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Local fractional integral inequalities of Hermite-Hadamard type involving local fractional integral operators with Mittag-Leffler kernel have been previously studied for generalized convexities and preinvexities. In this article, we analyze Hermite-Hadamard-type local fractional integral inequalities via generalized (h˜1,h˜2)\left({\tilde{h}}_{1},{\tilde{h}}_{2})-preinvex function comprising local fractional integral operators and Mittag-Leffler kernel. In addition, two examples are discussed to ensure that the derived consequences are correct. As an application, we construct an inequality to establish central moments of a random variable.

The fundamental objective of this article is to investigate about the boundary value problem with the uses of a generalized conformable fractional derivative introduced by Zarikaya et al. (On generalized the conformable calculus, TWMS J. App. Eng. Math. 9 (2019), no. 4, 792–799, http://jaem.isikun.edu.tr/web/images/articles/vol.9.no.4/11.pdf). In the development of the this article, by using classical methods of fractional calculus, we find a definition of the generalized fractional Wronskian according to the fractional differential operator defined by Zarikaya, a fractional version of the Sturm-Picone theorem, and in addition, the stability criterion given by the Hyers-Ulam theorem is studied with the use of the aforementioned fractional derivatives.

In this article, we introduce the notions of generalized fractional integrals for the interval-valued functions (IVFs) of two variables. We establish Hermite-Hadamard (H-H) type inequalities and some related inequalities for co-ordinated convex IVFs by using the newly defined integrals. The fundamental benefit of these inequalities is that these can be turned into classical H-H inequalities and Riemann-Liouville fractional H-H inequalities, and new kk-Riemann-Liouville fractional H-H inequalities can be obtained for co-ordinated convex IVFs without having to prove each one separately.

This article is aimed at establishing some results concerning integral inequalities of the Opial type in the fractional calculus scenario. Specifically, a generalized definition of a fractional integral operator is introduced from a new Raina-type special function, and with certain results proposed in previous publications and the choice of the parameters involved, the established results in the work are obtained. In addition, some criteria are established to obtain the aforementioned inequalities based on other integral operators. Finally, a more generalized definition is suggested, with which interesting results can be obtained in the field of fractional integral inequalities.

Fractional integrals and inequalities have gained a lot of attention in recent years. By introducing innovative analytical approaches and applications, and by applying these approaches, numerous forms of inequalities have been examined. In this paper, we establish new identity for the twice differentiable function where the absolute value is convex. By utilizing this identity, numerous Corrected Euler-Maclaurin-type inequalities are developed for the Caputo-Fabrizio fractional integral operator. Based on this identity, the Corrected Euler-Maclaurin-type inequalities for $s$ s -convex function are obtained. By employing well-known inequalities such as Hölder's and Power -Mean, we are introduced several new error bounds and estimates for Corrected Euler-Maclaurin-type inequalities. Additionally, special cases of the present results are applied to obtain the previous well-known results. [ABSTRACT FROM AUTHOR]

We establish new conformable fractional Hermite-Hadamard (H–H) Mercer type inequalities for harmonically convex functions using the concept of support line. We introduce two new conformable fractional auxiliary equalities in the Mercer sense and apply them to differentiable functions with harmonic convexity. We also use Power-mean, Hölder's and improved Hölder inequality to derive new Mercer type inequalities via conformable fractional integrals. The accuracy and superiority of the offered technique are clearly depicted through impactful visual illustrations. We also use our technique to derive new estimates for hypergeometric functions and special means of real numbers that are more precise than existing ones. Some applications are provided as well. Our results generalize and extend some existing ones in the literature. [ABSTRACT FROM AUTHOR]

Applying the operator of q -difference, we examine the convolution properties of the subclasses MS ζ , q r , p (A , B) and M K ζ , q r , p (A , B) of p-valent meromorphic functions defined in the punctured open-unit disc. We derived specific inclusion features and coefficient estimates for functions that fall into these subclasses. Additionally, connections between the results presented here and those discovered in earlier papers are emphasized. [ABSTRACT FROM AUTHOR]

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

The perturbed Gerdjikov–Ivanov equation has immersed applications and significance in photonic crystal fibers and fiber optics. Extracting soliton solutions of the proposed equation has always been a challenging task for researchers. In this article, soliton solutions of governing equation are formed by using two very good and reliable analytical techniques, two variables (G′/G, 1/G) expansion method and generalized projective Riccati equation method. These methods are well‐known and widely employed. The constraint conditions for newly constructed solutions are also specified. Furthermore, the obtained solutions can be utilized in nonlinear physical situations. Surface plots of the solution functions are also given in this paper to highlight the physical significance. Fractional effects are also discussed through various line graphs. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the generalized Langevin-Sturm-Liouville differential problems involving Caputo-Atangana-Baleanu fractional derivatives of higher orders with respect to another positive, increasing function denoted by ρ. The fixed point theorems in the framework of Kransnoselskii and Banach are utilized to discuss the existence and uniqueness of the results. In addition, the stability criteria of Ulam-Hyers, generalize Ulam-Hyers, Ulam-Hyers-Rassias, and generalize Ulam-Hyers-Rassias are investigated by non-linear analysis besides fractional calculus. Finally, illustrative examples are reinforced by tables and graphics to describe the main achievements. [ABSTRACT FROM AUTHOR]

Nonlinear distinct models have wide applications in various fields of science and engineering. The present research uses the mapping and generalized Riccati equation mapping methods to address the exact solutions for the nonlinear Klein–Gordon equation. First, the travelling wave transform is used to create an ordinary differential equation form for the nonlinear partial differential equation. This work presents the construction of novel trigonometric, hyperbolic and Jacobi elliptic functions to the nonlinear Klein–Gordon equation using the mapping and generalized Riccati equation mapping methods. In the fields of fluid motion, plasma science, and classical physics the nonlinear Klein–Gordon equation is frequently used to identify of a wide range of interesting physical occurrences. It is considered that the obtained results have not been established in prior study via these methods. To fully evaluate the wave character of the solutions, a number of typical wave profiles are presented, including bell-shaped wave, anti-bell shaped wave, W-shaped wave, continuous periodic wave, while kink wave, smooth kink wave, anti-peakon wave, V-shaped wave and flat wave solitons. Several 2D, 3D and contour plots are produced by taking precise values of parameters in order to improve the physical description of solutions. It is noteworthy that the suggested techniques for solving nonlinear partial differential equations are capable, reliable, and captivating analytical instruments. [ABSTRACT FROM AUTHOR]

This paper focused on establishing results regarding the existence of solutions for a class of nonlocal terminal value problems involving hybrid implicit nonlinear fractional differential equations with the (k,φ)-Hilfer fractional derivative, which includes both finite delay and anticipation arguments. Our analysis was based on the Banach fixed point technique, and the Schauder and Krasnoselskii fixed point theorems. Moreover, illustrative examples were considered to support our new results. [ABSTRACT FROM AUTHOR]

The internal atmospheric waves are gravity waves and occur in the inner part of the fluid system. In this study, a time-fractional model for internal atmospheric waves is investigated with the Caputo-Fabrizio time-fractional differential operator. The analytical solution of the considered model is retrieved by the Elzaki Adomian decomposition method. The variation in the solution is examined for increasing order of the fractional parameter α through numerical and graphical simulations. The accuracy of the obtained results is established by comparing the obtained solution of considered fractional model with the results available in the literature. [ABSTRACT FROM AUTHOR]

This paper's findings are related to geometric function theory (GFT). We employ one of the most recent methods in this area, the fuzzy admissible functions methodology, which is based on fuzzy differential subordination, to produce them. To do this, the relevant fuzzy admissible function classes must first be defined. This work deals with fuzzy differential subordinations, ideas borrowed from fuzzy set theory and applied to complex analysis. This work examines the characteristics of analytic functions and presents a class of operators in the open unit disk J η , ς κ (a , e , x) for ς > − 1 , η > 0 , such that a , e ∈ R , (e − a) ≥ 0 , a > − x . The fuzzy differential subordination results are obtained using (GFT) concepts outside the field of complex analysis because of the operator's compositional structure, and some relevant classes of admissible functions are studied by utilizing fuzzy differential subordination. [ABSTRACT FROM AUTHOR]

The main aim of this study is to obtain soliton solutions of the generalized reaction Duffing model, which is a generalization for a collection of prominent models describing various key phenomena in science and engineering. The equation models the motion of a damped oscillator with a more complex potential than in basic harmonic motion. Two effective techniques, the mapping method and Bernoulli sub-ODE technique, are used for the first time to obtain the soliton solutions of the proposed model. Initially, the traveling wave transform, which comes from Lie symmetry infinitesimals, is applied, and a nonlinear ordinary differential equation form is derived. These approaches effectively retrieve a hyperbolic, Jacobi function as well as trigonometric solutions while the appropriate conditions are applied to the parameters. Numerous innovative solutions, including the kink wave, anti-kink wave, bell shape, anti-bell shape, W-shape, bright, dark and singular shape soliton solutions, were produced via the mapping and Bernoulli sub-ODE approaches. The research includes comprehensive 2D and 3D graphical representations of the solutions, facilitating a better understanding of their physical attributes and proving the effectiveness of the proposed methods in solving complex nonlinear equations. It is important to note that the proposed methods are competent, credible and interesting analytical tools for solving nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

We propose a new definition of the γ -convex stochastic processes (C SP) using center and radius (CR) order with the notion of interval valued functions (C. R I. V) . By utilizing this definition and Mean-Square Fractional Integrals, we generalize fractional Hermite–Hadamard–Mercer-type inclusions for generalized   C. R I. V versions of convex, tgs-convex, P-convex, exponential-type convex, Godunova–Levin convex, s-convex, Godunova–Levin s-convex, h-convex, n-polynomial convex, and fractional n-polynomial (C SP) . Also, our work uses interesting examples of   C. R I. V (C SP) with Python-programmed graphs to validate our findings using an extension of Mercer's inclusions with applications related to entropy and information theory. [ABSTRACT FROM AUTHOR]

In this manuscript, we present the concept of neutrosophic controlled pentagonal metric space (NCPMS), and prove some new fixed point results. Furthermore, we established many interesting outcomes for contraction maps. At last, we show the uniqueness and existence results for fractional differential and integral equations to illustrate the validity of the main outcomes. [ABSTRACT FROM AUTHOR]

This research manuscript aims to study a novel implicit differential equation in the non-singular fractional derivatives sense, namely Atangana-Baleanu-Caputo (ABC) of arbitrary orders belonging to the interval (2, 3] with respect to another positive and increasing function. The major results of the existence and uniqueness are investigated by utilizing the Banach and topology degree theorems. The stability of the Ulam-Hyers (UH) type is analyzed by employing the topics of nonlinear analysis. Finally, two examples are constructed and enhanced with some special cases as well as illustrative graphics for checking the influence of major outcomes. [ABSTRACT FROM AUTHOR]

In this work the concept of exponentially (h1, h2)-convex function is introduced and using it, the Hermite-Hadamard inequality and some bounds for the right side of this inequality, via Raina's fractional integral operator and generalized convex functions, are established .

This research work is devoted to investigating new common fixed point theorems on bipolar fuzzy b -metric space. Our main findings generalize some of the existence outcomes in the literature. Furthermore, we illustrate our findings by providing some applications for fractional differential and integral equations. [ABSTRACT FROM AUTHOR]

The goal of this research is to utilize some ansatz forms of solutions to obtain novel forms of soliton solutions for the Benney–Luke equation. It is a mathematically valid approximation that describes the propagation of two-way water waves in the presence of surface tension. By using ansatz forms of solutions, with an appropriate set of parameters, the lump soliton, periodic cross-kink waves, multi-waves, breather waves, Ma-breather, Kuznetsov–Ma-breather, periodic waves and rogue waves solutions can be obtained. Breather waves are confined, periodic, nonlinear wave solutions that preserve their amplitude and shape despite alternating between compression and expansion. For some integrable nonlinear partial differential equations, a lump soliton is a confined, stable solitary wave solution. Rogue waves are unusually powerful and sharp ocean surface waves that deviate significantly from the surrounding wave pattern. They pose a threat to maritime safety. They typically show up in solitary, seemingly random circumstances. Periodic cross-kink waves are a particular type of wave pattern that has frequent bends or oscillations that cross at right angles. These waves provide insights into complicated wave dynamics and arise spontaneously in a variety of settings. In order to predict the wave dynamics, certain 2D, 3D and contour profiles are also analyzed. Since these recently discovered solutions contain certain arbitrary constants, they can be used to describe the variation in the qualitative characteristics of wave phenomena. [ABSTRACT FROM AUTHOR]

This research focuses on exploring the significance of chemical reactions and thermal radiation on the magnetohydrodynamic (MHD) flow of a Casson–Williamson nanofluid (CWNF) over a stretching sheet. The objective is to comprehend how these factors influence the flow and heat transfer. A mathematical model, comprising partial differential equations adjusted into ordinary differential equations (ODEs) via utilizing some transformation. These ODEs are then tackled by MATLAB's BVP4C method, which is part of the finite difference technique. Results are verified by comparison with existing literature and are depicted visually and in tabular format. Additionally, the study explores the effects of external factors such as magnetic fields and the Lewis number on parameters like Nusselt number, friction factor, and Sherwood number. Furthermore, heat generation in MHD CWNF is analyzed, along with a thorough evaluation of heat transfer near a stretching sheet with a permeable layer. The findings suggest that growing Brownian motion factor (N b) and thermophoresis coefficient (N t) enhance the rate of heat transfer, signifying improved heat transfer rates. Similarly, higher N t values are associated with enhanced Sherwood numbers, indicating better mass transfer. Conversely, higher N b values lead in lower local Sherwood numbers. Physically, an increase in Brownian motion causes significant displacement of nanofluid particles, boosting their kinetic energy and thereby enhancing heat generation within the boundary layer. It is noted that the Eckert number (Ec) reflects the impact of different Ec values on temperature distribution. As Ec increases, there is a proportional increase in fluid temperature due to frictional heating, which stores heat energy within the fluid. This effect becomes more pronounced for non-linear stretching surfaces, demonstrating the response of the thermal region to viscous dissipation. Viscous dissipation has the potential to enhance convective heat transfer, leading to amplified temperature distribution and thickening of the thermal layer. [ABSTRACT FROM AUTHOR]

To analyze and study the behaviour of the shallow water waves, the perturbed Boussinesq equation has acquired fundamental importance. The principal objective of this paper is to manifest the exact traveling wave solution of the perturbed Boussinesq equation by two well known techniques named as, two variables (G′G,1G) expansion method and generalized projective Riccati equations method. A diverse array of soliton solutions, encompassing periodic, bright solitons, singular solitons and bright singular solitons are obtained by the applications of proposed techniques. The constraint conditions for newly constructed solutions are also specified. To enhance comprehension, the numerical illustrations of constructed solutions have been represented using surface plots, 2D plots and density plots. The results delineated in this paper transcend existing analysis, offering a novel, well-structured, and modern perspective. The solutions obtained not only enrich understanding of shallow water wave models but also exhibit efficacy in providing detailed descriptions of their dynamics. [ABSTRACT FROM AUTHOR]