Fractional Differential Equations Research Articles

Abstract Fractional partial differential equations (FPDEs) are used as tools in the mathematical modeling of the natural phenomena and interpretation of many life problems in the fields of engineering and applied science. Mathematical models that include different types of partial differential equations are used in some fields of applied sciences such as biology, diffusion, electronic circuits, damping laws, and fluid mechanics. The derivation of modern analytical or numerical methods for solving FPDEs is a significant problem. In this article an efficient numerical method is proposed for solving a class of sixth-order FPDEs in which created by combining numerical Runge–Kutta–Mohammed (RKM) techniques with the method of lines. We have applied the modified approach to solve some problems involving the sixth-order FPDEs, and then, the numerical and analytical solutions for these problems have been compared. The comparisons in the implementations have proved the efficiency and accuracy of the developed RKM method. Also, the wave equation of fractional order as an application of the wave equation has been presented. Additionally, by contrasting the numerical implementations with exact answers, the effectiveness and accuracy of the suggested methodologies are shown.

This study presents a novel family of operators, referred to as tempered quantum fractional operators, and investigates the well-posedness of related tempered quantum fractional differential equations. The q-Adams predictor–corrector method is employed to conduct the analysis. By carefully adjusting the scheme’s parameters, the convergence rate can be controlled, while computational expenses increase linearly with time. Numerical simulations confirm the effectiveness and precision of the introduced algorithm.

Feature selection in high-dimensional datasets presents significant computational challenges, particularly in domains with large feature spaces and limited sample sizes. This paper introduces FL-SBA, a novel metaheuristic algorithm integrating fractional calculus enhancements with Laguerre operators into the Secretary Bird Optimization Algorithm framework for binary feature selection. The methodology incorporates fractional opposition-based learning utilizing Laguerre operators for enhanced population initialization with non-local memory characteristics, and a Laguerre-based binary transformation function replacing conventional sigmoid mechanisms through orthogonal polynomial approximation. Fractional calculus integration introduces memory effects that enable historical search information retention, while Laguerre polynomials provide superior approximation properties and computational stability. Comprehensive experimental validation across ten high-dimensional gene expression datasets compared FL-SBA against standard SBA and five contemporary methods including BinCOA, BAOA, BJSO, BGWO, and BMVO. Results demonstrate FL-SBA’s superior performance, achieving 96.06% average classification accuracy compared to 94.41% for standard SBA and 82.91% for BinCOA. The algorithm simultaneously maintained exceptional dimensionality reduction efficiency, selecting 29 features compared to 40 for competing methods, representing 27% improvement while achieving higher accuracy. Statistical analysis reveals consistently lower fitness values (0.04924 averages) and stable performance with minimal standard deviation. The integration addresses fundamental limitations in integer-based computations while enhancing convergence behavior. These findings suggest FL-SBA represents significant advancement in metaheuristic-based feature selection, offering theoretical innovation and practical performance improvements for high-dimensional optimization challenges.

The article presents a study of the computational complexity and efficiency of various parallel algorithms that implement the numerical solution of the equation in the hereditary α(t)-model of radon volumetric activity (RVA) in a storage chamber. As a test example, a problem based on such a model is solved, which is a Cauchy problem for a nonlinear fractional differential equation with a Gerasimov–Caputo derivative of a variable order and variable coefficients. Such equations arise in problems of modeling anomalous RVA variations. Anomalous RVA can be considered one of the short-term precursors to earthquakes as an indicator of geological processes. However, the mechanisms of such anomalies are still poorly understood, and direct observations are impossible. This determines the importance of such mathematical modeling tasks and, therefore, of effective algorithms for their solution. This subsequently allows us to move on to inverse problems based on RVA data, where it is important to choose the most suitable algorithm for solving the direct problem in terms of computational resource costs. An analysis and an evaluation of various algorithms are based on data on the average time taken to solve a test problem in a series of computational experiments. To analyze effectiveness, the acceleration, efficiency, and cost of algorithms are determined, and the efficiency of CPU thread loading is evaluated. The results show that parallel algorithms demonstrate a significant increase in calculation speed compared to sequential analogs; hybrid parallel CPU–GPU algorithms provide a significant performance advantage when solving computationally complex problems, and it is possible to determine the optimal number of CPU threads for calculations. For sequential and parallel algorithms implementing numerical solutions, asymptotic complexity estimates are given, showing that, for most of the proposed algorithm implementations, the complexity tends to be n2 in terms of both computation time and memory consumption.

In the present study, we aimed to derive analytical solutions of the homotopy analysis method (HAM) for the time-fractional Navier–Stokes equations in cylindrical coordinates in the form of a rapidly convergent series. In this work, we explore the time-fractional Navier–Stokes equations by replacing the standard time derivative with the Katugampola fractional derivative, expressed in the Caputo form. The homotopy analysis method is then employed to obtain an analytical solution for this time-fractional problem. The convergence of the proposed method to the solution is demonstrated. To validate the method’s accuracy and effectiveness, two examples of time-fractional Navier–Stokes equations modeling fluid flow in a pipe are presented. A comparison with existing results from previous studies is also provided. This method can be used as an alternative to obtain analytic and approximate solutions of different types of fractional differential equations applied in engineering mathematics.

Modelling of PEEK crystallisation kinetics using fractional differential equations

Measurement of changes in muscle viscoelasticity during static stretching using stress-relaxation data.

Analysis and discretization of nonlinear generalized fractional stochastic differential equations

Existence and stability of mixed type Hilfer fractional differential equations with impulses and time delay

This paper presents a novel B-spline wavelet-based scheme for solving multi-term time–space variable-order fractional nonlinear diffusion-wave equations. By combining semi-orthogonal B-spline wavelets with a collocation approach and a quasilinearization technique, we transform the original problem into a system of algebraic equations. To enhance the computational efficiency, we derive the operational matrix formulation of the proposed scheme. We provide a rigorous convergence analysis of the method and demonstrate its accuracy and effectiveness through numerical experiments. The results confirm the robustness and computational advantages of our approach for solving this class of fractional differential equations.

This study employs fractional analysis to investigate the existence and uniqueness of weak solutions to the fractional NavierStokes equations in three-dimensional turbulent flows, thereby addressing the complexity inherent in turbulent regimes. Turbulence presents significant challenges in fluid dynamics, characterised by chaotic and erratic motion that confounds conventional modelling techniques. By reformulating the NavierStokes equations using fractional derivatives, we capture non-local effects and memory phenomena, thereby enhancing the mathematical representation of fluid behaviour. First, we transform the Navier-Stokes equations to demonstrate the utility of fractional analysis. The second thing we do is show that there are weak answers in some situations. This means that we can use our models with starting data that isn’t stable. This is the third thing we do. We show that these answers are unique. This proves that our models are right. Ultimately, we demonstrate that weak solutions remain bounded over time under specific conditions regarding the initial data and external forces. When the Reynolds number approaches a critical level, we investigate its stability. This helps us understand how smooth flow can become rough flow. The findings of this research not only advance the theoretical understanding of weak solutions to the fractional Navier-Stokes equations but also have practical implications for modeling complex fluid systems. By linking fractional derivatives with turbulent flows, this work contributes to the broader field of fluid dynamics, paving the way for future investigations in applied mathematics and engineering. Ultimately, this exploration enhances our understanding of turbulence and its mathematical foundations, emphasising the importance of fractional calculus in accurately modelling fluid dynamics.

This article investigates the state estimation of fractional-order memristive systems with discrete-time terms. By considering discrete fractional calculus, we propose a novel and efficient criterion for ensuring the global Mittag–Leffler stability of the estimation error system. Additionally, by utilizing a functional that incorporates a discrete fractional sum element, we derive the stability condition for the concerned system. It is noteworthy that the proposed approach integrates a vector optimization method, which enhances the understanding of how to construct a meaningful convex closure formed by quaternions. Finally, numerical simulations are conducted to validate the theoretical results.

In this paper, we introduce notion of extended fuzzy bipolar metric space and prove some fixed point results in this space. Our results generalize and expand some of the literature’s well-known results. We also explore some of the applications of our key results to integral equation and fractional differential equation.

Fascioliasis, a zoonotic disease, involves complex transmission across humans, animals, snails, and the environment. This study presents a fractional-order model using the Caputo-Katugampola derivative to capture memory effects in disease dynamics. The model, a system of nine fractional differential equations, is solved using the Chebyshev spectral method, with convergence ensured through rigorous analysis. We prove the existence and uniqueness of solutions via the Banach fixed-point theorem. Stability analysis derives the basic reproduction number R_0, assessing disease-free and endemic equilibria. Sensitivity analysis identifies key parameters influencing transmission, informing control strategies. Numerical simulations illustrate the time evolution of all compartments, providing insights into fascioliasis dynamics. This framework enhances understanding of zoonotic disease epidemiology and demonstrates the utility of fractional calculus in modeling memory-dependent systems, offering a robust tool for studying infectious diseases.

This paper studies the existence theorems and Ulam stability results of solutions for implicit (p, q)-fractional difference equations. By applying Banach and Schauder fixed-point principles, we derive results related to the existence and uniqueness of solutions. Additionally, we analyze generalized Ulam-Hyers stability under (p, q)-Gronwall inequality. Key results are supported with illustrative examples, demonstrating the applicability of the proposed framework. Compared to previous studies restricted to the standard q-calculus, the present work introduces the (p, q)-Caputo fractional difference setting, which offers a more flexible and generalized approach. This novelty extends existing results and provides new perspectives for the analysis of stability and solvability of fractional systems.

The continuous search for new mathematical models that accurately represent real-world phenomena is a constant goal in the scientific community. To achieve this objective, the complexity of a large number of mathematical models has increased, such that certain considerations or restrictions and the use of numerical methods with greater computational requirements are required for their solution. For this, theories such as fractional calculus have been used, which have demonstrated an adequate characterization of physical phenomena, mainly in the area of biomechanics. However, there is no unique definition of the derivative concept, as in the case of integers, because it is a “new theory”. In this article, the kernel of the Atangana-Baleanu fractional derivative is used, which satisfies the most common properties of the classical derivative, and addresses the existing problem when considering initial conditions of the model, which in biomechanics is associated with material memory.

Statistical analysis for fractional differential equation models

In the presents investigation a fractional-order SIR mathematical model formulated with the Caputo-Fabrizio derivative to investigate the transmission dynamics of Lumpy Skin Disease (LSD) in domestic cattle. Model parameter characteristic included as positivity, boundedness and the basic reproduction number are analytically established along with existence and uniqueness proven by the fixed-point theory. Optimal control strategies derived using Pontryagin's maximum principle target infection reduction through prevention, pre-treatment and enhanced treatment measures. Numerical simulations have been carried out the fractional-order dynamics which capture the real-world complexities with lower fractional order accelerating disease spread. Comparative analysis with integer-order models confirms the superior applicability of fractional calculus. The obtained results demonstrate that the proposed control measures can substantially reduce infection levels, offering practical guidance for LSD mitigation in cattle populations.

This paper provides a comprehensive examination of the pivotal roles played by Partial Differential Equations (PDE), Stochastic Differential Equations (SDE), and Backward Stochastic Differential Equations (BSDE) in financial asset pricing. It traces their theoretical evolution from the foundational Black-Scholes PDE model, through flexible SDE capturing market randomness, to advanced BSDE adept at handling complex, nonlinear problems with terminal conditions. The study details their diverse applications in pricing interest rate derivatives, exotic options, XVAs, and portfolio optimization. It also analyzes key challenges: the curse of dimensionality for PDE, unrealistic assumptions in SDE, and computational intensity for BSDE. Future research directions are explored, focusing on integration with fractional calculus, artificial intelligence, and high-performance computing to enhance efficiency and adaptability. This work concludes that PDE, SDE, and BSDE form a synergistic, indispensable toolkit for modern quantitative finance.

Globally, breast cancer represents the leading cancer type, with millions of women impacted annually. The success of breast cancer treatment relies heavily on timely detection and precise tumor classification. The classification of breast cancer has gained considerable importance in Deep Learning (DL) and medical research with the development of medical imaging techniques, like histopathological imaging. Many existing DL schemes suffer from overfitting and endure difficulties in effectively mining the key features from high-resolution images with subtle variations. Hence, the Xception Convolutional Deep Maxout Network (Xcov-DMN) is developed to classify breast cancer. At the initial stage, the Mean-Shift Filter is applied to the input histopathological image. Following this, the White Blood Cell Network (WBC-Net) is employed for blood cell segmentation with the Balanced Cross-Entropy (BCE) and Focal Loss for ensuring precise segmentation. Next, Colored Histograms, shape features, Haralick Texture Features, and Complete Local Binary Pattern (CLBP) features are excerpted. Consequently, the developed Xcov-DMN is utilized to classify breast cancer. Xcov-DMN is the combination of the Deep Maxout Network (DMN), Fractional Calculus (FC), and Xception Convolutional Neural Network (XCovNet). Moreover, with learning data at 90%, the Xcov-DMN achieved the highest accuracy of 92.755%, True Negative Rate (TNR) of 91.977%, and True Positive Rate (TPR) of 94.765%.