The existence of solutions for the given problem is established through the use of a nonlinear alternative of the Leray-Schauder theorem, while the uniqueness of its solutions is shown with the aid of Banach's fixed-point theorem. We also discuss the stability criteria, icluding Ulam-Hyers, generalized Ulam-Hyers, Ulam-Hyers-Rassias, and generalized Ulam-Hyers-Rassias stability, for solutions of the problem at hand. For illustration of the abstract results, we present examples. Our results are new and useful for the discipline of multi-term fractional differential equations related to hydrodynamics. The paper concludes with some interesting observations. [ABSTRACT FROM AUTHOR]

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 FROM AUTHOR]

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 ≤ c < d and fixed points at the end of the interval [0, d]. For i = 0, 1, the boundary conditions are as follows: ai, bi > 0, a0p(c) = -b0p(d), a1p'(c) = -b1p'(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 fixedpoint 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. [ABSTRACT FROM AUTHOR]

The aim of this paper is to numerically solve the Fokker-Planck-Kolmogorov fractional-time differential equations using the Legendre wavelet. Also, we analyzed the convergence of function approximation using Legendre wavelets. Introduced the absolute value between the exact answer and the approximate answer obtained by the given numerical methods, and analyzed the error of the numerical method. This method has the advantage of being simple to solve. The results revealed that the suggested numerical method is highly accurate and effective. The results for some numerical examples are documented in table and graph form to elaborate on the efficiency and precision of the suggested method. The simulation was carried out using MATLAB software. In this paper and for the first time, the authors presented results on the numerical simulation for classes of time-fractional differential equations. The authors applied the reproducing Legendre wavelet method for the numerical solutions of nonlinear Fokker-Planck-Kolmogorov time-fractional differential equation. The method presented in the present study can be used by programmers, engineers, and other researchers in this field. [ABSTRACT FROM AUTHOR]

The existence of solutions of non-instantaneous impulsive Hilfer–Katugampola fractional differential equations of order $ 1/2 \lt \alpha \lt 1 $ 1 / 2 < α < 1 and parameter $ 0\leq \beta \leq 1 $ 0 ≤ β ≤ 1 with fractional Brownian motion (fBm) and Poisson jumps is investigated in this paper. The required results are obtained based on fractional calculus, stochastic analysis, semigroups, and the fixed point theorem. In the end of the paper, an example is provided to illustrate the applicability of the theoretical results. [ABSTRACT FROM AUTHOR]

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

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

In the present paper, we discuss the application of the Laplace decomposition method (LDM) to find approximate solutions for some classes of time and space-conformable fractional evolution equations. Some illustrative examples are presented showing that the LDM is an efficient method for finding solutions of nonlinear conformable fractional problems. [ABSTRACT FROM AUTHOR]

This paper devotes to consider the local existence of the strong solutions to the generalized MHD system with fractional dissipative terms 2au for the velocity field and 2ab for the magnetic field, respectively. We construct the approximate solutions by the Fourier truncation method, and use energy method to obtain the local existence of strong solutions in Hs(Rn) (s > max n 2 + 1 - 2a, 0) for any a = 0. [ABSTRACT FROM AUTHOR]

Grain-oriented silicon steel (GO FeSi) laminations are vital components for efficient energy conversion in electromagnetic devices. While traditionally optimized for power frequencies of 50/60 Hz, the pursuit of higher frequency operation (f ≥ 200 Hz) promises enhanced power density. This paper introduces a model for estimating GO FeSi laminations' magnetic behavior under these elevated operational frequencies. The proposed model combines the Maxwell diffusion equation and a material law derived from a fractional differential equation, capturing the viscoelastic characteristics of the magnetization process. Remarkably, the model's dynamical contribution, characterized by only two parameters, achieves a notable 4.8% Euclidean relative distance error across the frequency spectrum from 50 Hz to 1 kHz. The paper's initial section offers an exhaustive description of the model, featuring comprehensive comparisons between simulated and measured data. Subsequently, a methodology is presented for the localized segregation of magnetic losses into three conventional categories: hysteresis, classical, and excess, delineated across various tested frequencies. Further leveraging the model's predictive capabilities, the study extends to investigating the very high-frequency regime, elucidating the spatial distribution of loss contributions. The application of proportional–iterative learning control facilitates the model's adaptation to standard characterization conditions, employing sinusoidal imposed flux density. The paper deliberates on the implications of GO FeSi behavior under extreme operational conditions, offering insights and reflections essential for understanding and optimizing magnetic core performance in high-frequency applications. [ABSTRACT FROM AUTHOR]

This paper deals with a class of backward stochastic differential equations driven by fractional Brownian motion (with Hurst parameter H greater than 1/2) with time-delayed generators. In this type of equation, a generator at time t can depend on the values of a solution in the past, weighted with a time-delay function, for instance, of the moving average type. We establish an existence and uniqueness result of solutions for a sufficiently small time horizon or for a sufficiently small Lipschitz constant of a generator. The stochastic integral used throughout the paper is the divergence operator-type integral. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of solution and extremal solutions for an initial-value problem of perturbed functional fractional differential equations with Caputo-Hadamard derivative. Our analysis relies on the fixed point theorem of Burton and Kirk and the concept of upper and lower solutions combined with a fixed point theorem in ordered Banach space established by Dhage and Henderson. [ABSTRACT FROM AUTHOR]

This paper was concerned with the existence and uniqueness results for a coupled system of nonlinear generalized fractional differential equations supplemented with a new class of nonlocal coupled multipoint boundary conditions containing Riemann-Stieltjes and generalized fractional integrals. The nonlinearities in the given system depend on the unknown functions as well as their lower order generalized fractional derivatives. We made use of the Leray-Schauder alternative and Banach contraction mapping principle to obtain the desired results. An illustrative example was also discussed. The paper concluded with some interesting observations. [ABSTRACT FROM AUTHOR]

This paper focuses on the analysis of a coupled system governed by a Caputo-fractional derivative with q-integral-coupled boundary conditions. This system is particularly relevant in modeling multi-atomic systems, including scenarios involving adsorbed atoms or clusters on crystalline surfaces, surface–atom scattering, and atomic friction. To investigate this system, we introduce an operator that exhibits fixed points corresponding to the solutions of the problem, effectively transforming the system into an equivalent fixed-point problem. We established the necessary conditions for the existence and uniqueness of solutions using the Leray–Schauder nonlinear alternative and the Banach contraction mapping principle, respectively. Stability results in the Ulam sense for the coupled system are also discussed, along with a sensitivity analysis of the range parameters. To support the validity of their findings, we provide illustrative examples. Overall, the paper offers a thorough examination and analysis of the considered coupled system, making important contributions to the understanding of multi-atomic systems and their mathematical modeling. [ABSTRACT FROM AUTHOR]

In this paper, we study the approximate controllability of certain class of impulsive evolution stochastic functional differential equations, with variable delays, driven by a fractional Brownian motion in a separable real Hilbert space. We derive a new set of sufficient conditions for approximate controllability using a stochastic analysis of fractional Brownian motion with Hurst parameter H ∈ (1/2,1) and a Schaefer's fixed point theorem. An example is considered at the end of the paper to illustrate the obtained abstract results. [ABSTRACT FROM AUTHOR]

Nonexistence theorems constitute an important part of the theory of differential and partial differential equations. Motivated by the numerous applications of fractional differential equations in diverse fields, in this paper, we studied sufficient conditions for the nonexistence of solutions (or, equivalently, necessary conditions for the existence of solutions) for nonlinear fractional differential inequalities and systems in the sense of Erdélyi-Kober. Our approach is based on nonlinear capacity estimates specifically adapted to the Erdélyi-Kober fractional operators and some integral inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we developed a novel numerical method for solving general nonlinear fractional ordinary differential equations (FODEs). First, we transformed the nonlinear FODEs into the equivalent Volterra integral equations. We then developed a time-stepping algorithm for the numerical solution of the Volterra integral equations based on the third-order Taylor expansion for approximating the integrands in the Volterra integral equations on a chosen mesh with the mesh parameter h. This approximation led to implicit nonlinear algebraic equations in the unknowns at each given mesh point, and an iterative algorithm based on Newton's method was developed to solve the resulting implicit equations. A convergence analysis of this numerical scheme showed that the error between the exact solution and numerical solution at each mesh point is O(h³), independent of the fractional order. Finally, four numerical examples were solved to verify the theoretical results and demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

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

It is well known that investigation on exact solutions of nonlinear fractional partial differential equations (PDEs) is a very difficult work compared with integer‐order nonlinear PDEs. In this paper, based on the separation method of semifixed variables and integral bifurcation method, a combinational method is proposed. By using this new method, a class of generalized time‐fractional thin‐film equations are studied. Under two kinds of definitions of fractional derivatives, exact solutions of two generalized time‐fractional thin‐film equations are investigated respectively. Different kinds of exact solutions are obtained and their dynamic properties are discussed. Compared to the results in the existing references, the types of solutions obtained in this paper are abundant and very different from those in the existing references. Investigation shows that the solutions of the model defined by Riemann–Liouville differential operator converge faster than those defined by Caputo differential operator. It is also found that the profiles of some solutions are very similar to solitons, but they are not true soliton solutions. In order to visually show the dynamic properties of these solutions, the profiles of some representative exact solutions are illustrated by 3D graphs. [ABSTRACT FROM AUTHOR]

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

In this research paper, we investigate the existence and uniqueness of solutions for neutral functional differential equations with sequential fractional orders, specifically involving the G -Caputo operator. To obtain the desired results, we employ the Banach fixed point theorem (BFPT), a nonlinear variation of the Leray-Schauder fixed point theorem (SFPT), and the Krasnoselski fixed point theorem (KFPT). Additionally, we provide illustrative examples that demonstrate the key findings. Furthermore, we address a scenario where an initial value integral condition is considered. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying a class of fuzzy fractional switched implicit differential equations (FFSIDEs) with non-instantaneous impulses that there are few papers considering this issue. Considering switching law and the memory property of fractional calculus, we first present a formula of solution for FFSIDEs with non-instantaneous impulses. Subsequently, based on a sequence of Picard functions, we explore the existence of solutions for the addressed equations by successive approximation. Furthermore, Ulam–Hyers (U–H) stability for this considered equations is derived. The main results are obtained using fuzzy-valued fractional calculus and nonlinear analysis. Finally, two numerical examples illustrating the theoretical result are given. [ABSTRACT FROM AUTHOR]

Herein, we present an algorithm for handling fractional delay differential equations (FDDEs). Chebyshev polynomials (CPs) class of the seventh kind is a subclass of the generalized Gegenbauer (ultraspherical) polynomials. The members of this class make up the basis functions in this paper. Our suggested numerical algorithm is derived using new theoretical findings about these polynomials and their shifted counterparts. We will use the Tau method to convert the FDDE with the governing conditions into a linear algebraic system, which can then be solved numerically using a suitable procedure. We will give a detailed discussion of the convergence and error analysis of the shifted Chebyshev expansion. Lastly, some numerical examples are provided to verify the precision and applicability of the proposed strategy. [ABSTRACT FROM AUTHOR]

After the initiation of Jachymski's contraction principle via digraph, the area of metric fixed point theory has attracted much attention. A number of outcomes on fixed points in the context of graph metric space employing various types of contractions have been investigated. The aim of this paper is to investigate some fixed point theorems for a class of nonlinear contractions in a metric space endued with a transitive digraph. The outcomes presented herewith improve, extend and enrich several existing results. Employing our findings, we describe the existence and uniqueness of a singular fractional boundary value problem. [ABSTRACT FROM AUTHOR]

In this paper, we present a highly efficient analytical method that combines the Laplace transform and the residual power series approach to approximate solutions of nonlinear time-fractional partial differential equations (PDEs). First, we derive the analytical method for a general form of fractional partial differential equations. Then, we apply the proposed method to find approximate solutions to the time-fractional coupled Berger equations, the time-fractional coupled Korteweg–de Vries equations and time-fractional Whitham–Broer–Kaup equations. Secondly, we extend the proposed method to solve the two-dimensional time-fractional coupled Navier–Stokes equations. The proposed method is validated through various test problems, measuring quality and efficiency using error norms E 2 and E ∞ , and compared to existing methods. [ABSTRACT FROM AUTHOR]

This paper describes a new class of boundary value fractional-order differential equations of the q-Hilfer and q-Caputo types, with separated boundary conditions. The presented problem is converted to an equivalent integral form, and fixed-point theorems are used to prove the existence and uniqueness of solutions. Moreover, several special cases demonstrate how the proposed problems advance beyond the existing literature. Examples are provided to support the analysis presented. [ABSTRACT FROM AUTHOR]

In this paper, an SIRS epidemic model using Grunwald–Letnikov fractional-order derivative is formulated with the help of a nonlinear system of fractional differential equations to analyze the effects of fear in the population during the outbreak of deadly infectious diseases. The criteria for the spread or extinction of the disease are derived and discussed on the basis of the basic reproduction number. The condition for the existence of Hopf bifurcation is discussed considering fractional order as a bifurcation parameter. Additionally, using the Grunwald–Letnikov approximation, the simulation is carried out to confirm the validity of analytic results graphically. Using the real data of COVID-19 in India recorded during the second wave from 15 May 2021 to 15 December 2021, we estimate the model parameters and find that the fractional-order model gives the closer forecast of the disease than the classical one. Both the analytical results and numerical simulations presented in this study suggest different policies for controlling or eradicating many infectious diseases. [ABSTRACT FROM AUTHOR]

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

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]

The notion of A -condensing operators via the measure of noncompactness is proposed, which retains the existing classes of condensing operators. Results concerning the existence of the best proximity point (pair) of cyclic (noncyclic) A -condensing operators along with the coupled best proximity-point theorem for cyclic A -condensing operators have been formulated. An application to a (k , ℷ) -Hilfer fractional differential equation of order 2 < p < 3 , type q ∈ [ 0 , 1 ] satistfying some boundary conditions is presented. The paper is the first to investigate the optimum solution of such a generalized fractional differential equation. The hypothesis involved in the investigation is independent of the incorporated measure of noncompactness, thereby making our result better in application than that present in the literature. Moreover, added numerical examples validate the theoretical conclusions. [ABSTRACT FROM AUTHOR]

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

This research paper introduces a novel approach to deriving traveling wave solutions (TWSs) for the Caputo-fractional Klein–Gordon equations. This research presents a distinct methodological advancement by introducing TWSs of a particular time-fractional partial differential equation, utilizing a non-local fractional operator, specifically the Caputo derivative. To achieve our goal, a novel transformation is considered, that converts a time-fractional partial differential equation into fractional ordinary differential equations, enabling analytical solutions through various analytical methods. This paper employs the homotopy analysis method to achieve the target objectives. To demonstrate the efficiency and applicability of the proposed transform and method, two examples are discussed and analyzed in figures. [ABSTRACT FROM AUTHOR]

This paper studies the analytical solution to the Fractional Differential Equations (FDEs) under uncertainty. Fuzzy differential equations are one of the emerging topics in the present era of research, where we have found an effective combination of FDEs with fuzziness. The Homotopy Analysis Transform Method (HATM) with the Caputo fractional derivative is applied in this work to find the analytical fuzzy solution of two fuzzy FDEs. One of the interesting parts of this study is that we have found upper and lower fuzzy solutions for both fuzzy FDEs. The different graphical representations that have been presented for both examples show that there is a symmetry relation between the upper and lower-cut fuzzy solutions. In this method, the region and rate of convergence of the solution series are controlled by the auxiliary parameter r-cut. This paper shows that the proposed method is reliable and efficient in determining the fuzzy solutions of the FDEs in applied mathematics and engineering [ABSTRACT FROM AUTHOR]

In this paper, we investigated n-dimensional fractional hybrid differential equations (FHDEs) with nonlinear boundary conditions in a nonlinear coupled system. For this purpose, we used Dhage's fixed point theory, and applied the Krasnoselskii-type coupled fixed point theorem to construct existence conditions of the solution of the FHDEs. To illustrated this idea, suitable examples are presented in 3-dimensional space at the end of the paper. [ABSTRACT FROM AUTHOR]

We examine a biological population model of fractional order (FBPM) in this paper using the Riccati-Bernoulli sub-ODE approach. Many scenarios in computational biology make use of this fundamental fractional model. Of particular note is that our study's FBPM uses fractional derivatives to track changes in the density populations. The study is concerned with the construction of new solitary wave solutions for the FBPM, a system of two nonlinear fractional ordinary differential equations. In this investigation, we use the conformable derivative as the fractional derivative. The Backlund transformation is the foundation of the solution process. We create a variety of families of soliton wave solutions and explain different physical behaviours that are inherent in the problems we explore. In particular, we apply the suggested methods to investigate rational, periodic, and hyperbolic solutions. The solutions found in various classes provide insightful information about the underlying physical mechanisms. To sum up, our current methods are superior instruments for analyzing different families of solutions in fractional-order issues. [ABSTRACT FROM AUTHOR]

Caputo-Hadamard-type fractional calculus involves the logarithmic function of an arbitrary exponent as its convolutional kernel, which causes challenges in numerical approximations. In this paper, we construct and analyze a spectral collocation approach using mapped Jacobi functions as basis functions and construct an efficient algorithm to solve systems of fractional pantograph delay differential equations involving Caputo-Hadamard fractional derivatives. What we study is the error estimates of the derived method. In addition, we tabulate numerical results to support our theoretical analysis. [ABSTRACT FROM AUTHOR]

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 Volterratype 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 d, 0 < d < 1. [ABSTRACT FROM AUTHOR]

In this paper, we apply the fractional homotopy perturbation transform method (FHPTM) to deliver an effective semi-analytical technique for determining fractional-order Kuramoto–Sivashinsky equations. The project technique combines the Laplace transform with the Caputo–Fabrizio fractional derivative of order α where α ∈ (0 , 1 ] . Fractional-order Kuramoto–Sivashinsky equation is indeed important in the field of nonlinear physics and mathematics. It is a fractional partial differential equation that describes the behaviour of waves in certain dissipative media, such as flames and chemicals. The FHPTM is described to be fast and accurate. Illustrative examples are included to demonstrate the efficiency and reliability of the presented techniques. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of at least two weak solutions to a class of singular two-phase problems with variable exponents involving a ψ-Hilfer fractional operator and Dirichlet-type boundary conditions when the term source is dependent on one parameter. Here, we use the fiber method and the Nehari manifold to prove our results. [ABSTRACT FROM AUTHOR]

The two-parameter Mittag–Leffler function E α , β is of fundamental importance in fractional calculus, and it appears frequently in the solutions of fractional differential and integral equations. However, the expense of calculating this function often prompts efforts to devise accurate approximations that are more cost-effective. When α > 1 , the monotonicity property is largely lost, resulting in the emergence of roots and oscillations. As a result, current rational approximants constructed mainly for α ∈ (0 , 1) often fail to capture this oscillatory behavior. In this paper, we develop computationally efficient rational approximants for E α , β (− t) , t ≥ 0 , with α ∈ (1 , 2) . This process involves decomposing the Mittag–Leffler function with real roots into a weighted root-free Mittag–Leffler function and a polynomial. This provides approximants valid over extended intervals. These approximants are then extended to the matrix Mittag–Leffler function, and different implementation strategies are discussed, including using partial fraction decomposition. Numerical experiments are conducted to illustrate the performance of the proposed approximants. [ABSTRACT FROM AUTHOR]