This review paper focuses on the numerical solution of the time-fractional diffusion equation using various discretization techniques. For the time-fractional derivative, we consider methods such as L-type approximations and Grünwald-Letnikov-based formulas, while for the spatial diffusion term, we utilize the compact finite difference method, finite element method, spectral element method, meshless method, Chebyshev spectral method, and finite block method. In addition, stability and convergence theorems are presented, accompanied by numerical examples that confirm the theoretical results.

In this article, we construct a new basis by twice integrating linear B-Spline functions. This new basis can expand functions similarly to linear B-Spline functions, but it also possesses the capability to exactly approximate third-order polynomials. We investigate the properties of these functions and apply them to solve fractional optimal control problems. By using this basis in the collocation method, the original problem is reduced to a nonlinear programming problem, enabling us to obtain approximate solutions through appropriate methods. The numerical results demonstrate the effectiveness of the new basis.

In the realm of engineering, one frequently encounters coupled linear matrix equations. In this work, based on the study in [1] a new gradient-based parameter free iterative scheme is presented to find solutions for a coupled matrix equations. The proposed iterative method has been proven to attain solutions for the general coupled matrix equations, regardless of the initial matrices. In the following some applications for generalized Sylvester matrix equation such as solving transient heat conduction problem by moving least squares (MLS) scheme as trial approximation in time and space and a new novel transform domain steganography technique-hiding an image message in a high order vector are studied. Lastly, computational experiments are presented to illustrate that the introduced iterative algorithm with optimal parameter selection outperforms the gradient-based iterative scheme in terms of speed, number of iterates, and elapsed times.

This work presents a series of computational experiments to evaluate the performance and convergence of various iterative methods for solving a minimization problem min _x∈ℝ^n‖x‖ _2 subject to Ax=b, where A∈ℝ^m × n (m ≤ n) ( A is full row rank) and b∈ℝ^m are known and x∈ℝ^n must be determined. The proposed methods are compared with Craig’s method, and the direct pseudo-inverse approach using MATLAB. The efficiency of these methods is assessed based on the convergence behavior and CPU time required for each iteration. Numerical results for different problem sizes indicate that one of the new methods achieves the fastest convergence and lowest computational cost. The effectiveness of these methods is also demonstrated through an image face decryption example. Finally a minimum-norm steganography method is proposed for covert message transmission using digital images. By solving a constrained optimization problem, a secret message is securely embedded and transmitted through the image, and experimental results confirm accurate decoding and high imperceptibility.

This paper presents a high-order, stable numerical method for simulating a system of fractional partial differential equations, specifically focusing on the Gray-Scott reaction-diffusion model. Known for its autocatalytic nature and smooth, bounded solutions, the model exhibits a rich variety of steady-state patterns arising from diffusion-driven instabilities. To discretize the system, the Crank-Nicolson (C-N) method is employed for time integration, while a spatial discretization scheme based on the Backward Differentiation Formula (BDF) is implemented. Additionally, the potential of the Runge-Kutta method is explored to improve computational efficiency and reduce processing time. A rigorous analysis of the finite difference scheme is conducted using the discrete energy method, establishing its boundedness, existence, uniqueness, and stability. Convergence analysis of the semi-discrete scheme confirms fourth-order accuracy in space and second-order accuracy in time, supported by comprehensive numerical experiments.

This work develops a weak Galerkin (WG) finite element method for the chemo-repulsion-Navier-Stokes (CR-NS) system, a coupled nonlinear model describing the interaction between chemorepulsive cell transport and incompressible fluid dynamics. The proposed SWG formulation incorporates the chemotactic flux and Navier-Stokes nonlinearities through a stabilizer and numerical flux designed to balance accuracy, stability, and computational efficiency. For temporal discretization, we construct a scheme that treats nonlinear couplings consistently while ensures energy stability under a CFL-type condition that depends only on bounded solution quantities. In space, the method captures sharp gradients and boundary-layer structures induced by chemorepulsion without requiring mesh refinement. A series of numerical experiments demonstrates that the method delivers high accuracy and robustness across a broad range of parameters. The results suggest that SWG provides a reliable and practical tool for simulating strongly coupled chemotaxis-fluid systems.

The Physics-Informed Deep Operator Network (PI-DeepONet) is a class of neural networks that incorporates physics-based constraints through regularization mechanisms. However, its application to complex nonlinear parametric PDEs is often constrained by computational inefficiency and limited accuracy. In this paper, we propose a hybrid model that integrates the standard PI-DeepONet with a lightweight, single-layer artificial neural network (ANN) that employs Chebyshev (ChNN) or Legendre (LeNN) polynomial bases. The key idea behind this combination is to leverage the intrinsic benefits of these orthogonal polynomial networks–namely, their ability to handle sharp gradients via input expansion, their convergence and stability for capturing nonlinear behaviors, and their requirement for fewer trainable parameters–to complement and enhance the physics-informed learning of the DeepONet. Numerical results demonstrate that our proposed hybrid method achieves significant accuracy improvements compared to the standard PI-DeepONet, while maintaining computational efficiency without introducing significant extra cost. In fact, it reduces the number of training iterations required to achieve comparable accuracy, thereby improving overall computational efficiency. Extensive validations on a nonlinear parametric reaction-diffusion equation and two-dimensional (2D) Eikonal equations with circular and airfoil geometries confirm the superior performance of the proposed scheme in terms of accuracy, stability, generalization ability and training convergence.

The primary motivation of the current article is to study an approximate algorithm to solve a type of Volterra integro-differential equations involving nonlinear terms. These integro-differential equations have been utilized to simulate several realistic modeling arising in biological sciences, for example, the growth model of the toxins’ cumulative effects on a population residing in a closed system. Thin plate splines (TPSs), known as a subclass of radial basis functions (RBFs) independent of the shape parameter, create an efficient scheme to interpolate a function. Due to this desirable property, we use the discrete collocation approach with the TPS basis for estimating the solution of the mentioned integro-differential equations. Furthermore, the Gauss-Legendre integration formula is utilized to calculate the integrals emerging in the scheme. Remarkably, the algorithm of the offered approach can be comfortably performed on a PC with typical specifications derived from the simplicity of using TPSs. The researchers also investigate the convergence of the presented scheme. Some test examples including Volterra integro-differential equations are considered to make sure the veracity of the proposed scheme and the validity of the theoretical error analysis.

A simplified weak Galerkin (SWG) method is proposed to solve the solute transport problem. The SWG method makes the possibility of using the arbitrarily shaped polygonal meshes which leads to significant advantages such as fewer elemental calculations compared with the standard finite element method (FEM). This method has important advantages over polygonal finite element method (PFEM), virtual element method (VEM) and usual WGM including no requirement for explicit form of the basis functions, enabling better treatment of problems with discontinuous solutions because of the use of totally discontinuous basis functions and simplicity in constructing discrete schemes. Time discretization is performed using the Crank-Nicolson finite difference method. Convergence error analysis is established in the H1 and L2 norms, showing optimal spatial convergence rates of order O(h) and O(h2), respectively. Furthermore, we combine the SWG method with the proper orthogonal decomposition (POD) technique (POD-SWG) to reduce the dimension of full scheme and the computational complexity of the method. Additionally, a convergence analysis of the POD-SWG plan is established. So, POD-SWG method proposed a computational efficient approach and also had sufficiently high accuracy. Numerical results are given that verify the theoretical analysis and show the computational efficiency and acceptable accuracy of the proposed method.

The Lyapunov matrix equations occur in many branches of control theory, such as stability analysis and optimal control. In this work, we introduce a novel iterative approach to address the generalized Lyapunov matrix equation within the framework of complex matrices. At each iteration, the procedure involves solving two conventional Lyapunov equations with real-valued coefficient matrices. The scheme incorporates two positive parameters, for which we establish sufficient conditions to guarantee the convergence of the method under certain assumptions. Then we solve the Lyapunov equation arising by applying a finite difference procedure to Helmholtz equation by proposed method.

This manuscript presents a reduced-order meshless approach for efficiently simulating shallow water dynamics, leveraging the Integrated Radial Basis Function based on Generalized Moving Least Squares (IRBF-GMLS) and Proper Orthogonal Decomposition (POD) techniques. The proposed method addresses the computational challenges associated with large-scale shallow water problems by employing a semi-implicit scheme for temporal discretization and optimizing shape parameters to minimize condition numbers in the resulting matrices. By integrating the Discrete Empirical Interpolation Method (DEIM) with POD, we significantly reduce the computational complexity of the reduced-order model while maintaining accuracy. Numerical results demonstrate the effectiveness of the IRBF-GMLS-POD/DEIM approach, showcasing its ability to efficiently handle nonlinear systems of algebraic equations and providing a robust framework for environmental modeling and fluid dynamics applications.

This study introduces a method based on a gradient neural network (GNN), a type of artificial neural network (ANN), for solving time-varying matrix inversion problems. Detailed step-by-step algorithms are provided along with a convergence theorem to guide the selection of optimal learning parameters. The momentum acceleration method has been integrated into the iterative process to improve the convergence rate. Additionally, the optimal value of the momentum parameter has been calculated to further enhance the algorithm’s performance. The proposed methods are demonstrated to be efficient and reliable through a series of numerical examples, which also include performance comparisons with existing algorithms. Furthermore, the practical utility of these methods is explored, specifically in image encryption and decryption based on matrix inversion and in solving the one-dimensional second-order wave equation.

This paper presents a novel and efficient numerical methodology for solving time-fractional partial integro-differential equations with time delay, employing a combination of finite difference, spectral, and finite block methods on both square and non-rectangular regions. The Caputo derivative is discretized employing the L2-1_σ scheme, while a L–type approximation is applied to the Riemann–Liouville fractional integral at the n+σ time level. For square domains, a fully–discrete scheme is constructed using Chebyshev spectral methods via the differentiation matrix. For irregular geometries, the finite block method is developed to handle domain complexity effectively. Theoretical analyses of stability and convergence are investigated, and a series of numerical experiments demonstrate the accuracy and computational efficiency of the proposed method.

This study refines the SIDARTHE model for Italy's COVID-19 outbreak using a hybrid, data-driven framework. A two-stage approach compares Maximum Likelihood Estimation (MLE) with Physics-Informed Neural Networks (PINNs) for parameter estimation, then applies Symbolic Regression (via gplearn and PySR) to optimize the governing equations. Results show PINNs surpass MLE in accuracy, and PySR outperforms gplearn in deriving robust expressions. The final integrated model-combining PINN estimation with Symbolic Regression-significantly reduces predictive uncertainty and aligns closely with observed data, providing a resilient tool for public health planning.

In this work, we propose and analyze a mixed virtual element method within a Banach space framework to numerically investigate the unsteady motion of non-Newtonian pseudoplastic Stokes flows. Motivated by the growing interest in non-Newtonian fluid problems where the stress response plays a pivotal role, our formulation introduces additional unknowns, including the rate of strain and the symmetric stress tensor. This leads to a mixed variational formulation that incorporates the velocity, strain rate, and stress within a Banach space setting. We establish the well-posedness of the weak solution and derive stability estimates using classical results from the theory of nonlinear monotone operators. The discretization in both space and time is performed using an H(div)-conforming virtual element method and the implicit Euler scheme, respectively. Specifically, the pseudostress is approximated using a virtual element subspace of H(div; Omega), while piecewise polynomial subspaces of degree j are employed for approximating the velocity and the rate of strain tensor. The nonlinear term is treated implicitly in the time discretization, and the resulting scheme is shown to be well-posed and unconditionally stable. A rigorous convergence analysis is provided for all unknowns in their natural norms, establishing optimal convergence rates with respect to both the spatial mesh size and the time step. Finally, a series of numerical experiments is presented to validate the accuracy and effectiveness of the proposed method.

This work investigates a nonlinear conservative finite difference scheme of second-order accuracy for the Davey-Stewartson (DS) equations, a model of nonlinear dispersive wave equations. The proposed numerical scheme is shown to satisfy key properties including conservativeness, uniqueness of solutions, and stability. The existence of approximate solutions and convergence of the difference scheme are established, with the convergence rate proven to be under the uniform norm without restrictions on mesh sizes. As an application, fundamental line rogue waves of DS equations are studied numerically in detail, depicting emergence from and decay back to a constant background. Finally, several numerical experiments are reviewed to validate the theoretical results and compare performance against existing methods.

Dissipative solitons are localized wave structures that emerge in nonlinear systems characterized by the presence of both energy gain and loss. Such structures arise in a variety of important physical models, particularly in nonlinear optics. One of the fundamental equations used to describe these phenomena is the cubic-quintic complex Ginzburg-Landau equation, which models the propagation of dissipative nonlinear waves in optical media. In this work, we develop a numerical method for computing solutions of this equation based on the local discontinuous Galerkin framework combined with a time-splitting strategy. The stability of the resulting semi-discrete scheme is analyzed, and optimal error estimates are established. A series of numerical experiments is performed to verify the theoretical convergence rates. Beyond the numerical analysis, we also investigate the dynamical behavior of stable and dissipative solitons. In two dimensions, we examine the evolution of stable vortex solitons with topological charges one, two, three, and four, together with a stable fundamental soliton. We further explore how variations in the gain and loss parameters affect the dynamics of both the fundamental and vortex solitons. Finally, the interaction dynamics of dissipative structures in three dimensions are studied by simulating coaxial collisions between two three-dimensional dissipative solitons.

The Fokker-Planck (FP) equation is a fundamental component in the mathematical modeling of stochastic processes which governs the time development of probability density functions in a variety of fields, including psychology, neuroscience, statistical physics, quantitative finance, and systems biology. Despite its significance, finding exact solutions to nonlinear FP equations with time and space dependent coefficients is still very difficult, especially in high dimensions when using classical discretization methods becomes computationally expensive. In this work, we introduce an enhanced Physics-Informed Neural Network (PINN) framework with added Dynamic Sample Adaptation (DSA) to address these difficulties. The proposed method by adaptively moving collocation points during training, concentrating computational effort in point where the solution has sharp gradients, high variability, or localized peaks. Unlike standard PINNs, where collocation points remain fixed throughout training, the proposed DSA strategy dynamically relocates collocation points during the optimization process based on the PDE residual. This adaptive sampling technique works by updating the sampling points using either the Adam optimizer or gradient descent after a few training epochs to ensure both global exploration and fine-scale accuracy. By integrating physics-informed loss functions with dynamically refined training samples, the framework achieves improved stability and convergence when compared to standard PINNs. An example set of seven problems is used to thoroughly evaluate the effectiveness of the proposed approach. This set includes two and three dimensional challenging nonlinear FP equations with stiff initial conditions and time varying variables. A comparison with standard PINN and the traditional numerical approach is carried out. The DSA-PINN method exhibits better accuracy, quicker convergence, and increased robustness in capturing intricate solution structures in most test cases.

The finite element method is widely used in engineering computations due to its flexibility in handling complex geometries. However, it encounters challenges when solving wave equations at high frequencies. As dispersion errors increase, additional nodes are required to maintain accuracy, leading to significantly higher computational costs. To address this issue, high-order methods are essential for achieving greater accuracy while maintaining efficiency. In this study, we investigate the performance of the matrix-free local discontinuous Galerkin spectral element method for solving equations in computational acoustics. The matrix-free formulation reduces storage costs, making it particularly suitable for large-scale simulations. In the numerical process for linear wave propagation, precise evaluation of volume and surface integrals is employed to define semi-discrete operators and their role in controlling dispersion and dissipation errors is analyzed. Additionally, time discretization is carried out using a low-storage explicit fourth-order Runge-Kutta method. Finally, several numerical examples are presented to assess the performance of the proposed method. The results are compared with those obtained using the finite element method and the spectral element method. The findings demonstrate that the matrix-free local discontinuous Galerkin spectral element method offers superior accuracy, particularly in problems involving non-smooth solutions.

In this article, we introduce a preconditioned minimal residual (PMR) algorithm designed to address a wide range of matrix equations and linear systems. We illustrate the efficacy of this algorithm through several numerical examples, including the solution of matrix equations. Notably, we tackle various significant problems such as the minimization of Frobenius norms, least squares optimization, and the computation of the Moore-Penrose pseudo-inverse. Convergence analysis shows that it converges without any constraints and for any initial guess, although this algorithm is more efficient when the matrices are sparse. To validate the effectiveness of our proposed iterative algorithm, we offer various numerical examples by large matrices. As an application of the matrix equation, we explore a method for encrypting and decrypting color images.