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The (2+1)-dimensional time-fractional chiral non-linear Schrödinger equation in physical media is considered in this paper. At the outset, the Lie group analysis is applied to build a set of infinitesimal generators for this equation with the aid of the Riemann–Liouville fractional derivatives. Consequently, the reduction for the considered equation into an ordinary differential equation of fractional order is obtained by using these generators and the ErdLélyi–Kober fractional operator. As a result of this reduction, we use power series analysis to get an analytical solution provided by a convergence analysis of the obtained solution. Furthermore, we construct a numerical solution based on hyperbolic functions using the fractional reduced differential transform method in the sense of Caputo fractional derivatives. Also, we detect absolute errors by performing a comparison between the exact and numerical solutions of the equation under study, while investigating the effect of fractional order α on the numerical solution. Finally, conservation laws are derived using the the formal Lagrangian and new conservation theorem. [ABSTRACT FROM AUTHOR]

This paper proposes a new operational numerical method based on Touchard wavelets for solving fractional pantograph differential equations. First, we present an operational matrix of fractional integration as well as the fractional derivative of the Touchard wavelets. Then, by approximating the fractional derivative of the unknown function in terms of the Touchard wavelets and also by using collocation method, the original problem is reduced to a system of algebraic equations. Finally, to show the accuracy and the validity of the proposed technique, we provide some numerical examples. [ABSTRACT FROM AUTHOR]

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

We introduce the concept of quantum integration for interval-valued functions and establish new q-Hermite–Hadamard and q-Hermite–Hadamard–Fejér inequalities for left and right log ⁢ - ⁢ h -convex interval-valued functions. Our results generalize the known ones in the literature and serve as a foundation for future studies in inequalities for interval-valued functions and interval differential equations. We illustrate our results with examples. [ABSTRACT FROM AUTHOR]

This paper is devoted to the existence and uniqueness of solution for a large class of perturbed sweeping processes formulated by fractional differential inclusions in infinite dimensional setting. The normal cone to the (mildly non-convex) prox-regular moving set C(t) is supposed to have a Hölder continuous variation, is perturbed by a continuous mapping, which is both time and state dependent. Using an explicit catching-up algorithm, we show that the fractional perturbed sweeping process has one and only one Hölder continuous solution. Then this abstract result is applied to provide a theorem on the weak solvability of a fractional viscoelastic frictionless contact problem. The process is quasistatic and the constitutive relation is modeled with the fractional Kelvin–Voigt law. This application represents an additional novelty of our paper. [ABSTRACT FROM AUTHOR]

Jeffreys equation and its fractional generalizations provide extensions of the classical diffusive laws of Fourier and Fick for heat and particle transport. In this work, a class of multi-term time-fractional generalizations of the classical Jeffreys equation is studied. Restrictions on the parameters are derived, which ensure that the fundamental solution to the one-dimensional Cauchy problem is a spatial probability density function evolving in time. The studied equations are recast as Volterra integral equations with kernels represented in terms of multinomial Mittag-Leffler functions. Applying operator-theoretic approach, we establish subordination results with respect to appropriate evolution equations of integer order, depending on the considered range of parameters. Analyticity of the corresponding solution operator is also discussed. The main tools in the proofs are Laplace transform and the Bernstein functions' technique, especially, some properties of the sets of real powers of complete Bernstein functions. [ABSTRACT FROM AUTHOR]

A new approach to the transformations of the matrices of the fractional linear systems with desired eigenvalues is proposed. Conditions for the existence of the solution to the transformation problem of the linear system to its asymptotically stable controllable and observable canonical forms with desired eigenvalues are given and illustrated by numerical examples of fractional linear systems. [ABSTRACT FROM AUTHOR]

In this paper, our focus is on exploring the relative controllability of systems governed by linear fractional differential equations incorporating state delay. We introduce a novel counterpart to the Cayley-Hamilton theorem. Leveraging a delayed perturbation of the Mittag-Leffler function, along with a determining function and an analog of the Cayley-Hamilton theorem, we establish an algebraic Kalman-type rank criterion for assessing the relative controllability of fractional differential equations with state delay. Moreover, we articulate necessary and sufficient conditions for relative controllability criteria concerning linear fractional time-delay systems, expressed in terms of a new α -Gramian matrix and define a control which transfer the system from any initial state to any final state within a given time. The theoretical findings are exemplified through the presentation of illustrative examples. [ABSTRACT FROM AUTHOR]

In this paper, we apply the steepest descent method to the Schläfli-type integral representation of the three-parameter Mittag-Leffler function (well-known as the Prabhakar function). We find the asymptotic expansions of this function for its large parameters with respect to the real and complex saddle points. For each parameter, we separately establish a relation between the variable and parameter of function to determine the leading asymptotic term. We also introduce differentiations of the three-parameter Mittag-Leffler functions with respect to parameters and modify the associated asymptotic expansions for their large parameters. As an application, we derive the leading asymptotic term of fundamental solution of the time-fractional sub-diffusion equation including the Bessel operator with large order. [ABSTRACT FROM AUTHOR]

This article introduces a measure of noncompactness in the generalized Morrey space. We study the applications of our new definition in investigating the conditions for the existence of solutions for systems of nonlinear integral equations. We can extend many useful theorems in L p (R N) for functions belonging to the generalized Morrey spaces. Compared to the L p (R N) spaces, the advantage of studying in the Morrey spaces is that we can research no compact support functions in our problems. Finally, significant examples are presented to show the efficiency of the main results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the backward problem of recovering initial value for a homogeneous time-space fractional pseudo parabolic differential equation. Since the problem is ill-posed, a version of modified quasi boundary value method is used as the method of regularization for obtaining stable approximations. Error analysis and parameter choice strategies are done for both a priori and a posteriori cases. It is shown that, the order of the convergence rate can exceed 2 3. Hence, the obtained rate improves upon the previously known rates of order 2 3 for the a priori case, and 1 2 for the a posteriori case for the considered problem. [ABSTRACT FROM AUTHOR]

We consider weak solutions of the nonlinear time-fractional biharmonic diffusion equation ∂ t α u + ∂ t β u + u x x x x = h (t , x) | u | p in (0 , ∞) × (0 , 1) subject to the initial conditions u (0 , x) = u 0 (x) , u t (0 , x) = u 1 (x) and the Navier boundary conditions u (t , 1) = u x x (t , 1) = 0 , where α ∈ (0 , 1) , β ∈ (1 , 2) , ∂ t α (resp. ∂ t β ) is the fractional derivative of order α (resp. β) with respect to the time-variable in the Caputo sense, p > 1 and h is a measurable positive weight function. Using nonlinear capacity estimates specifically adapted to the fourth-order differential operator ∂ 4 ∂ x 4 , the domain, the initial conditions and the boundary condition, a general nonexistence result is established. Next, some special cases of weight functions h are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we first construct an integral identity associated with tempered fractional operators. By using this identity, we have found the error bounds for Simpson's second formula, namely Newton–Cotes quadrature formula for differentiable convex functions in the framework of tempered fractional integrals and classical calculus. Furthermore, it is also shown that the newly established inequalities are the extension of comparable inequalities inside the literature. [ABSTRACT FROM AUTHOR]

In the present paper, some inequalities regarding Csiszár divergence are generalized for diamond integrals by utilizing Green function and Fink’s identity. By selecting various kinds of convex functions, bound of different entropies and divergence measures are obtained. Furthermore, in seek of applications, bounds of different divergence measures are estimated on diverse fixed time scales. These divergence measures are fundamental in problems pertaining to source and channel coding, large deviations theory, and other sorts of problems. [ABSTRACT FROM AUTHOR]

This article investigates sufficient conditions for the existence and uniqueness of solutions to the ψ-Hilfer sequential type pantograph fractional boundary value problem. Considering the system depends on a lower-order fractional derivative of an unknown function, the study is carried out in a special working space. Standard fixed point theorems such as the Banach contraction principle and Krasnosel'skii's fixed point theorem are applied to prove the uniqueness and the existence of a solution, respectively. Finally, an example demonstrating our results with numerical simulations is presented. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following perturbed fractional Hamiltonian systems { t D ∞ α (− ∞ D t α u (t)) + L (t) u (t) = ∇ u W (t , u (t)) + ∇ u G (t , u (t)) , t ∈ ℝ , u ∈ H α (ℝ , ℝ N) , where α ∈ (1 / 2 , 1 ] , L ∈ C (ℝ , ℝ N × N) is symmetric and not necessarily required to be positive definite, W ∈ C 1 (ℝ × ℝ N , ℝ) is locally subquadratic and locally even near the origin, and perturbed term G ∈ C 1 (ℝ × ℝ N , ℝ) maybe has no parity in u. Utilizing the perturbed method improved by the authors, a sequence of nontrivial homoclinic solutions is obtained, which generalizes previous results. [ABSTRACT FROM AUTHOR]

In this paper, we mainly explore the averaging principle of Caputo-type fractional delay stochastic differential equations with Brownian motion. Firstly, the solutions of this considered system are derived with the aid of the Picard iteration technique along with the Laplace transformation and its inverse. Secondly, we obtain the unique result by using the contradiction method. In addition, the averaging principle is discussed by means of the Burkholder-Davis-Gundy inequality, Jensen inequality, Hölder inequality and Grönwall-Bellman inequality under some hypotheses. Finally, an example with numerical simulations is carried out to prove the relevant theories. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a biparametrized multiplicative integral identity and employ it to establish a collection of inequalities for multiplicatively convex mappings. These inequalities encompass several novel findings and refinements of established results. To enhance readers' comprehension, we offer illustrative examples that highlight appropriate choices of multiplicatively convex mappings along with graphical representations. Finally, we demonstrate the applicability of our results to special means of real numbers within the realm of multiplicative calculus. [ABSTRACT FROM AUTHOR]

This paper aims to derive the images of the product of the generalized q-analogue of Mittag-Leffler functions and the basic binomial function, under the fractional q-integral operators of Kober and Weyl type. The derived findings are of a general nature and include, as specific examples, the Weyl q-integral operator, the q-derivative operators, and the Riemann–Liouville q-integral operator. [ABSTRACT FROM AUTHOR]

The paper is designed to highlight the presence of an optimal solution to a differential system comprising fractional order derivatives, precisely the ψ-Caputo derivative, under the prescribed initial constraints. This is attained through a new variant of cyclic (noncyclic) condensing operators incorporating a pair of maps. The altered notion facilitates in evolving best proximity point (pair) results as well as coupled best proximity point results with supportive corollaries. [ABSTRACT FROM AUTHOR]

The objective of this research is to present new fixed point theorems for two separate families of fuzzy-dominated mappings. These mappings must satisfy a unique locally contraction in a complete b-multiplicative metric space. Also, we have obtained novel results for families of fuzzy-dominated mappings on a closed ball that meet the requirements of a generalized locally contraction. This research introduces new and challenging fixed-point problems for families of ordered fuzzy-dominated mappings in ordered complete b-multiplicative metric spaces. Moreover, we demonstrate a new concept for families of fuzzy graph-dominated mappings on a closed ball in these spaces. Additionally, we present novel findings for graphic contraction endowed with graphic structure. These findings are groundbreaking and provide a strong foundation for future research in this field. To demonstrate the uniqueness of our novel findings, we provide evidence of their applicability in obtaining the common solution of integral and fractional differential equations. Our findings have resulted in modifications to several contemporary and classical results in the research literature. This provides further evidence of the originality and impact of our work. [ABSTRACT FROM AUTHOR]

We introduce a novel multiparameterized fractional multiplicative integral identity and utilize it to derive a range of inequalities for multiplicatively s-convex mappings in connection with different quadrature rules involving one, two, and three points. Our results cover both new findings and established ones, offering a holistic framework for comprehending these inequalities. To validate our outcomes, we provide an illustrative example with visual aids. Furthermore, we highlight the practical significance of our discoveries by applying them to special means of real numbers within the realm of multiplicative calculus. [ABSTRACT FROM AUTHOR]

This paper investigates the existence and uniqueness of solution for a new class of ψ -Hilfer-type fractional differential equation with two fractional derivatives of different order. By making use of topological degree theory for condensing maps we established the existence result, for the reason that the previously mentioned theory softens the criteria of strong compactness by weakening it. To deal with the uniqueness result we use Banach's contraction principle. As application, we give an example to demonstrate our theoretical result. [ABSTRACT FROM AUTHOR]

The fractional Lindley distribution is used to model the distribution of perturbations in count data regressions, which allow for dealing with widely dispersed data. It is obtained from the non-fractional Lindley distribution by replacing the support $\mathbb{T} = {\mathbb{R}^ + }$ T = R + by ${\mathbb{T}} = {\mathbb{N}}\backslash \{ 0\} $ T = N ∖ { 0 } and applying time scale theory, whose ambition is to unify the theories of difference equations and differential equations, integral and differential calculus, and the calculus of finite differences. It thus provides a framework for the study of dynamical systems in discrete-continuous time. Delta moments are discrete-time Laplace transforms of the frequency function of the fractional Lindley distribution. The parameter of the fractional Lindley distribution is estimated by least squares, weighted least squares, maximum likelihood, moments, and proportions. The moment estimator always exists, so that delta moments result from the nabla Laplace transform of the frequency function of the fractional Lindley distribution. The maximum likelihood estimates have the least mean-square errors. The proportion method works satisfactorily only when the mode of the distribution is null and the proportion of zeros is high. A simulation allows for quantifying the mean-square errors associated with the estimators. A count regression based on the fractional Lindley distribution with data on the total number of stays after hospital admission among U.S. residents aged 65 and over shows that the Akaike information criteria is significantly lower than with the uniform Poisson and Poisson regressions. [ABSTRACT FROM AUTHOR]

In this paper, we prove an equality for twice-differentiable convex functions involving the conformable fractional integrals. Moreover, several Bullen-type inequalities are established for twice-differentiable functions. More precisely, conformable fractional integrals are used to derive such inequalities. Furthermore, sundry significant inequalities are obtained by taking advantage of the convexity, Hölder inequality, and power-mean inequality. Finally, we provide our results by using special cases of obtained theorems. [ABSTRACT FROM AUTHOR]

We develop a new generalized form of the fractional kinetic equation involving the generalized Mathieu series. By using the Sumudu transform, a solution of these generalized fractional kinetic equation is obtained in terms of the Mittag-Leffler function. The numerical results and graphical interpretation are also presented. [ABSTRACT FROM AUTHOR]

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

The direct and two inverse problems defined for an integro-differential equation on a bounded domain have been considered. The spectral problem of the integro-differential equation constitutes the Legendre differential equation in space variable. Finding a space-dependent source term whenever the data at some time, say T, as over-specified condition, constitutes the Ist inverse problem. The 2nd inverse problem consists of recovering a time-dependent coefficient in the source term from an integral type over-specified condition. The Fourier approach is used to have the analytical series solution of the problems. The existence and uniqueness results for the direct and inverse problems under certain regularity conditions on the data are presented. [ABSTRACT FROM AUTHOR]

We study the approximate optimal control for a class of fractional stochastic hemivariational inequalities with non-instantaneous impulses driven by Rosenblatt process in a Hilbert space. Firstly, a suitable definition of piecewise continuous mild solution is introduced, and by using stochastic analysis, properties of α -order sine and cosine family and Picard type approximate sequences, we show the existence and uniqueness of approximate mild solutions for the inequality problems of fractional order (1, 2] under the non-Lipschitz conditions. Secondly, we provide the existence conditions of approximate solutions to optimal control problems driven by the presented control systems with the help of a new minimizing sequence method. Finally, an example is provided to illustrate the theory. [ABSTRACT FROM AUTHOR]

This work explores the possibility of developing the analog of some classic results from elliptic PDEs for a class of fractional ODEs involving the composition of both left- and right-sided Riemann-Liouville (R-L) fractional derivatives, D a + α D b - β , 1 < α + β < 2 . Compared to one-sided non-local R-L derivatives, these composite operators are completely non-local, which means that the evaluation of D a + α D b - β u (x) at a point x will have to retrieve the information not only to the left of x all the way to the left boundary but also to the right up to the right boundary, simultaneously. Therefore, only limited tools can be applied to such a situation, which is the most challenging part of the work. To overcome this, we do the analysis from a non-traditional perspective and eventually establish elliptic-type results, including Hopf's Lemma and maximum principles. As α → 1 - or α , β → 1 - , those operators reduce to the one-sided fractional diffusion operator and the classic diffusion operator, respectively. For these reasons, we still refer to them as "elliptic diffusion operators", however, without any physical interpretation. [ABSTRACT FROM AUTHOR]

In this work we introduce discrete convolution operators and study their most basic properties. We then solve linear difference equations depending on such operators. The theory herein developed generalizes, in particular, the theory of discrete fractional calculus and fractional difference equations. To that matter we make use of the so-called Sonine pairs of kernels. [ABSTRACT FROM AUTHOR]

In this work we generalize some results of the Riemann-Liouville fractional calculus for the tempered case, namely, we deal with some boundedness results of Riemann-Liouville tempered fractional integrals on continuous function space and Lebesgue spaces in bounded intervals and on the real line. Moreover, the limit behavior of the Riemann-Liouville tempered fractional integrals approaching to the Riemann-Liouville fractional integrals is considered. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of generalized Krätzel functions, which are the kernel functions of type-1 and type-2 pathway transforms. Various analytical properties such as Lipschitz continuity, fixed point property and integrability of these functions are investigated. Furthermore, the paper introduces two new inequalities associated with generalized Krätzel functions. The composition formulae for various fractional operators, such as Grünwald-Letnikov fractional differential operator, Riemann-Liouville fractional integral and differential operators with these functions are also obtained. Additionally, it has been demonstrated that these kernel functions are related to the Heaviside step function, the Dirac delta function, and the Riemann zeta function. A computable series representation of the kernel function is also obtained. Further scope of the work is pointed out by modifying the kernel function with the Mittag-Leffler function. [ABSTRACT FROM AUTHOR]

The aim of the paper is to study a coupled system which consists of a fractional differential hemivariational inequality (FDHVI, for short) combined with a fractional hemivariational inequality (FHVI, for short). By utilizing Rothe method and surjectivity result, the existence of coupled system is established. In addition, we apply the results to a new quasistatic contact problem, in which the constitutive equations are described by the fractional Kelvin-Voigt laws. [ABSTRACT FROM AUTHOR]

This paper deals with some existence and uniqueness results for a class of deformable fractional differential equations. These problems encompassed nonlinear implicit fractional differential equations involving boundary conditions and various types of delays, including finite, infinite, and state-dependent delays. Our approach to proving the existence and uniqueness of solutions relied on the application of the Banach contraction principle and Schauder's fixed-point theorem. In the last section, we provide different examples to illustrate our obtained results. [ABSTRACT FROM AUTHOR]

In this work, we consider fractional variational problems depending on higher order fractional derivatives. We obtain optimality conditions for such problems and we present and discuss some examples. We conclude with possible research directions. [ABSTRACT FROM AUTHOR]

The purpose of this article is to obtain appropriate tools for solving fractional differential equations that are flexible enough to adapt to different purposes. We thus look for a very general fractional Fourier transform with a phase function which can be appropriately chosen according to the problem you want to face. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce and study the linear canonical Fourier–Bessel wavelet transform. We prove an orthogonality relation, inversion formula and some inequality for linear canonical Fourier–Bessel wavelet transform. We first present a direct relationship between the linear canonical Bessel wavelet transform and ordinary Bessel wavelet transform. Based on this relation, we provide an alternative proof of the orthogonality relation for the linear canonical Bessel wavelet transform. Some of its essential properties are also studied in detail. Finally, we explicitly derive several versions of inequalities associated with the linear canonical Bessel wavelet transform. [ABSTRACT FROM AUTHOR]

Conformable fractional calculus is a relatively new branch of mathematics that seeks to extend traditional calculus to include non-integer order derivatives and integrals. This new form of calculus allows for a more precise description of physical phenomena, such as those related to fractal geometry and chaos theory. It also offers new tools for solving problems in physics, engineering, finance, and other areas. By using conformable fractional calculus, researchers can gain insight into the behavior of systems that would otherwise be difficult or impossible to analyze. In this article, we will discuss the fundamentals of conformable fractional calculus and its applications in various fields. For this purpose ruled surfaces are studied with conformable fractional calculus. The ruled surface is rearranged concerning the conformable surface definition and geometric properties are investigated. [ABSTRACT FROM AUTHOR]

Let D denote the unit disc in C . We define the generalized Cesàro operator as follows: C ω (f) (z) = ∫ 0 1 f (t z) 1 z ∫ 0 z B t ω (u) d u ω (t) d t , where { B ζ ω } ζ ∈ D are the reproducing kernels of the Bergman space A ω 2 induced by a radial weight ω in the unit disc D . We study the action of the operator C ω on weighted Hardy spaces of analytic functions H γ , γ > 0 and on general weighted Bergman spaces A μ 2 . [ABSTRACT FROM AUTHOR]

In mathematics and the applied sciences, as a very useful tool, fractional calculus is a basic concept. Furthermore, in many areas of mathematics, it is better to use a new hybrid fractional operator, which combines the proportional and Caputo operators. So we concentrate on the proportional Caputo-hybrid operator because of its numerous applications. In this research, we introduce a novel extension of the Hermite–Hadamard-type inequalities for proportional Caputo-hybrid operator and establish an identity. Then, taking into account this novel generalized identity, we develop some integral inequalities associated with the left-side of Hermite–Hadamard-type inequalities for proportional Caputo-hybrid operator. Moreover, to illustrate the newly established inequalities, we give some examples with the help of graphs. [ABSTRACT FROM AUTHOR]

This article presents an efficient implicit spline-based numerical technique to solve the time-fractional generalized coupled Burgers' equation. The time-fractional derivative is considered in the Caputo sense. The time discretization of the fractional derivative is discussed using the quadrature formula. The quasilinearization process is used to linearize this non-linear problem. In this work, the formulation of the numerical scheme is broadly discussed using cubic B-spline functions. The stability of the proposed method is proved theoretically through Von-Neumann analysis. The reliability and efficiency are demonstrated by numerical experiments that validate theoretical results via tables and plots. [ABSTRACT FROM AUTHOR]

The application of fractional calculus for solving nonlinear equations through iterative techniques is an emerging area of research. In recent times, several Newton-type methods have been proposed which particularly utilize the concept of fractional order derivatives. However, convergence of such methods essentially require the existence of at least first order derivative. Accordingly, in the case where derivative is not feasible to obtain, the derivative-free methods are of much significance. In this paper, a fractional Traub-Steffensen-type method is developed, the formulation of which is based on the idea of conformable fractional derivatives of order α ∈ (0 , 1 ] . In addition, the scheme is extended to its multi-dimensional case to solve the systems of equations. The proposed derivative-free scheme is further investigated for its dynamical aspects and convergence characteristics by varying the value of α in given range. In this concern, the convergence planes are presented in a well-defined region which in general provide the fundamental information about the stability of given method. Furthermore, the numerical performance is analyzed by locating the solutions of variety of nonlinear equations, including some applied problems. [ABSTRACT FROM AUTHOR]

A rotation-two-component Camassa–Holm (R2CH) system was proposed recently to describe the motion of shallow water waves under the influence of gravity. This is a highly nonlinear and strongly coupled system of partial differential equations. A crucial issue in designing numerical schemes is to preserve invariants as many as possible at the discrete level. In this paper, we present a provable implicit nonlinear difference scheme which preserves at least three discrete conservation invariants: energy, mass, and momentum, and prove the existence of the difference solution via the Browder theorem. The error analysis is based on novel and refined estimates of the bilinear operator in the difference scheme. By skillfully using the energy method, we prove that the difference scheme not only converges unconditionally when the rotational parameter diminishes, but also converges without any step-ratio restriction for the small energy case when the rotational parameter is nonzero. The convergence orders in both settings (zero or nonzero rotation parameter) are O (τ 2 + h 2) for the velocity in the L ∞ -norm and the surface elevation in the L 2 -norm, where τ denotes the temporal stepsize and h the spatial stepsize, respectively. The theoretical predictions are confirmed by a properly designed two-level iteration scheme. Compared with existing numerical methods in the literature, the proposed method demonstrates its effectiveness for long-time simulation over larger domains and superior resolution for both smooth and non-smooth initial values. [ABSTRACT FROM AUTHOR]

In this work, we analyze a q-fractional jerk problem having anti-periodic boundary conditions. The focus is on investigating whether a unique solution exists and remains stable under specific conditions. To prove the uniqueness of the solution, we employ a Banach fixed point theorem and a mathematical tool for establishing the presence of distinct fixed points. To demonstrate the availability of a solution, we utilize Leray–Schauder's alternative, a method commonly employed in mathematical analysis. Furthermore, we examine and introduce different kinds of stability concepts for the given problem. In conclusion, we present several examples to illustrate and validate the outcomes of our study. [ABSTRACT FROM AUTHOR]

Breast cancer is the most common type of cancer in women. Chemotherapy is primarily used for patients with stage 2 to 4 breast cancer. Most chemotherapy drugs are effective at destroying rapidly growing and proliferating cancer cells. However, drugs also damage normal, rapidly growing cells, which can lead to serious side effects. Breast cancer treatment with chemotherapy can affect heart health. Side effects of chemotherapy on the heart are called cardiotoxicity. Therefore, we have constructed a mathematical model from the breast cancer patient population. In this article, we utilize the Caputo–Fabrizio fractional order derivative for mathematical modeling of the breast cancer stages in chemotherapy patients. The use of Caputo–Fabrizio fractional derivative provides a more valuable insight into the complexity of the breast cancer model. The stability of the fractional order model is also proven by the P -stable approach of the fixed point theorem. Also, the numerical simulations are performed via Laplace Adomian decomposition method to establish the dependence of the breast cancer dynamics on the order of the fractional derivatives. Based on the geometric results in the figures, we can conclude that the magnitude of the fractional order has a considerable impact on the days, which the maximum or minimum of the system solutions are reached, with a shift in the time at which this happens as the fractional order decreases from 1. However, it is obvious that the solutions of Caputo–Fabrizio fractional model approach the relevant results of the classical integer order system, when the fractional order approaches to 1. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to study the asymptotic behaviour and the finite extinction time of weak solutions to some time-fractional parabolic equations. Moreover, we improve some results in [5, 10] by dropping out some conditions assumed there. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study an abstract system which consists of a parabolic hemivariational inequality with a Hilfer fractional evolution equation involving history-dependent operators, which is called a Hilfer fractional differential hemivariational inequality. We first show existence and a priori estimates for the parabolic hemivariational inequality. Then, by using the well-known Bohnenblust-Karlin fixed point theorem, we prove an existence theorem for the Hilfer fractional differential hemivariational inequality. [ABSTRACT FROM AUTHOR]

In this work we introduce a general formulation of resolvent family, also named resolvent operator, to describe explicit formulas for the solutions of dynamic equations on time scales of order 0 < α ≤ 1 . The treatment developed here is based on a formulation of Laplace Transform on time scales that includes continuous and discrete cases, to obtain concise expressions for such explicit formulas. Moreover, this formulation of Laplace Transform allows to obtain discrete counterparts of some important properties in the context of fractional calculus. As main results in this work, we study the relationship between an abstract resolvent family and its infinitesimal generator, along with the main properties of resolvent families, and the existence of solutions for abstract dynamic equations on time scales. In addition, we introduce formulas for resolvent families as solutions of several types of dynamic equations of order 0 < α ≤ 1 on continuous, discrete and quantum time scales. [ABSTRACT FROM AUTHOR]

In this study, a new robust predictive control technique is investigated for uncertain fractional-order descriptor systems. Using the properties of fractional calculus and the construction of an appropriate Lyapunov function, the sufficient conditions to guarantee the existence of a robust predictive controller are given by minimizing the worst-case optimization problem. The new robust predictive controller is presented by using Shur complement, linear matrix inequality toolbox and cone-complement linear algorithm, and it is proved that the feasible solution of the optimization problem at the initial time can guarantee the admissibility of fractional-order descriptor closed-loop system. [ABSTRACT FROM AUTHOR]