In this study, a variational model is proposed for filtering additive and multiplicative noise in wrapped phase maps. The model includes the Pythagorean identity, which is a fundamental requirement for phase maps, as well as the total variation regularizer to maintain discontinuities. We prove that our model is well-posed, compute the Euler-Lagrange equations, and present a fixed-point convergent algorithm for solving it. The quality of the proposed model is confirmed by conducting experiments on both synthetic and real data. Additionally, the implementation of the parallel fixed-point algorithm to accelerate the model's solution is presented. Implementations consist of OpenMP and CUDA execution on a multicore CPU and GPU, respectively. A comparison of the performance of parallel implementations with state-of-the-art methods is presented using synthetic and real data. According to the findings, our parallel implementations achieve speedups of 12x for multi-core CPUs and 125x for GPUs compared to the serial implementation.

Deep learning has proven to be a powerful approach for solving complex problems in physical and environmental systems. Inspired by its success, we propose a novel framework called the Wavelet Integrated Lateral Neural Network (WLNN) for approximating solutions of time-fractional differential equations that describe the dynamics of oil pollution. The proposed architecture incorporates lateral connections between hidden layers to enhance spatiotemporal learning and integrates wavelet transforms to efficiently capture multi-scale and non-local features inherent in fractional dynamics. To ensure theoretical soundness, the existence and uniqueness of the solution to the time-fractional differential equation are rigorously established. The framework is validated using a time-fractional Allen–Cahn equation, a representative model for oil dispersion in aquatic environments. Comparative analyses with Physics-Informed Neural Networks (PINNs) and gradient-enhanced PINNs (gPINNs) demonstrate that the proposed WLNN achieves higher accuracy in solving time-fractional equations related to oil pollution.

This paper develops a numerical method to study the patterns and chaos dynamic behaviour of the fractional-order-in-time Lengyel-Epstein reaction-diffusion system (LERDS). We combine theoretical analysis with numerical simulation to conduct stability, Turing bifurcation, and critical wave number for the fractional-order-in-time LERDS. The numerical results show some chaos and patterns.

This article proposes a numerical scheme based on finite difference approximation for solving the time-fractional advection-diffusion equation on a metric star graph comprising initial, boundary, and transmission conditions on the nodes of the graph. The choice of considering a star graph is motivated by the fact that any arbitrary graph can be decomposed into a star graph. The fractional derivative is considered in the Caputo sense and the well-known L 1 method is applied for its discrete approximation. The unconditional stability of the difference scheme for Δ x i ≤ 2 is analyzed by employing the discrete energy method. The unique solution behaviour of the scheme is also studied. An overall convergence of order O ( Δ t 2 − α + ∑ i = 1 k Δ x i 2 ) is obtained from the proposed scheme. Finally, a few experiments over test examples are demonstrated to verify the accuracy and convergence of the proposed scheme.

This paper introduces an efficient method that hybridizes an accelerated derivative-free approach with the Picard-Krasnoselskii iterative process to solve systems of nonlinear equations. The proposed method integrates a correction parameter into the search direction using the hybrid of the Picard-Krasnoselskii approach and approximates the Jacobian matrix with an acceleration parameter. We establish the global convergence of the method under favourable conditions. Numerical experiments demonstrate the proposed method's superior performance compared to existing techniques. Additionally, the method has been effectively applied to the three-degree-of-freedom (3DOF) model of robotic motion control, showcasing its practical applicability in science and engineering.

This research examines a system of differential-difference equations, emphasizing the impact of small shifts, which are crucial in delayed feedback control systems. By employing Taylor series expansions, we approximate the delay and advance parameters within these equations, enabling a detailed exploration of their complex dynamics with both accuracy and practical applicability. To discretize the problem, we employ the Crank-Nicolson finite difference method along the time dimension using a uniform mesh. In the spatial dimension, we apply a Shishkin-type mesh alongside a cubic B-spline collocation method. This integrated approach takes advantage of the strengths of each discretization method across different dimensions, resulting in a robust and highly accurate numerical solution. The convergence analysis demonstrates that the method attains nearly second-order uniform convergence with respect to the perturbation parameters, and the numerical results are in close agreement with the theoretical predictions.

Chemical potential and temperature gradients cause energy/heat and diffusion/mass flows, which advance electrical power production, solar energy technologies, chemical engineering, hydrology, nuclear waste management, high-temperature processes, petrology and geology. This study presents an unsteady mathematical model for magneto-convective viscoelastic Casson nanofluid flow over a stretching surface with nano-coating and multiple slip effects. The governing partial differential equations capture the transient behaviour of momentum, thermal and solutal transport under the influence of an external magnetic field, thermal radiation and thermophoresis. To improve thermal conductivity in the fluid, a nano-coating with carbon nanotubes has been primarily applied on the stretching sheet surface. The viscoelasticity of the Casson fluid has been investigated through analysing the effect of Deborah number on skin friction against Reynold number. Since the partial differential equations are very nonlinear, they must be converted into ordinary differential equations (ODEs) using the appropriate correlation transformation. A numerical technique using a shooting procedure is utilized to solve the nonlinear and coupled ODEs. The graphical representation of numerical solutions for momentum and energy descriptions is explored with various parameters.

Dealing with extreme values is a key challenge in probabilistic modeling, relevant to economics, engineering, and life sciences. Standard GANs, built on light-tailed noise, fail to capture heavy-tail behavior and dependence in extreme regions. We propose HTGAN, a modified GAN with heavy-tailed input noise. Using the stable tail dependence function (stdf) from extreme-value theory, we derive a worst-case error bound for approximating the stdf of the target, scaling as N − 1 / ( d − 1 ) , where N is the latent noise dimension and the data dimension. Thus, higher latent dimension improves dependence estimation. Experiments on synthetic and real heavy-tailed datasets confirm that HTGAN better reproduces extreme dependence than classical light-tailed GANs, with accuracy improving as latent dimension increases.

This paper provides an analysis of the regularity and Hölder continuity of the analytical solution to the stochastic convolution equation for the fractional Brownian motion. It also attempts to analyse the strong convergence of a fully discrete approximation for such equations, implemented by the spectral Galerkin in space and the classical Euler–Maruyama (EM) method in time. Furthermore, it conducts a convergence analysis of the numerical solution, obtaining the corresponding strong convergence order and finally presents numerical an example to verify the theoretical analysis.

The goal of this work is investigating the existence and uniqueness of solutions for a class of non-local Riemann-Liouville fractional q-integro-differential equations, in addition to finding the approximate solutions to the proposed equation. The problem incorporates Riemann-Liouville fractional derivatives, Riemann-Liouville fractional q integral, and a q-nonlocal boundary condition. By leveraging the Schauder fixed-point theorem and the Banach contraction principle, we establish rigorous conditions under which solution exists and is unique. These mathematical methods are vital for determining the existence and uniqueness of a solution for a class of non-local Riemann-Liouville fractional q-integro-differential equation. Five illustrative examples validate the theoretical results, demonstrating the applicability of the proposed framework. Furthermore, a neural network-based approach is employed to approximate solutions, achieving high accuracy with minimal error, as evidenced by detailed simulations. The integration of theoretical analysis and computational methods highlights the effectiveness of the approach in addressing complex fractional systems.