Search Results (666)

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 presents a fully implicit finite difference scheme for the numerical approximation of a wave equation featuring strong damping and a distributed delay term. The discretization employs second-order accurate approximations in both time and space. Although implicit, the scheme does not ensure unconditional stability due to the nonlocal nature of the delayed damping. To address this, we perform a stability analysis based on Rouché’s theorem from complex analysis and derive a sufficient condition for asymptotic stability of the discrete system. The resulting criterion highlights the interplay among the discretization parameters, the damping coefficient, and the delay kernel. Two quadrature techniques, the composite trapezoidal rule (CTR) and the Gaussian quadrature rule (GQR), are employed to approximate the convolution integral. Numerical experiments validate the theoretical predictions and illustrate both stable and unstable dynamics across different parameter regimes. Full article

Elastic solutions of a differential system of vibrational responses of a bidirectional porous functionally graded plate (BPFG) are described by employing high-order normal and shear deformation theory, in the present study. Natural frequency values are computed for the plates with simply supported boundary conditions and taking into consideration the thickness stretching effect. Grading of the effective material property for the BPFG plate is defined according to a power-law distribution. Navier’s approach is applied to determine the governing differential equations solution of the studied model derived by Hamilton’s principle. To confirm the reliability of the solution and the model accuracy, a comparison study with various studies that are presented in the literature is carried out. Numerical illustrations are presented to discuss the influences of the plate geometry, the porosity, the volume fraction distribution, and the nonlocality on the vibration behaviors of the model. The dynamic responses of unidirectional and bidirectional porous functionally graded nanoplates are analyzed in detail, employing two parametric numerical examples. Numerical results show the sensitivity of frequencies to the studied parametric factors and their dependence on porosity and nonlocality coefficients. Frequencies of BPFG with uneven/even distribution porosity decrease when increasing the transverse and axial power-law indexes ($P\ne 0$), and the same effect appears when increasing the nonlocal parameter. Full article

In this paper, we establish the global boundedness of weak solutions to fractional nonlocal equations using the fractional Moser iteration argument and some other ideas. Our results not only extend the boundedness result of Ros-Oton-Serra to general fractional nonlocal equations under a weaker assumption can but also be viewed as a generalization of the boundedness of weak solutions of second-order elliptic equations to nonlocal equations. Full article

In this paper, solvability of some inverse problems for a nonlocal analog of a pseudoparabolic equation is studied. The nonlocal analog of a pseudoparabolic equation is formed using transformations that have the involution property. Two types of inverse problems are considered. In the first problem, in addition to the solution, a function in the right-hand side of the equation depending on the spatial variable is determined. In the second problem, a function depending on the time variable is found. The first problem is solved using the Fourier method, and the second problem is solved by reducing to the integral Volterra equation. 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 focuses on exploring an impulsive stochastic differential variational inequality (ISDVI), which combines an impulsive stochastic differential equation and a stochastic variational inequality. Innovatively, our work incorporates two key aspects: first, our stochastic differential equation contains an impulsive term, enabling better handling of sudden event impacts; second, we utilize a non-local condition $z\left(0\right)={\chi }_0+\vartheta \left(z\right)$ that integrates measurements from multiple locations to construct superior models. Methodologically, we commence our analysis by using the projection method, which ensures the existence and uniqueness of the solution to ISDVI. Subsequently, we showcase the practical applicability of our theoretical findings by implementing them in the investigation of a stochastic consumption process and electrical circuit model. Full article

This paper proposes a novel analytical method to address the Helmholtz fractional differential equation by combining the Aboodh transform with the Adomian Decomposition Method, resulting in the Aboodh–Adomian Decomposition Method (A-ADM). Fractional differential equations offer a comprehensive framework for describing intricate physical processes, including memory effects and anomalous diffusion. This work employs the Caputo–Fabrizio fractional derivative, defined by a non-singular exponential kernel, to more precisely capture these non-local effects. The classical Helmholtz equation, pivotal in acoustics, electromagnetics, and quantum physics, is extended to the fractional domain. Following the exposition of fundamental concepts and characteristics of fractional calculus and the Aboodh transform, the suggested A-ADM is employed to derive the analytical solution of the fractional Helmholtz equation. The method’s validity and efficiency are evidenced by comparisons of analytical and approximation solutions. The findings validate that A-ADM is a proficient and methodical approach for addressing fractional differential equations that incorporate Caputo–Fabrizio derivatives. Full article

In this paper, we investigate the existence of multiple solutions for a Kirchhoff-type equation with Dirichlet boundary conditions defined on locally finite graphs. Our study extends some previous results on nonlinear Laplacian equations to the more complex Kirchhoff equation which incorporates a nonlocal term. By employing an abstract three critical points theorem that is based on Morse theory, we provide sufficient conditions that guarantee the existence of at least three distinct solutions, including two strictly positive solutions. We also present an example to verify our results. Full article

This paper investigates the initial-boundary value problem for a fourth-order pseudo-parabolic equation with a nonlocal source: $u_t+{\Delta }^{2}u-\Delta u_t={\left|u\right|}^{q-1}u-{\displaystyle \frac{1}{\left|\Omega \right|}}{\displaystyle {\int }_{\Omega }{\left|u\right|}^{q-1}udx}$. By employing the Galerkin method, the potential well method, and the construction of an energy functional, we establish threshold conditions for both the global existence and finite-time blow-up of solutions. Additionally, under the assumption of low initial energy $J\left(u_0\right)<d$, an upper bound for the blow-up time is derived. Full article

This study investigates the influence of rotation on wave propagation in a semiconducting nanostructure thermoelastic solid subjected to a ramp-type heat source within a two-temperature model. The thermoelastic interactions are modeled using the two-temperature theory, which distinguishes between conductive and thermodynamic temperatures, providing a more accurate description of thermal and mechanical responses in semiconductor materials. The effects of rotation, ramp-type heating, and semiconductor properties on elastic wave propagation are analyzed theoretically. Governing equations are formulated and solved analytically, with numerical simulations illustrating the variations in thermal and elastic wave behavior. The key findings highlight the significant impact of rotation, nonlocal parameters $e_{0}a$, and time derivative fractional order (FO) $\alpha$ on physical quantities, offering insights into the thermoelastic performance of semiconductor nanostructures under dynamic thermal loads. A comparison is made with the previous results to show the impact of the external parameters on the propagation phenomenon. The numerical results show that increasing the rotation rate $\mathsf{\Omega }=5$ causes a phase lag of approximately 22% in thermal and elastic wave peaks. When the thermoelectric coupling parameter ${\mathsf{\epsilon }}_3$ is increased from $-0.8\times {10}^{-42}$ to $1.2\times {10}^{-42}$. The temperature amplitude rises by 17%, while the carrier density peak increases by over 25%. For nonlocal parameter values $\mathsf{\epsilon }=0.3\to 0.6$, high-frequency stress oscillations are damped by more than 35%. The results contribute to the understanding of wave propagation in advanced semiconductor materials, with potential applications in microelectronics, optoelectronics, and nanoscale thermal management. Full article

This study focuses on the numerical solution and dynamics analysis of fractional governing equations related to double-layer nanoplates based on the shifted Legendre polynomials algorithm. Firstly, the fractional governing equations of the complicated mechanical behavior for bilayer nanoplates are constructed by combining the Fractional Kelvin–Voigt (FKV) model with the Caputo fractional derivative and the theory of nonlocal elasticity. Then, the shifted Legendre polynomial is used to approximate the displacement function, and the governing equations are transformed into algebraic equations to facilitate the numerical solution in the time domain. Moreover, the systematic convergence analysis is carried out to verify the convergence of the ternary displacement function and its fractional derivatives in the equation, ensuring the rigor of the mathematical model. Finally, a dimensionless numerical example is given to verify the feasibility of the proposed algorithm, and the effects of material parameters on plate displacement are analyzed for double-layer plates with different materials. Full article

This article discusses the transformation of a continuous-time model of the fractional system into a discrete-time model of the fractional system. For the continuous-time model, the exact solution of the nonlinear equation with fractional derivatives (FDs) that has the form of the damped rotator type with power non-locality in time is obtained.This equation with two FDs and periodic kicks is solved in the general case for the arbitrary orders of FDs without any approximations. A three-stage method for solving a nonlinear equation with two FDs and deriving discrete maps with memory (DMMs) is proposed. The exact solutions of the nonlinear equation with two FDs are obtained for arbitrary values of the orders of these derivatives. In this article, the orders of two FDs are not related to each other, unlike in previous works. The exact solution of nonlinear equation with two FDs of different orders and periodic kicks are proposed. Using this exact solution, we derive DMMs that describe a kicked damped rotator with power-law non-localities in time. For the discrete-time model, these damped DMMs are described by the exact solution of nonlinear equations with FDs at discrete time points as the functions of all past discrete moments of time. An example of the application, the exact solution and DMMs are proposed for the economic growth model with two-parameter power-law memory and price kicks. It should be emphasized that the manuscript proposes exact analytical solutions to nonlinear equations with FDs, which are derived without any approximations. Therefore, it does not require any numerical proofs, justifications, or numerical validation. The proposed method gives exact analytical solutions, where approximations are not used at all. Full article

This paper studies the existence, regularity, and properties of normalized ground state solutions for the mixed fractional Schrödinger equations. For subcritical cases, we establish the boundedness and Sobolev regularity of solutions, derive Pohozaev identities, and prove the existence of radial, decreasing ground states, while showing nonexistence in the $L^2$-critical case. For $L^2$-supercritical exponents, we identify parameter regimes where ground states exist, characterized by a negative Lagrange multiplier. The analysis combines variational methods, scaling techniques, and the careful study of fibering maps to address challenges posed by competing nonlinearities and nonlocal interactions. Full article

This paper investigates the nonlinear bending analysis of nano-beams using the physics-informed neural network (PINN) method. The nonlinear governing equations for the bending of size-dependent nano-beams are derived from Hamilton’s principle, incorporating nonlocal strain gradient theory, and based on Euler–Bernoulli beam theory. In the PINN method, the solution is approximated by a deep neural network, with network parameters determined by minimizing a loss function that consists of the governing equation and boundary conditions. Despite numerous reports demonstrating the applicability of the PINN method for solving various engineering problems, tuning the network hyperparameters remains challenging. In this study, a systematic approach is employed to fine-tune the hyperparameters using hyperparameter optimization (HPO) via Gaussian process-based Bayesian optimization. Comparison of the PINN results with available reference solutions shows that the PINN, with the optimized parameters, produces results with high accuracy. Finally, the impacts of boundary conditions, different loads, and the influence of nonlocal strain gradient parameters on the bending behavior of nano-beams are investigated. Full article