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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.

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.

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.

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.

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.

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.

The KdV and mKdV equations have many significant applications in the field of plasma physics and ocean dynamics. In this work, the traveling wave behavior of the newly proposed (2+1)-dimensional combined KdV–mKdV equation has been examined using two direct analytical approaches. The two variable G′G,1G-expansion method and generalized projective Riccati equation method are used to determine a variety of solitons and solitary waves for the governing model. The graphical observations are made for suitable choice of parametric values. The obtained results will provide useful insight in further explorations of the wave phenomena depicted by the considered equation.