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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 [Formula: see text]-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.

ABSTRACTWe 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.

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 $ ({\mathsf{k}}, {\rm{\mathsf{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.

In mathematical analysis theory of inequalities has considerable influence due to its massive utility in various fields of physical sciences. These are investigated via multiple approaches to acquire more precise and rectified forms of already celebrated consequences. Integral inequalities are investigated to compute the error bounds for quadrature schemes. Among all of them, one is Hermite-Hadamard inequality, which has mighty efficacy. Numerous generalizations have been proposed in the literature based on different novel and innovative procedures. In recent years, Bullen inequality has been very commonly studied inequality. The main objective of our progressive study is to establish a new set of Bullen-type inequalities concerning the Jensen-Mecer inequality. For the completion of the current investigation, we derive a new general Bullen-Mecer equality, which is beneficial to achieve our primary consequences. Furthermore, Considering the Bullen-Mecer equation, we employ the convexity property together with famous Hölder's type and Young's inequalities, bounding, and Lipschitz characteristics of functions to conclude new variants of generalized upper bounds of Bullen inequality. Also, we deliver some applications of outcomes to means, special functions, error bounds, and iterative methods to solve non-linear problems. Lastly, we verify our findings through various simulations. The advantage of the current study is that several results concerning Bullen's inequality can be retrieved from our proposed results and various new results can be achieved by choosing the values for γ and δ. By utilizing the similar technique that we have adopted new iterative schemes can be established from integral inequalities.

This work aims to motivate the bipolar parametric metric space introduced by Pasha et al. We introduce the concept of bipolar parametric ν-metric space. Afterward, we state and investigate new fixed-point theorems. Suitable examples are given based on our outcomes. An application is provided to strengthen the findings we obtained.

There has been a worldwide epidemic of heroin that has affected people, families, societies, and cultures across the world. Now, the heroin epidemic has transitioned from heroin abuse to the overuse of synthetic narcotics, which are widely accessible and inexpensively produced. In this work, a novel mathematical approach is applied to investigate the dynamics of the heroin epidemic model and its harmful effect on society with different population data. A heroin model has been constructed with the importance of a non-singular kernel in the sense of a generalized Mittag-Leffler kernel. The well-posedness of the proposed model is proven via fixed-point theory. To examine the heroin model, two equilibrium states have been determined. These equilibrium states are proven to be locally and globally asymptotically stable. To analyze the behavior of heroin, a basic reproduction number and sensitivity analysis are used to determine the impact of different parameters mathematically as well as through simulations. To find the approximate solution, we implement the Toufik–Atangana numerical method at different fractional order values. The sensitivity of the heroin model is carried out, and 3-D graphs show the significance of the parameter involved in the model. Finally, the numerical outcomes are presented with different values of fractional parameters.

In this paper, we investigate Bullen-type inequalities applicable to functions that are twice-differentiable. To explore these advanced inequalities, we utilize generalized convexity and Riemann-type fractional integrals. A comparative analysis is provided to highlight the more refined inequalities from among the explored results. By exploring the limiting cases, a relation with existing literature is established. Several examples are also presented to illustrate the outcomes and their accuracy is validated through graphical analysis. Additionally, applications in generalized means are also discussed.

This research investigates the characteristics and attributes of a new (3+ 1)-dimensional Boussinesq equation that describes shallow water waves in higher dimensions. By utilizing the Hirota bilinear representation, a bilinear Bäcklund transformation is provided for the proposed model to get soliton solutions. Then, the extended transform rational function method is applied to calculate the complexitons type solutions. The results demonstrate various exact solutions with different structures, including periodic, singular, and bright solitons. Comprehensive graphical representations in 2D, 3D, and density plots are provided to highlight the physical properties of these solutions. Our approach is distinguished by the unique nature of the problem and the use of previously untested methods in this context, leading to many new and original optical soliton solutions. These results highlight the effectiveness of the proposed method in tackling nonlinear problems in engineering and the natural sciences, exceeding previous work found in the literature.

Abstract In the current research, some midpoint-type inequalities are generalized for co-ordinated convex functions with the help of generalized conformable fractional integrals. Moreover, some findings of this paper include results based on Riemann–Liouville fractional integrals and Riemann integrals.

In this research, we focus on a novel class of mean inequalities involving Riemann-Liouville fractional integrals. We employ these integrals to investigate various fundamental identities that help us to explore mean inequalities. By utilizing a generalized concept of convexity, we establish a unique set of these problems. To ensure the accuracy of our findings, we generate 2D and 3D graphs accompanied by corresponding numerical data using specific functions, effectively illustrating the inequalities. Furthermore, it is easy to observe that some known results from previous studies manifest as special cases of our primary outcomes. This approach enables us to substantiate the validity of our findings and strengthen our conclusions. The connection of the main finding with the context of statistics and mathematics is provided, playing a significant role in addressing real-life problems.

In this study, some new Hermite-Hadamard type inequalities for co-ordinated convex functions were obtained with the help of conformable fractional integrals. We have presented some remarks to give the relation between our results and earlier obtained results. Moreover, an identity for partial differentiable functions has been established. By using this equality and concept of co-ordinated convexity, we have proven a trapezoid type inequality for conformable fractional integrals.

This study investigates the dynamic characteristics of the dual-mode resonant non-linear Schrodinger equation with a Bhom potential. Hydrodynamics, nonlinear optical fibre communication, elastic media, and plasma physics are just a few of the mathematical physics and engineering applications for this model. The study aims to achieve two main objectives: first, to discuss bifurcation analysis, and second, to extract optical soliton solutions using the extended hyperbolic function method. The study successfully derives various wave solutions, including bright, singular, periodic singular and dark solitons, based on the governing model. The findings conferred in this article show a crucial advancement in understanding the propagation of waves in non-linear media. Additionally, bifurcation of phase portraits of ordinary differential equation consistent with the partial differential equation under consideration is conducted. We also highlight specific constraint conditions that ensure the presence of these obtained solutions. The existing literature shows that these methods are first time applied on this model.

The Levin-Nohel equations play key roles in the interpretation of real phenomena and have interesting applications in engineering and science areas, such as mathematical physics, mathematical biology, image processing, and numerical analyses. This article investigates a new structure for the delay neutral Levin-Nohel integrodifferential (NLNID) system via a Hilfer fractional derivative and is supplemented by initial and instantaneous impulse conditions. A fractional integral equation corresponding to the proposed system is derived and used to prove the existence and uniqueness of the solution with the help of the Banach contraction principle. Additionally, the Ulam-Hyers-Mittag-Leffler (UHML) stability is studied by utilizing the generalized Gronwall's inequality and nonlinear analysis issues. As a consequence, the Ulam-Hyers (UH) stability and generalized UH are also deduced. Furthermore, the Riemann-Liouville ($ \mathcal{R.L.} $) and Caputo fractional versions of the proposed system are discussed. Finally, numerical applications supported with tables and graphics are provided to test the exactitude of the findings.

The features of analytical functions were mostly studied using a fuzzy subset and a $ \mathfrak{q} $-difference operator in this study, as we investigate many fuzzy differential subordinations related to the $ \mathfrak{q} $-analogue multiplier-Noor integral operator. By applying fuzzy subordination to univalent functions whose range is symmetric with respect to the real axis, we create a few new subclasses of analytical functions. We define numerous classes related to the family of linear $ \mathfrak{q} $ -operators and introduce them. Here, we focus on the inclusion results and other integral features.

The fractional Bagley-Torvik system (FBTS) is initially created by utilizing fractional calculus to study the demeanor of real materials. It can be described as the dynamics of an inflexible plate dipped in a Newtonian fluid. In the present article, we aim for the first time to discuss the existence and uniqueness (E&U) theories of an unbounded solution for the proposed generalized FBTS involving Riemann-Liouville fractional derivatives in the half-line $ (0, \infty) $, by using fixed point theorems (FPTs). Moreover, the Hyers-Ulam stability (HUS), Hyers-Ulam-Rassias stability (HURS), and semi-Hyers-Ulam-Rassias stability (sHURS) are proved. Finally, two numerical examples are given for checking the validity of major findings. By investigating unbounded solutions for the FBTS, engineers gain a deeper understanding of the underlying physics, optimize performance, improve system design, and ensure the stability of the motion of real materials in a Newtonian fluid.

This manuscript aims to study the effectiveness of two different levels of fractional orders in the frame of Caputo-Hadamard (CH)-derivatives on a special type class of delay problem supplemented by Dirichlet boundary conditions. The corresponding Hadamard fractional integral equation is derived for a proposed CH-fractional pantograph system. The Banach, Schaefer, and Krasnoselskii fixed point theorems (FPTs), are used to investigate sufficient conditions of the existence and uniqueness theorems for the proposed system. Furthermore, the Green functions properties are investigated and used to discuss the Ulam-Hyers (UH) stability and its generalized by utilizing nonlinear analysis topics. Finally, three mathematical examples are provided with numerical results and figures by using Matlab software to illustrate the validity of our findings.

The significance of the study lies in illuminating the consequences of thermophoresis and Brownian motion on heat and mass transfer, offering valuable insights crucial for practical applications. The purpose of this study is to look into how Casson–Williamson and Maxwell nanofluids flow over a stretching sheet in order to understand how heat and mass move in that situation. The mathematical structure comprises a set of partial differential equations (PDEs) converted into ordinary differential equations (ODEs) through a similarity transformation. Subsequently, the well-established MATLAB BVP4C method, integral to the finite difference approach, is applied to solve these ODEs. For the assurance of the method’s reliability and precision, the acquired outcomes are cross-referenced with existing literature, which is a crucial step in validation. The numerical results are depicted visually and in tables, specifically highlighting the impulse of different influencing elements on the Nusselt number, friction factor, and Sherwood number. Notably, an enhancement within the Brownian motion parameter (Nb) and the thermophoresis parameter (Nt) is found to correspond to higher local Nusselt numbers, indicating an enhancement in heat transfer. Conversely, elevated quantities of the Brownian motion parameter (Nb) result in a reduction in local Sherwood numbers. Physically, a higher Brownian motion parameter causes significant nanofluid particle displacement, enhancing their kinetic energy and intensifying heat generation in the boundary layer. The magnetic parameter (M) influences speed profile decline caused by the Lorentz force, which hinders fluid motion.

In this paper, we establish a novel identity by leveraging fractional integral operators of Riemann-type. Subsequently, we employ this identity to investigate Bullen's fractional integral inequalities within the framework of classical convex functions. The validity of these innovative inequalities is verified through the visualization of 2D and 3D graphs. Furthermore, we obtain certain well-known results as special cases to our main findings and express them as remarks. Lastly, we derive differences in mean estimates, which serve as valuable tools for representing complex information, facilitating decision-making and aiding analysis in various fields.

We study the invariance properties of the fractional time version of the nonlinear class of equations $ u_{t}^{\alpha}-g(u)\; u_{x}-f(u)\; u_{xxx} = 0 $, where $ 0 < \alpha < 1 $ using some recently developed symmetry-based techniques. The equations reduce to ordinary fractional Airy type, Korteweg-de Vries (KdV) and modified KdV equations through the change of variables provided by the symmetries. Furthermore, we utilize the symmetries to construct conservation laws for the fractional partial differential equations.

Abstract This paper aims to extend the application of the Ostrowski inequality, a crucial tool for figuring out the error bounds of various numerical quadrature rules, including Simpson’s, trapezoidal, and midpoint rules. Specifically, we develop a more comprehensive class of Ostrowski-type inequalities by utilizing the weighted version of Riemann–Liouville (RL) fractional integrals on an increasing function. We apply our findings to estimate the error bounds of Hadamard-type inequalities. Our results are more comprehensive, since we obtain the results of the existing literatures as particular cases for certain parameter values. This research motivates researchers to apply this concept to other fractional operators.

We aim to investigate an integro-differential inclusion using a novel computational approach in this research. The use of quantum calculus, and consequently the creation of discrete space, allows the computer and computational algorithms to solve our desired problem. Furthermore, to guarantee the existence of the solution, we use the endpoint property based on fixed point methods, which is one of the most recent techniques in fixed point theory. The above will show the novelty of our work, because most researchers use classical fixed point techniques in continuous space. Moreover, the sensitivity of the parameters involved in controlling the existence of the solution can be recognized from the heatmaps. For a better understanding of the issue and validation of the results, we presented numerical algorithms, tables and some figures in our examples that are presented at the end of the work.

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.

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.

We propose a new definition of the γ-convex stochastic processes (CSP) using center and radius (CR) order with the notion of interval valued functions (C.RI.V). By utilizing this definition and Mean-Square Fractional Integrals, we generalize fractional Hermite–Hadamard–Mercer-type inclusions for generalized C.RI.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 (CSP). Also, our work uses interesting examples of C.RI.V(CSP) with Python-programmed graphs to validate our findings using an extension of Mercer’s inclusions with applications related to entropy and information theory.

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 Jη,ςκ(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.

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.

This article aims to establish sufficient conditions for qualitative properties of the solutions for a new class of a pantograph implicit system in the framework of Atangana-Baleanu-Caputo ($ \mathcal{ABC} $) fractional derivatives with respect to another function under integral boundary conditions. The Schaefer and Banach fixed point theorems (FPTs) are utilized to investigate the existence and uniqueness results for this pantograph implicit system. Moreover, some stability types such as the Ulam-Hyers $ (\mathbb{UH}) $, generalized $ \mathbb{UH} $, Ulam-Hyers-Rassias $ (\mathbb{UHR}) $ and generalized $ \mathbb{UHR} $ are discussed. Finally, interpretation mathematical examples are given in order to guarantee the validity of the main findings. Moreover, the fractional operator used in this study is more generalized and supports our results to be more extensive and covers several new and existing problems in the literature.

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.

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.

This research work is devoted to investigating new common fixed point theorems on bipolar fuzzy [Formula: see text]-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.

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 [Formula: see text] 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.

This study focuses on the extended (3+1)-dimensional Sakovich equation, which is utilized for the description of nonlinear wave motion and manifests increased dispersion and nonlinear effects within the realm of nonlinear dynamics. The Lie symmetries are initially identified, followed by the construction of the Abelian algebra corresponding to these symmetries. Through the utilization of this Abelian algebra, the governing equation is transformed into an ordinary differential equation. Precise solutions are obtained using the unified technique, and the results are visually represented through 3D, 2D, and contour plots. Subsequently, the qualitative behavior of the equation is explored from multiple perspectives, encompassing bifurcation, chaos, and sensitivity analysis. Bifurcation is examined at critical points, while the dynamical system is subjected to an external force, leading to the detection of chaotic behavior employing diverse tools such as 3D and 2D phase portraits, time series, Poincaré maps, and multistability analysis. Furthermore, sensitivity analysis is conducted for various initial values, revealing the considerable sensitivity of the presented model, wherein even slight changes in the initial conditions result in significant variations. The reported outcomes are intriguing, showcasing the efficacy and applicability of the suggested methodologies for the assessment of soliton solutions and phase patterns in a diverse range of nonlinear models.

In this paper, we consider a coupled snap system in a fractional G-Caputo derivative sense with integral boundary conditions. Hyers-Ulam stability criterion is investigated, and a numerical simulation will be supplied to some applications. Some numerical simulations are presented to guarantee the theoretical results.

The concept of subordination is expanded in this study from the fuzzy sets theory to the geometry theory of analytic functions with a single complex variable. This work aims to clarify fuzzy subordination as a notion and demonstrate its primary attributes. With this work’s assistance, new fuzzy differential subordinations will be presented. The first theorems lead to intriguing corollaries for specific aspects chosen to exhibit fuzzy best dominance. The work introduces a new integral operator for meromorphic functions and uses the newly developed integral operator, which is starlike and convex, respectively, to obtain conclusions on fuzzy differential subordination.

This study uses Raina’s function to obtain a new coordinated pq-integral identity. Using this identity, we construct several new pq-Simpson’s type inequalities for generalized convex functions on coordinates. Setting p1=p2=1 in these inequalities yields well-known quantum Simpson’s type inequalities for coordinated generalized convex functions. Our results have important implications for the creation of post quantum mathematical frameworks.

A B-spline function is a series of flexible elements that are managed by a set of control points to produce smooth curves. By using a variety of points, these functions make it possible to build and maintain complicated shapes. Any spline function of a certain degree can be expressed as a linear combination of the B-spline basis of that degree. The flexibility, symmetry and high-order accuracy of the B-spline functions make it possible to tackle the best solutions. In this study, extended cubic B-spline (ECBS) functions are utilized for the numerical solutions of the generalized nonlinear time-fractional Klein–Gordon Equation (TFKGE). Initially, the Caputo time-fractional derivative (CTFD) is approximated using standard finite difference techniques, and the space derivatives are discretized by utilizing ECBS functions. The stability and convergence analysis are discussed for the given numerical scheme. The presented technique is tested on a variety of problems, and the approximate results are compared with the existing computational schemes.

This paper explores the double-chain model of DNA known as deoxyribonucleic acid, which has a significant role in the preservation and conversion of genetic data in biological fields. The setup consists of two rods that represent polynucleotide chains found in the DNA and are interconnected by an elastic membrane, showing the hydrogen bonds existing between the base pairs of the chains. The extended hyperbolic function method (EHFM) is analyzed to extract the solutions, including singular, bright, periodic, and kink soliton. The 3D, 2D, and contour graphs provide information on longitudinal and transverse displacements within the DNA structure. The visual exploration of some results reveals the presence of solitary waves in the DNA strands, and these findings have the potential to evaluate various applications. This study represents a significant advancement in our understanding of DNA dynamics, and illustrates the fundamental nature of genetic information transmission and processing.

Symmetric derivatives and integrals are extensively studied to overcome the limitations of classical derivatives and integral operators. In the current investigation, we explore the quantum symmetric derivatives on finite intervals. We introduced the idea of right quantum symmetric derivatives and integral operators and studied various properties of both operators as well. Using these concepts, we deliver new variants of Young’s inequality, Hölder’s inequality, Minkowski’s inequality, Hermite–Hadamard’s inequality, Ostrowski’s inequality, and Gruss–Chebysev inequality. We report the Hermite–Hadamard’s inequalities by taking into account the differentiability of convex mappings. These fundamental results are pivotal to studying the various other problems in the field of inequalities. The validation of results is also supported with some visuals.

In recent years, integral inequalities are investigated due to their extensive applications in several domains. The aim of the paper is to investigate certain new fractional integral inequalities which include Hermite-Hadamard inequality and different forms of trapezoid type inequalities related to Hermite-Hadamard inequality for h-Godunova-Levin preinvex function. Moreover, we compare our obtained results with the existing work in the literature and are represented by corollaries.

In this paper, we introduce the class of generalized strongly convex functions using Raina's function. We derive two new general auxiliary results involving first and second order (p,q)-differentiable functions and Raina's function. Essentially using these identities and the generalized strongly convexity property of the functions, we also found corresponding new generalized post-quantum analogues of Dragomir-Agarwal's inequalities. We discuss some special cases about generalized convex functions. To support our main results, we offer applications to special means, to hypergeometric functions, to Mittag-Leffler functions and also to (p,q)-differentiable functions of first and second order that are bounded in absolute value.

As we know, Atangana and Baleanu developed great fractional integral operators which used the generalized Mittag-Leffler function as non-local and non-singular kernel. Inspired by these integral operators, we derive in this paper two new fractional integral identities involving Atangana-Baleanu fractional integrals. Using these identities as auxiliary results, we establish new fractional counterparts of classical inequalities essentially using first and second order differentiable higher order strongly n-polynomial convex functions. We also discuss several important special cases of the main results. In order to show the efficiency of our main results, we offer applications for special means and for differentiable functions of first and second order that are in absolute value bounded.

The main objective of this paper is to derive some new variants of trapezium like inequalities involving k-fractional integrals with exponential kernel essentially using the definition of preinvex functions. Fractional bounds associated with some new fractional integral identities are also obtained. In order to show the significance of our main results, we also discuss applications to means and q-digamma functions. We discuss special cases which show that our results are quite unifying.

In this work, we introduce generalized Raina fractional integral operators and derive Chebyshev-type inequalities involving these operators. In a first stage, we obtain Chebyshev-type inequalities for one product of functions. Then we extend those results to account for arbitrary products. Also, we establish some inequalities of the Chebyshev type for functions whose derivatives are bounded. In addition, we derive an estimate for the Chebyshev functional by applying the generalized Raina fractional integral operators. As corollaries of this study, some known results are recaptured from our general Chebyshev inequalities. The results of this work may prove useful in the theoretical analysis of numerical models to solve generalized Raina-type fractional-order integro-differential equations.

The objective of this research is to study in detail the fractional variants of Ostrowski–Mercer-type inequalities, specifically for the first and second order differentiable s-convex mappings of the second sense. To obtain the main outcomes of the paper, we leverage the use of conformable fractional integral operators. We also check the numerical validations of the main results. Our findings are also validated through visual representations. Furthermore, we provide a detailed discussion on applications of the obtained results related to special means, q-digamma mappings, and modified Bessel mappings.

Singular singularly-perturbed problems (SSPPs) are a powerful mathematical tool for modelling a variety of real phenomena, such as nuclear reactions, heat explosions, mechanics, and hydrodynamics. In this paper, the numerical solutions to fourth-order singular singularly-perturbed boundary and initial value problems are presented using a novel quintic B-spline (QBS) approximation approach. This method uses a quasi-linearization approach to solve SSPNL initial/boundary value problems. And the non-linear problems are transformed into a sequence of linear problems by applying the quasi-linearization approach. The QBS functions produce more accurate results when compared to other existing approaches because of their local support, symmetry, and partition of unity features. This method can be applied to immediately solve the SSPPs without reducing the order in which they are presented. It has been demonstrated that the suggested numerical approach converges uniformly over the whole domain. The proposed approach is implemented on a few problems to validate the scheme. The computational results are compared, and they illustrate that the proposed approach performs better.

In this research, some novel Hermite–Hadamard–Fejér-type inequalities using Raina fractional integrals for the class of ϑ-convex functions are obtained. These inequalities are more comprehensive and inclusive than the corresponding ones present in the literature.

Our paper explores Hermite–Hadamard inequalities through the application of Abel–Gontscharoff Green’s function methodology, which involves interpolating polynomials and Riemann-type generalized fractional integrals. While establishing our main results, we explore new identities. These identities are used to estimate novel findings for functions, such that the second derivative of the functions is monotone, absolutely convex, and concave. A section relating the results of exploration to generalized means and trapezoid formulas is included in the applications. We anticipate that the method presented in this study will inspire further research in this field.

The Ostrowski inequality expresses bounds on the deviation of a function from its integral mean. The Ostrowski’s type inequality is frequently used to investigate errors in numerical quadrature rules and computations. In this work, Ostrowski-type inequality is demonstrated using the generalized fractional integral operators. From an application perspective, we present the bounds of the fractional Hadamard inequalities. The results that are being presented involve a number of fractional inequalities that are already known and have been published.