Search Results (510)

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

This research develops a novel coupled system of nonlinear hybrid stochastic fractional differential equations that integrates neutral effects, stochastic perturbations, and hybrid switching mechanisms. The system is formulated using the Atangana–Baleanu–Caputo fractional operator with a non-singular Mittag–Leffler kernel, which enables accurate representation of memory effects without singularities. Unlike existing approaches, which are limited to either neutral or hybrid stochastic structures, the proposed framework unifies both features within a fractional setting, capturing the joint influence of randomness, history, and abrupt transitions in real-world processes. We establish the existence and uniqueness of mild solutions via the Picard approximation method under generalized Carathéodory-type conditions, allowing for non-Lipschitz nonlinearities. In addition, mean-square Mittag–Leffler stability is analyzed to characterize the boundedness and decay properties of solutions under stochastic fluctuations. Several illustrative examples are provided to validate the theoretical findings and demonstrate their applicability. Full article

The main objective of this work is to discuss the generalized anti-periodic boundary conditions of the generalized Caputo fractional differential equations with $p\left(t\right)$-Laplacian operators. By applying the Schaefer fixed point theorem, the existence of mild solutions to this problem is obtained, which generalizes and enriches the anti-periodic boundary value problem of Caputo fractional hybrid differential equations. Finally, a numerical example is given to verify our main results. The anti-periodic boundary value condition imparts a form of symmetric inversion with respect to the original state, so it exhibits an anti-symmetric structural feature. Full article

In this study, we examine the error bounds related to Milne-type inequalities and a widely recognized Newton–Cotes method, originally developed for three-times-differentiable convex functions within the context of Jensen–Mercer inequalities. Expanding on this foundation, we explore Milne–Mercer-type inequalities and their application to a more refined class of three-times-differentiable s-convex functions. This work introduces a new identity involving such functions and Jensen–Mercer inequalities, which is then used to improve the error bounds for Milne-type inequalities in both Jensen–Mercer and classical calculus frameworks. Our research highlights the importance of convexity principles and incorporates the power mean inequality to derive novel inequalities. Furthermore, we provide a new lemma using Caputo–Fabrizio fractional integral operators and apply it to derive several results of Milne–Mercer-type inequalities pertaining to $(\alpha ,m)$-convex functions. Additionally, we extend our findings to various classes of functions, including bounded and Lipschitzian functions, and explore their applications to special means, the q-digamma function, the modified Bessel function, and quadrature formulas. We also provide clear mathematical examples to demonstrate the effectiveness of the newly derived bounds for Milne–Mercer-type inequalities. Full article

This paper undertakes a rigorous analytical exposition of the approximate controllability of a novel class of Sobolev-type stochastic impulsive differential inclusions, incorporating the Atangana–Baleanu fractional derivative in the Caputo configuration under the influence of Wiener process and Poissonian discontinuities. The system’s analytical landscape is further enriched by the incorporation of Clarke sub-differentials, facilitating the treatment of nonsmooth, nonconvex, and multivalued dynamics. The inherent complexity arising from the confluence of fractional memory, stochastic perturbations, and impulsive phenomena necessitates the deployment of a sophisticated apparatus from variational analysis, measurable selection theory, and multivalued fixed point frameworks within infinite-dimensional Banach spaces. This study delineates rigorous sufficient conditions, ensuring controllability under such hybrid influences, thereby generalizing classical paradigms to encompass nonlocal and discontinuous dynamical regimes. A precisely articulated exemplar is included to validate the theoretical constructs and demonstrate the operational efficacy of the proposed analytical methodology. Full article

One of the important techniques for solving several partial differential equations is the residual power series method, which provides the approximate solutions of differential equations in power series form.In this work, we use Aboodh transform in the analogical structure of the residual power series method to obtain a new method called the Aboodh residual power series method (ARPSM). By using this technique, we calculate the coefficients of some power series solutions of time Caputo fractional Kawahara equations. To obtain analytical and numerical solutions for the TCFKEs, we use ARPSM, first with the approximate initial condition and then with the exact initial condition. We present ARPSM’s reliability, efficiency, and capability by graphically describing the numerical results for analytical solutions and by comparing our solutions with other solutions for the TCFKEs obtained using two alternative methods, namely, the residual power series method and the natural transform decomposition method. Full article

This paper introduces a dynamic model that explores smoking and optimal control strategies. It shows how fractional-order (FO) analysis has uncovered hidden parts of complex systems and provides information about previously ignored elements. This paper uses the Bernoulli wavelet operational matrix method and the Adam–Bashforth–Moulton (ABM) method to analyse this model numerically. The mathematical model is segmented into five sub-classes: susceptible smokers, ingestion class, unusual smokers, regular smokers, and ex-smokers. It considers four optimal control measures: an anti-smoking education campaign, distribution of anti-smoking gum, administration of anti-nicotine drugs, and governmental restrictions on smoking in public areas. We show in this model how to control smoking in society strategically. Full article

This work develops a Boundary Element Method (BEM) formulation for simulating bioheat transfer in perfused biological tissues using the Atangana–Baleanu fractional derivative in the Caputo sense (ABC). The ABC operator incorporates a nonsingular Mittag–Leffler kernel to model thermal memory effects while preserving compatibility with standard boundary conditions. The formulation combines boundary discretization with cell-based domain integration to account for volumetric heat sources, and a recursive time-stepping scheme to efficiently evaluate the fractional term. The model is applied to a one-dimensional cylindrical tissue domain subjected to metabolic heating and external energy deposition. Simulations are performed for multiple fractional orders, and the results are compared with classical BEM $(a=1.0)$, Caputo-based fractional BEM, and in vitro experimental temperature data. The fractional order $a\approx 0.894$ yields the best agreement with experimental measurements, reducing the maximum temperature error to 1.2% while maintaining moderate computational cost. These results indicate that the proposed BEM–ABC framework effectively captures nonlocal and time-delayed heat conduction effects in biological tissues and provides an efficient alternative to conventional fractional models for thermal analysis in biomedical applications. Full article

This paper proposes a novel fractional-order asset flow model based on the Atangana–Baleanu–Caputo (ABC) derivative to analyze asset price dynamics in financial markets. Compared to classical models, the proposed model incorporates a nonlocal and non-singular fractional operator, allowing for a more accurate representation of investor behavior and market adjustment processes. The model captures both short-term trend-driven responses and long-term valuation-based decisions. We establish key theoretical properties of the system, including the existence and uniqueness of solutions, positivity, boundedness, and both local and global stability using Lyapunov functions. Numerical simulations under varying fractional orders demonstrate how the ABC derivative governs the convergence speed and equilibrium behavior of the system. Compared to classical integer-order models, the ABC-based approach provides smoother dynamics, greater flexibility in modeling behavioral heterogeneity, and better alignment with observed long-term financial phenomena. Full article

This work investigates the solutions of fractional integro-differential equations (FIDEs) using a unique kernel operator within the Caputo framework. The problem is addressed using both analytical and numerical techniques. First, the two-step Adomian decomposition method (TSADM) is applied to obtain an exact solution (if it exists). In the second part, numerical methods are used to generate approximate solutions, complementing the analytical approach based on the Adomian decomposition method (ADM), which is further extended using the Sumudu and Shehu transform techniques in cases where TSADM fails to yield an exact solution. Additionally, we establish the existence and uniqueness of the solution via fixed-point theorems. Furthermore, the Ulam–Hyers stability of the solution is analyzed. A detailed error analysis is performed to assess the precision and performance of the developed approaches. The results are demonstrated through validated examples, supported by comparative graphs and detailed error norm tables ($L_{\infty }$, $L_2$, and $L_1$). The graphical and tabular comparisons indicate that the Sumudu-Adomian decomposition method (Sumudu-ADM) and the Shehu-Adomian decomposition method (Shehu-ADM) approaches provide highly accurate approximations, with Shehu-ADM often delivering enhanced performance due to its weighted formulation. The suggested approach is simple and effective, often producing accurate estimates in a few iterations. Compared to conventional numerical and analytical techniques, the presented methods are computationally less intensive and more adaptable to a broad class of fractional-order differential equations encountered in scientific applications. The adopted methods offer high accuracy, low computational cost, and strong adaptability, with potential for extension to variable-order fractional models. They are suitable for a wide range of complex systems exhibiting evolving memory behavior. Full article

In this work, we explore Wirtinger-type inequalities involving tempered Ψ-Caputo fractional derivatives by utilizing Taylor’s formula. We establish more general inequalities for the same operator in $L_p$ norms for $p>1$ by using Hölder’s inequality. Special cases are discussed in the form of remarks by highlighting their relationships with the existing literature. The derived results are also verified through illustrative examples, including tables and graphs. Moreover, applications of the obtained inequalities are discussed in the context of arithmetic and geometric means. Full article

Objective: The accurate visualization of root canal systems on periapical radiographs is critical for successful endodontic treatment. This study aimed to evaluate and compare the effectiveness of several image enhancement algorithms—including a novel Total Variation–Contrast-Limited Adaptive Histogram Equalization (TV-CLAHE) technique—in improving the detectability of root canal configurations in mandibular incisors, using cone-beam computed tomography (CBCT) as the gold standard. A null hypothesis was tested, assuming that enhancement methods would not significantly improve root canal detection compared to original radiographs. Method: A retrospective analysis was conducted on 60 periapical radiographs of mandibular incisors, resulting in 420 images after applying seven enhancement techniques: Histogram Equalization (HE), Contrast-Limited Adaptive Histogram Equalization (CLAHE), CLAHE optimized with Pelican Optimization Algorithm (CLAHE-POA), Global CLAHE (G-CLAHE), k-Caputo Fractional Differential Operator (KCFDO), and the proposed TV-CLAHE. Four experienced observers (two radiologists and two dentists) independently assessed root canal visibility. Subjective evaluation was performed using an own scale inspired by a 5-point Likert scale, and the detection accuracy was compared to the CBCT findings. Quantitative metrics including Peak Signal-to-Noise Ratio (PSNR), Signal-to-Noise Ratio (SNR), image entropy, and Structural Similarity Index Measure (SSIM) were calculated to objectively assess image quality. Results: Root canal detection accuracy improved across all enhancement methods, with the proposed TV-CLAHE algorithm achieving the highest performance (93–98% accuracy), closely approaching CBCT-level visualization. G-CLAHE also showed substantial improvement (up to 92%). Statistical analysis confirmed significant inter-method differences (p < 0.001). TV-CLAHE outperformed all other techniques in subjective quality ratings and yielded superior SNR and entropy values. Conclusions: Advanced image enhancement methods, particularly TV-CLAHE, significantly improve root canal visibility in 2D radiographs and offer a practical, low-cost alternative to CBCT in routine dental diagnostics. These findings support the integration of optimized contrast enhancement techniques into endodontic imaging workflows to reduce the risk of missed canals and improve treatment outcomes. Full article

This paper focuses on the numerical modeling of drug diffusion in biological tissues using fractional time-dependent parabolic equations with non-local boundary conditions. The model includes a Caputo fractional derivative to capture the non-local effects and memory inherent in biological processes, such as drug absorption and transport. The theoretical framework of the problem is based on the work of Alhazzani, et al.,which demonstrates the solution’s goodness, existence, and uniqueness. Building on this foundation, we present a robust numerical method designed to deal with the complexity of fractional derivatives and non-local interactions at the boundaries of biological tissues. Numerical simulations reveal how fractal order and non-local boundary conditions affect the drug concentration distribution over time, providing valuable insights into drug delivery dynamics in biological systems. The results underscore the potential of fractal models to accurately represent diffusion processes in heterogeneous and complex biological environments. Full article

This paper introduces a novel fractional-order model using the Caputo derivative operator to investigate the virus dynamics of adaptive immune responses. Two infection routes, namely cell-to-cell and virus-to-cell transmissions, are incorporated into the dynamics. Our research establishes the existence and uniqueness of positive and bounded solutions through the application of the generalized mean-value theorem and Banach fixed-point theory methods. The fractional-order model is shown to be Ulam–Hyers stable, ensuring the model’s resilience to small errors. By employing the normalized forward sensitivity method, we identify critical parameters that profoundly influence the transmission dynamics of the fractional-order virus model. Additionally, the framework of optimal control theory is used to explore the characterization of optimal adaptive immune responses, encompassing antibodies and cytotoxic T lymphocytes (CTL). To assess the influence of memory effects, we utilize the generalized forward–backward sweep technique to simulate the fractional-order virus dynamics. This study contributes to the existing body of knowledge by providing insights into how the interaction between virus-to-cell and cell-to-cell dynamics within the host is affected by memory effects in the presence of optimal control, reinforcing the invaluable synergy between fractional calculus and optimal control theory in modeling within-host virus dynamics, and paving the way for potential control strategies rooted in adaptive immunity and fractional-order modeling. Full article