Search Results (2,253)

This paper investigates soliton solutions of the time-fractional Biswas–Arshed (BA) equation using the Extended Simplest Equation Method (ESEM). The model is analyzed under two distinct fractional derivative operators: the $\beta$-derivative and the M-truncated derivative. These approaches yield diverse solution types, including kink, singular, and periodic-singular forms. Also, in this work, a nonlinear second-order differential equation is reconstructed as a planar dynamical system in order to study its bifurcation structure. The stability and nature of equilibrium points are established using a conserved Hamiltonian and phase space analysis. A bifurcation parameter that determines the change from center to saddle-type behaviors is identified in the study. The findings provide insight into the fundamental dynamics of nonlinear wave propagation by showing how changes in model parameters induce qualitative changes in the phase portrait. The derived solutions are depicted via contour plots, along with two-dimensional (2D) and three-dimensional (3D) representations, utilizing Mathematica for computational validation and graphical illustration. This study is motivated by the growing role of fractional calculus in modeling nonlinear wave phenomena where memory and hereditary effects cannot be captured by classical integer-order approaches. The time-fractional Biswas–Arshed (BA) equation is investigated to obtain diverse soliton solutions using the Extended Simplest Equation Method (ESEM) under the $\beta$-derivative and M-truncated derivative operators. Beyond solution construction, a nonlinear second-order equation is reformulated as a planar dynamical system to analyze its bifurcation and stability properties. This dual approach highlights how parameter variations affect equilibrium structures and soliton behaviors, offering both theoretical insights and potential applications in physics and engineering. Full article

This study introduces a novel Fuzzy Piecewise Fractional Derivative (FPFD) framework to enhance epidemiological modeling, specifically for the multi-phasic co-infection dynamics of Omicron and malaria. We address the limitations of traditional models by incorporating two key realities. First, we use fuzzy set theory to manage the inherent uncertainty in biological parameters. Second, we employ piecewise fractional operators to capture the dynamic, phase-dependent nature of epidemics. The framework utilizes a fuzzy classical derivative for initial memoryless spread and transitions to a fuzzy Atangana–Baleanu–Caputo (ABC) fractional derivative to capture post-intervention memory effects. We establish the mathematical rigor of the FPFD model through proofs of positivity, boundedness, and stability of equilibrium points, including the basic reproductive number (R0). A hybrid numerical scheme, combining Fuzzy Runge–Kutta and Fuzzy Fractional Adams–Bashforth–Moulton algorithms, is developed for solving the system. Simulations show that the framework successfully models dynamic shifts while propagating uncertainty. This provides forecasts that are more robust and practical, directly informing public health interventions. Full article

This paper investigates a unique and stable numerical approximation of the Riemann–Liouville Fractional Derivative. We utilize diagonal norm finite difference-based time integration methods within the summation-by-parts framework. The second-order accurate discretizations developed in this study are proven to possess eigenvalues with strictly positive real parts for non-integer orders of the fractional derivative. These results lead to provably invertible, fully discrete approximations of Riemann–Liouville derivatives. Full article

This study investigates the effectiveness of integrating explicit learning-strategy instruction into gatekeeper mathematics courses to foster a math growth mindset, self-regulated learning (SRL), and improved academic performance among underrepresented minority students. The intervention was implemented across four key courses—College Algebra I/II and Calculus I/II—and incorporated evidence-based cognitive, metacognitive, and behavioral learning strategies through course materials, class discussions, and reflective assignments. Grounded in a conceptual framework linking learning-strategy instruction, growth mindset, SRL, and performance—while accounting for students’ social identities—the study explores both direct and indirect effects of the intervention. Using an explanatory sequential mixed-methods design, we first collected quantitative data via pre- and post-surveys/tests and analyzed performance outcomes, followed by qualitative focus groups to contextualize the findings. Results showed no significant effects of the intervention on growth mindset or SRL, nor evidence of mediation through these constructs. The direct effect of the intervention on performance was negative, though baseline mindset, SRL, and pre-course preparedness strongly predicted outcomes. No moderation effects were detected by student identities. The findings suggest that while explicit learning-strategy instruction may not independently shift mindset or SRL in the short term, pre-existing differences in these areas are consequential for performance. Qualitative findings provided further context for understanding how students engaged with the strategies and how instructor implementation shaped outcomes. These insights inform how learning strategies might be more effectively embedded in introductory math to support success and equity in STEM pathways, particularly in post-COVID educational contexts. Full article

In this study, we explore the soliton solutions of the truncated M-fractional fifth-order Korteweg–de Vries (KdV) equation by applying the improved modified extended tanh function method (IMETM). Novel analytical solutions are obtained for the proposed system, such as brigh soliton, dark soliton, hyperbolic, exponential, Weierstrass, singular periodic, and Jacobi elliptic periodic solutions. To validate these results, we present detailed graphical representations of selected solutions, demonstrating both their mathematical structure and physical behavior. Furthermore, we conduct a comprehensive linear stability analysis to investigate the stability of these solutions. Our findings reveal that the fractional derivative significantly affects the amplitude, width, and velocity of the solitons, offering new insights into the control and manipulation of soliton dynamics in fractional systems. The novelty of this work lies in extending the IMETM approach to the truncated M-fractional fifth-order KdV equation for the first time, yielding a wide spectrum of exact analytical soliton solutions together with a rigorous stability analysis. This research contributes to the broader understanding of fractional differential equations and their applications in various scientific fields. Full article

The accurate estimation of parameters in dynamical systems stands for an open key research issue in modeling, control, and fault diagnosis. The presence of noise in input and output signals poses a serious challenge for accurate real-time dynamical system parameter estimation. This paper proposes a new robust algebraic parameter estimation methodology for integer-order dynamical systems that explicitly incorporates the signal filtering dynamics within the estimator structure and enhances noise attenuation through fractional differentiation in frequency domain. The introduced estimation methodology is valid for Liouville-type fractional derivatives and can be applied to estimate online the parameters of differentially flat, oscillating or vibrating systems of multiple degrees of freedom. The parametric estimation can be thus implemented for a wide class of oscillating or vibrating, nth-order dynamical systems under noise influence in measurement and control signals. Positive values are considered for the inertia, stiffness, and viscous damping parameters of vibrating systems. Parameter identification can be also used for development of actuators and control technology. In this sense, validation of the algebraic parameter estimation is performed to identify parameters of a differentially flat, permanent-magnet direct-current motor actuator. Parameter estimation for both open-loop and closed-loop control scenarios using experimental data is examined. Experimental results demonstrate that the new parameter estimation methodology combining signal filtering dynamics and fractional calculus outperforms other conventional methods under presence of significant noise in measurements. Full article

The theory of stochastic processes is the prominent part of advanced probability theory and very influential in various mathematical models having randomness. One of the potential aspects is to investigate the stochastic convex processes. Working in the following direction, this study explores the set-valued super-quadratic processes through a unified approach under the centre-radius order relation, which is a totally ordered relation. First, we discuss some captivating properties and important results, which serve as a criterion. Relying on the newly proposed class of super-quadratic processes, we develop several fundamental inequalities within the fractional framework. Moreover, we present some novel deductions to complement the theoretical results with the existing literature. Also, we have provided the graphical breakdown, applications to the means, information theory, and divergence measures of the main inequalities. Full article

This study presents an innovative numerical method for solving linear fractional differential equations (LFDEs) using modified Bernstein polynomial bases. The proposed approach effectively addresses the challenges posed by the nonlocal nature of fractional derivatives, providing a robust framework for handling multiple initial and boundary value constraints. By integrating the LFDEs and approximating the solutions with modified fractional-order Bernstein polynomials, we derive operational matrices to solve the resulting system numerically. The method’s accuracy is validated through several examples, showing excellent agreement between numerical and exact solutions. Comparative analysis with existing data further confirms the reliability of the approach, with absolute errors ranging from 10⁻¹⁸ to 10⁻⁴. The results highlight the method’s efficiency and versatility in modeling complex systems governed by fractional dynamics. This work offers a computationally efficient and accurate tool for fractional calculus applications in science and engineering, helping to bridge existing gaps in numerical techniques. Full article

In this study, we investigate implicit fractional differential equations subject to anti-periodic boundary conditions. The fractional operator incorporates two distinct generalizations: the Caputo tempered fractional derivative and the Caputo fractional derivative with respect to a smooth function. We investigate the existence and uniqueness of solutions using fixed-point theorems. Stability in the sense of Ulam–Hyers and Ulam–Hyers–Rassias is also considered. Three detailed examples are presented to illustrate the applicability and scope of the theoretical results. Several existing results in the literature can be recovered as particular cases of the framework developed in this work. Full article

Non-zero differential initial values hinder the application of fractional operator theory in practical systems. This paper proposes a differential initial values zeroing method, decomposing functions with non-zero differential initial values into a compensation function (Taylor polynomial) and a zeroing function (with all differential initial values being zero). A differential initial values “zeroing operator” is defined, with properties such as initial value annihilation and linearity, and operational rules compatible with unilateral Laplace transforms and Mikusinski calculus operators. Based on the zeroing operator, the “zeroing differential operator” is defined to extract the zero-initial-value differential intrinsic properties of the functions with non-zero differential initial values. Using the zeroing operator, fractional constitutive equations are reconstructed in both time and complex Laplace domains in the self-congruent physical space, introducing complex fractional operators and generalized fractional operators. Validated by the complex fractional constitutive model of bone, this method breaks the bottleneck of zero-initial-value assumption in fractional operator theory in the self-congruent physical space, providing a rigorous mathematical foundation and a standardized tool for modeling sophisticated fractional systems with non-zero differential initial values. Full article

Background/objectives: Aging is associated with a decline in physiological and cognitive functions. Periodontitis, a disease affecting the periodontal tissues, increases in prevalence with age. Bacteria and inflammatory mediators resulting from periodontitis can trigger neuroinflammation and potentially accelerate the progression of neurodegenerative diseases. This study aimed to evaluate the association between periodontal status, salivary flow rate, salivary cortisol levels, and cytokine levels with cognitive status in elderly Indonesian subjects. Methods: This cross-sectional study involved 70 participants aged ≥ 60 years from several social institutions in Jakarta and the Dental Hospital, Faculty of Dentistry, Universitas Indonesia. All participants provided written informed consent before the examination. Periodontal parameters, including plaque score, calculus index, bleeding on probing, number of remaining teeth, and functional tooth units, were assessed. Unstimulated salivary flow was collected over five minutes, and salivary cortisol levels were measured. Gingival crevicular fluid samples from the deepest periodontal pockets were collected to measure cytokine levels (TNF-α and IL-1β). Both cortisol and cytokine levels were analyzed using ELISA. Cognitive function was evaluated using the Hopkins Verbal Learning Test. Results: Plaque score, calculus index, and bleeding on probing were moderately associated with cognitive scores (p < 0.05). In contrast, the number of remaining teeth, functional tooth units, periodontitis severity, salivary flow rate, salivary cortisol, and cytokine levels were not significantly associated with cognitive scores (p > 0.05). Conclusions: These findings suggest that elderly individuals with cognitive impairment tend to have poorer periodontal health than those with normal cognitive function. Full article

In recent years, fractional-order controllers have garnered increasing attention due to their enhanced flexibility and superior dynamic performance in control system design. Among them, the fractional-order Proportional–Integral–Derivative (FOPID) controller offers two additional tunable parameters, $\lambda$ and $\mu$, expanding the design space and allowing for finer performance tuning. However, the increased parameter dimensionality poses significant challenges for optimisation. To address this, the present study investigates the application of FOPID controllers to a two-wheeled self-balancing robot (TWSBR), with controller parameters optimised using intelligent algorithms. A novel Multi-Strategy Improved Beluga Whale Optimization (MSBWO) algorithm is proposed, integrating chaotic mapping, elite pooling, adaptive Lévy flight, and a golden sine search mechanism to enhance global convergence and local search capability. Comparative experiments are conducted using several widely known algorithms to evaluate performance. Results demonstrate that the FOPID controller optimised via the proposed MSBWO algorithm significantly outperforms both traditional PID and FOPID controllers tuned by other optimisation strategies, achieving faster convergence, reduced overshoot, and improved stability. Full article

The aim of this manuscript is to introduce the fractional integral inequalities of H-H types via multiplicative (Antagana-Baleanu) A-B fractional operators. We also provide the fractional version of the H-H type of the product and quotient of multiplicative superquadratic and multiplicative subquadratic functions via the same operators. Superquadratic functions, have stronger convexity-like behavior. They provide sharper bounds and more refined inequalities, which are valuable in optimization, information theory, and related fields. The use of multiplicative fractional operators establishes a nonlinear fractional structure, enhancing the analytical tools available for studying dynamic and nonlinear systems. The authenticity of the obtained results are verified by graphical and numerical illustrations by taking into account some examples. Additionally, the study explores applications involving special means, special functions and moments of random variables resulting in new fractional recurrence relations within the multiplicative calculus framework. These contributions not only generalize existing inequalities but also pave the way for future research in both theoretical mathematics and real-world modeling scenarios. Full article

DNA methylation is an epigenetic modification where a methyl group is added to a DNA molecule, typically at the cytosine base within a CpG dinucleotide. This process can influence gene expression without changing the underlying DNA sequence. Essentially, methylation can act like a switch that regulates which genes are active in a cell. DNA methylation (DNAm) models often describe the dynamic changes of methylation levels at specific DNA sites, considering methylation and demethylation processes. A common approach involves representing the methylation state as a continuous variable, and modelling its change over time or in response to various factors using differential equations. These equations can incorporate parameters such as the methylation and demethylation rates, factors like DNA replication, the influence of regulatory proteins, and other related parameters. Understanding DNAm dynamics in relation to age is crucial for elucidating ageing processes and developing biomarkers. This work introduces a theoretical framework for modelling DNAm dynamics using a fractional calculus approach, extending standard models based on the integer-order differential equations. The proposed fractional-calculus representation of the methylation process, defined by the fractional-order differential equation and its solution based on the Mittag–Leffler function, provides improved results compared to the standard model that uses a first-order differential equation, which contains an exponential function in its solution, in terms of the comparison criteria (sum of absolute errors, sum of squared errors, mean absolute percentage error, R-squared, and adjusted R-squared). Moreover, the Mittag–Leffler model provides a more general representation of DNAm dynamics, making the standard exponential model only one specific case. Full article