Search Results (164)

The fractional PDE-ODE coupled system with Neumann boundary condition is considered in this paper. We design a boundary state feedback controller using the Backstepping method to stabilize the considered system. According to operator semigroup theory, we obtain a unique solution of the investigated system. Based on this, by using the fractional Lyapunov scheme, we prove that the system is Mittag-Leffler stable. Full article

This paper investigates a new class of fractional integral operators, namely, the exponentially damped Riesz-type operators within the framework of variable exponent Lebesgue spaces $L^{\mathtt{p}(·)}$. To the best of our knowledge, the boundedness of such operators has not been addressed in any existing functional setting. We establish their boundedness under appropriate log-Hölder continuity and growth conditions on the exponent function $\mathtt{p}(·)$. To highlight the novelty and practical relevance of the proposed operator, we conduct a comparative analysis demonstrating its effectiveness in addressing convergence, regularity, and stability of solutions to partial differential equations. We also provide non-trivial examples that illustrate not only these properties but also show that, under this operator, a broader class of functions becomes locally integrable. The exponential decay factor notably broadens the domain of boundedness compared to classical Riesz and Bessel–Riesz potentials, making the operator more versatile and robust. Additionally, we generalize earlier results on Sobolev-type inequalities previously studied in constant exponent spaces by extending them to the variable exponent setting through our fractional operator, which reduces to the classical Riesz potential when the decay parameter $\lambda =0$. Applications to elliptic PDEs are provided to illustrate the functional impact of our results. Furthermore, we develop several new structural properties tailored to variable exponent frameworks, reinforcing the strength and applicability of the proposed theory. Full article

This study introduces a novel analytical technique known as the double local fractional Yang–Laplace transform method ($LF{L}_{\zeta }^2$) and rigorously investigates its foundational properties, including linearity, differentiation, and convolution. The proposed method is formulated via double local fractional integrals, enabling a robust mechanism for addressing local fractional partial differential equations defined on fractal domains, particularly Cantor sets. Through a series of illustrative examples, we demonstrate the applicability and efficacy of the $LF{L}_{\zeta }^2$ transform in solving complex local fractional partial differential equation models. Special emphasis is placed on the local fractional Laplace equation, the linear local fractional Klein–Gordon equation, and other models, wherein the method reveals significant computational and analytical advantages. The results substantiate the method’s potential as a powerful tool for broader classes of problems governed by local fractional dynamics on fractal geometries. Full article

This paper proposes a computational procedure to resolve the temporal fractional financial option pricing partial differential equation (PDE) using a localized meshless approach via the multiquadric radial basis function (RBF). Given that financial market information is best characterized within a martingale framework, the resulting option pricing model follows a modified Black–Sholes (BS) equation, requiring efficient numerical techniques for practical implementation. The key innovation in this study is the derivation of analytical weights for approximating first and second derivatives, ensuring improved numerical stability and accuracy. The construction of these weights is grounded in the second integration of a variant of the multiquadric RBF, which enhances smoothness and convergence properties. The performance of the presented solver is analyzed through computational tests, where the analytical weights exhibit superior accuracy and stability in comparison to conventional numerical weights. The results confirm that the new approach reduces absolute errors, demonstrating its effectiveness for financial option pricing problems. Full article

The main objective of this study is to present a fundamental mathematical model for nerve impulse transport, based on the underlying physical phenomena, with a straightforward application in describing the functionality of prosthetic devices. The governing equation of the resultant model is a two-dimensional nonlinear partial differential equation with a time-fractional derivative of order $\alpha \in (0,1)$. novel and effective numerical approach for solving this fractional-order problem is constructed based on the virtual element method. Three basic technical building blocks form the basis of our methodology: the regularity theory related to nonlinearity, discrete maximal regularity, and a fractional variant of the Grünwald–Letnikov approximation. By utilizing these components, along with the energy projection operator, a fully discrete virtual element scheme is formulated in such a way that it ensures stability and consistency. We establish the uniqueness and existence of the approximate solution. Numerical findings confirm the convergence in the $L^2$–norm and $H^1$–norm on both uniform square and regular Voronoi meshes, confirming the effectiveness of the proposed model and method, and their potential to support the efficient design of sensory prosthetics. Full article

In this work, the novel general formula for a time-dependent nuclear reactor system of equations with delayed neutron effect has been formulated using a fractional calculus model. We explore the properties of this model, including two analytical approximation methods, the Temimi–Ansari method (TAM) and the Sumudu residual power series method (SRPSM), for solving the equation. These methods allow for the computation of approximate solutions at specific points. This is particularly useful for partial differential equations (PDEs) arising in various fields like physics, engineering, and finance. This work is hoped to improve the advancement of nuclear modeling and simulation, providing researchers and engineers with a powerful mathematical tool for studying the complex dynamics of these critical energy systems. Full article

In this study, we introduce an innovative approach for addressing fractional partial differential equations (fPDEs) by combining Monte Carlo-based physics-informed neural networks (PINNs) with the cuckoo search (CS) optimization algorithm, termed PINN-CS. There is a further enhancement in the application of quasi-Monte Carlo assessment that comes with high efficiency and computational solutions to estimates of fractional derivatives. By employing structured sampling nodes comparable to techniques used in finite difference approaches on staggered or irregular grids, the proposed PINN-CS minimizes storage and computation costs while maintaining high precision in estimating solutions. This is supported by numerous numerical simulations to analyze various high-dimensional phenomena in various environments, comprising two-dimensional space-fractional Poisson equations, two-dimensional time-space fractional diffusion equations, and three-dimensional fractional Bloch–Torrey equations. The results demonstrate that PINN-CS achieves superior numerical accuracy and computational efficiency compared to traditional fPINN and Monte Carlo fPINN methods. Furthermore, the extended use to problem areas with irregular geometries and difficult-to-define boundary conditions makes the method immensely practical. This research thus lays a foundation for more adaptive and accurate use of hybrid techniques in the development of the fractional differential equations and in computing science and engineering. Full article

Robust epidemiological models are essential for managing COVID-19, especially in diverse urban settings. In this study, we present a fractional advection–diffusion–reaction model to analyze COVID-19 spread in three major Turkish cities: Ankara, Istanbul, and Izmir. The model employs a Caputo-type time-fractional derivative, with its order dynamically determined by the Hurst exponent, capturing the memory effects of disease transmission. A nonlinear reaction term models self-reinforcing viral spread, while a Gaussian forcing term simulates public health interventions with adjustable spatial and temporal parameters. We solve the resulting fractional PDE using an implicit finite difference scheme that ensures numerical stability. Calibration with weekly case data from February 2021 to March 2022 reveals that Ankara has a Hurst exponent of 0.4222, Istanbul 0.1932, and Izmir 0.6085, indicating varied persistence characteristics. Distribution fitting shows that a Weibull model best represents the data for Ankara and Istanbul, whereas a two-component normal mixture suits Izmir. Sensitivity analysis confirms that key parameters, including the fractional order and forcing duration, critically influence outcomes. These findings provide valuable insights for public health policy and urban planning, offering a tailored forecasting tool for epidemic management. Full article

We introduce a novel class of coherent states, termed ${\mathcal{W}}_{\alpha ,\beta ,\nu }$-coherent states, constructed using a deformed boson algebra based on the generalised factorial ${\left[n\right]}_{\alpha ,\beta ,\nu }!$. This algebra extends conventional factorials, incorporating advanced special functions such as the Mittag-Leffler and Wright functions, enabling the exploration of a broader class of quantum states. The mathematical properties of these states, including their continuity, completeness, and quantum fluctuations, are analysed. A key aspect of this work is the resolution of the Stieltjes moment problem associated with these states, achieved through the inverse Mellin transformation method. The framework provides insights into the interplay between the classical and quantum regimes, with potential applications in quantum optics and fractional quantum mechanics. By extending the theoretical landscape of coherent states, this study opens avenues for further exploration in mathematical physics and quantum technologies. Full article

The combination of fractional derivatives (due to their global behavior) and the challenges related to hyperbolic PDEs pose formidable obstacles in solving fractional hyperbolic equations. Due to the importance and applications of the fractional telegraph equation, solving it and presenting accurate solutions via a novel and effective method can be useful. This work introduces and implements a method based on the spectral element method (SEM) that relies on interpolating scaling functions (ISFs). Through the use of an orthonormal projection, the method maps the equation to scaling spaces raised from multi-resolution analysis (MRA). To achieve this, the Caputo fractional derivative (CFD) is represented by ISFs as a square matrix. Remarkable efficiency, ease of implementation, and precision are the distinguishing features of the presented method. An analysis is provided to demonstrate the convergence of the scheme, and illustrative examples validate our method. Full article

In this work, we present various novelty methods by employing the fractional differential quadrature technique to solve the time and space fractional nonlinear Benjamin–Bona–Mahony equation and the Benjamin–Bona–Mahony–Burger equation. The novelty of these methods is based on the generalized Caputo sense, classical differential quadrature method, and discrete singular convolution methods based on two different kernels. Also, the solution strategy is to apply perturbation analysis or an iterative method to reduce the problem to a series of linear initial boundary value problems. Consequently, we apply these suggested techniques to reduce the nonlinear fractional PDEs into ordinary differential equations. Hence, to validate the suggested techniques, a solution to this problem was obtained by designing a MATLAB code for each method. Also, we compare this solution with the exact ones. Furthermore, more figures and tables have been investigated to illustrate the high accuracy and rapid convergence of these novel techniques. From the obtained solutions, it was found that the suggested techniques are easily applicable and effective, which can help in the study of the other higher-D nonlinear fractional PDEs emerging in mathematical physics. Full article

Fractional calculus plays a pivotal role in modern scientific and engineering disciplines, providing more accurate solutions for complex fluid dynamics phenomena due to its non-locality and inherent memory characteristics. In this study, Caputo’s time fractional derivative operator approach is employed for heat and mass transfer modeling in unsteady Maxwell fluid within a cylinder. Governing equations within a cylinder involve a system of coupled, nonlinear fractional partial differential equations (PDEs). A machine learning technique based on the Levenberg–Marquardt scheme with a backpropagation neural network (LMS-BPNN) is employed to evaluate the predicted solution of governing flow equations up to the required level of accuracy. The numerical data sheet is obtained using series solution approach Homotopy perturbation methods. The data sheet is divided into three portions i.e., $80\%$ is used for training, $10\%$ for validation, and $10\%$ for testing. The mean-squared error (MSE), error histograms, correlation coefficient (R), and function fitting are computed to examine the effectiveness and consistency of the proposed machine learning technique i.e., LMS-BPNN. Moreover, additional error metrics, such as R-squared, residual plots, and confidence intervals, are incorporated to provide a more comprehensive evaluation of model accuracy. The comparison of predicted solutions with LMS-BPNN and an approximate series solution are compared and the goodness of fit is found. The momentum boundary layer became higher and higher as there was an enhancement in the value of Caputo, fractional order α = 0.5 to α = 0.9. Higher thermal boundary layer (TBL) profiles were observed with the rising value of the heat source. Full article

This study contains a mathematical model for river pollution and its remediation for an unsteady state and investigates the effect of aeration on the degradation of pollutants. The governing equation is a pair of nonlinear time-fractional two-dimensional advection-diffusion equations for pollutant and dissolved oxygen (DO) concentration. The coupling of these equations arises due to the chemical interactions between oxygen and pollutants, forming harmless chemicals. The Fractional Reduced Differential Transform Method (FRDTM) is applied to provide approximate solutions for the given model. Also, the convergence of solutions is checked for efficacy and accuracy. The effect of longitudinal and transverse diffusion coefficients of pollutant and DO on the concentration of pollutant and DO is analyzed numerically and graphically. Also, we checked the effect of change in the river’s longitudinal and transverse seepage velocity on pollutant and DO concentration numerically and graphically. We analyzed the comparison of change in the value of half-saturated oxygen demand concentration for pollutant decay on pollutant and DO concentration numerically and graphically. Also, numerical and graphical analysis examined the effect of fractional parameters on pollution levels. Full article

In the present article, the method which was obtained from a combination of the conformable fractional double Laplace transform method (CFDLTM) and the homotopy perturbation method (HPM) was successfully applied to solve linear and nonlinear conformable fractional partial differential equations (CFPDEs). We included three examples to help our presented technique. Moreover, the results show that the proposed method is efficient, dependable, and easy to use for certain problems in PDEs compared with existing methods. The solution graphs show close contact between the exact and CFDLTM solutions. The outcome obtained by the conformable fractional double Laplace transform method is symmetrical to the gain using the double Laplace transform. Full article

This paper studies synchronization behaviors of two sorts of non-linear fractional-order complex spatio-temporal networks modeled by hyperbolic space-varying PDEs (FCSNHSPDEs), respectively, with time-invariant delays and time-varying delays, including one delayed coupling. One distributed controller with space-varying control gains is firstly designed. For time-invariant delayed cases, sufficient conditions for synchronization of FCSNHSPDEs are presented via LMIs, which have no relation to time delays. For time-varying delayed cases, synchronization conditions of FCSNHSPDEs are presented via spatial algebraic LMIs (SALMIs), which are related to time delay varying speeds. Finally, two examples show the validity of the control approaches. Full article