This paper presents new eighth-order numerical methods for solving singular boundary value problems arising from the Lane-Emden equation, which appears in theoretical chemistry, chemical physics, and biophysics. A major difficulty in solving this equation is the singularity at the origin. Consequently, existing numerical methods typically achieve at most seventh-order accuracy. To overcome this accuracy limitation, we propose a novel approach for this singular problem. We first establish the existence and uniqueness of the solution by reformulating the problem as a nonlinear operator equation and constructing a continuous iterative scheme. Based on this framework, we develop three eighth-order schemes using corrected trapezoidal quadrature formulas derived from the Euler-Maclaurin expansion. Rigorous theoretical analysis confirms convergence and establishes the eighth-order accuracy of the proposed methods. Extensive numerical experiments validate the theoretical results and demonstrate the eighth-order convergence of the proposed schemes compared with existing methods.
In this study, we introduce a fast algorithm based on an advanced relaxed iterated Tikhonov regularization technique designed for large ill-posed linear systems. It employs a series of projections onto Krylov subspaces, with the regularization parameter carefully selected based on the discrepancy principle. Insightful commentary is provided on the computational execution of the algorithm, accompanied by numerical tests that illustrate the effectiveness of this projected method.
This paper presents our contributions to the development and analysis of a well-established model for cholera. In infected individuals, the disease occurs either with mild, or severe, symptoms. Applying the next generation matrix, we find the basic reproduction number, R-0, endemic and disease-free equilibrium points. We state the existence conditions for equilibrium points, prove the local and the global asymptotic stability at disease-free equilibrium, using the Routh-Hurwitz criterion and finding a Lyapunov function. If R-0 < 1, there exists a disease-free equilibrium point which is stable. Otherwise, if R-0 > 1, an endemic equilibrium point exists and it is globally asymptotic stable. Sensitivity analysis is used to know the impact of each parameter on R-0, as well as on the control of illness. Validity of the obtained theoretical outcomes is verified through a numerical discussion.
The presence of irregularities and steep gradients in functions is known to degrade the accuracy and stability of numerical approximations. To overcome this, a filtering-based methodology is proposed in this study with minimal added complexity. An ultraspherical polynomial spectral collocation framework is employed, and filtered differentiation matrices are formulated at Gauss-Lobatto points. The centro-antisymmetric structure of these matrices is exploited, by which the computational effort is nearly halved. Two representative problems are examined: a discontinuous function and a variable-coefficient wave equation. The unfiltered approximation of the latter is found to be unstable at larger time levels. For the wave equation, the resulting system of ODEs is integrated using the classical fourth-order Runge-Kutta method, and a stability analysis is presented. The results show that stability and accuracy are markedly improved by spectral filtering. Faster and more effective convergence is achieved through Chebyshev filtering than through Legendre filtering.
Structured models have proven to be a valuable tool to understand population dynamics and how physiological variables affect them. Although numerous existing models incorporate only one variable, most commonly age (but it is not the only one; maturation, size, stage or other physical property), using two of these characteristics improves the modelling of the phenomenon to be analyzed (for example, in Epidemiology, when describing certain distinctive features of an infection). We propose an upwind numerical method for a general population model structured by several internal variables: a well-known technique for solving hyperbolic partial differential equations, as it uses information in the direction of flow to approximate derivatives. We demonstrate that the numerical approximations obtained using this method converge towards the exact solution of the problem, provided that a natural constraint on the discretization parameters is verified. Novel and specific numerical experiments have been designed to further support the theoretical results.
A level-set-free method is proposed for detecting image boundaries using the shape gradient of the Mumford-Shah energy for segmentation. Minimizing the variance in pixel intensities inside and outside a boundary set of points is the primary pursuit. The boundary set as a polygon of points evolves under the guidance of a shape gradient of the Mumford-Shah piece-wise constant energy by iteratively updating through gradient descent method. The proposed method has been tested on various images of both grey and colour, demonstrating its effectiveness in capturing intricate and concave type boundaries of texture images.
This paper presents a numerical approach for solving Fractional Volterra Integro-Differential Equations (FVIDEs) based on the Least Squares Method (LSM). Shifted Chebyshev polynomials are employed as basis functions to construct the approximate solution. The proposed scheme efficiently handles the combined challenges of fractional derivatives, integer-order derivatives, and a Volterra-type integral term. The accuracy and efficiency of the proposed scheme are demonstrated through several numerical examples, where the results are compared against known exact solutions. The findings indicate that the method achieves rapid convergence to the exact solution, confirming its effectiveness and potential as a powerful tool for handling this complex class of equations.
Generative Latent Optimization (GLO) is a non-adversarial framework for training deep convolutional generators using reconstruction losses and explicit per-image latent codes, without a discriminator. We propose Regularized GLO (RGLO), an extension of GLO that incorporates a variational regularization term to guide both training and inverse-problem reconstruction. We evaluate RGLO on image synthesis and on deconvolution/compressed sensing across MNIST, CelebA, BreCaHAD, and AFHQ, showing improved fidelity and stable optimization relative to GLO and a regularized GAN baseline.
In this paper, a novel parallel modulus-based synchronous multisplitting iteration method is designed for solving large sparse implicit complementarity problems. Subsequently, under the assumption that the system matrix is an H-matrix, globally convergent conditions for the proposed method are established, extending the existing convergence results for the corresponding serial method. Finally, numerical experiments demonstrate that the proposed method achieves favorable parallel efficiency.
In recent years, we have witnessed the application of a number of neural networks to the numerical solution of some partial differential equations (PDEs). In the present work, we present Kolmogorov-Arnold networks and their applications for complex-valued nonlinear PDEs (CNPDEs). These networks have demonstrated great potential for data-driven modelling, as an alternative to multilayer perceptrons (MLPs). We propose a new method called Complex Physics-Informed Kolmogorov-Arnold Network (CPIKAN) combining the learning capabilities of KANs with the rigour of physics equations. By directly integrating the equation into the network loss function, we can capture the complexity of nonlinear models and provide accurate numerical solutions. We evaluate the performance of CPIKAN by solving direct and inverse complex-valued problems, and compare the obtained results with analytical results. Our results demonstrate the potential of CPIKAN for solving complex-valued PDEs, opening new perspectives for the modelling and simulation of complex systems.
This work aims to investigate the numerical solution of homogeneous and nonhomogeneous heat-like problems. The numerical algorithm is developed in two phases. In the first phase, the space derivatives along with solution are approximated via Haar wavelet which yield a system of ordinary differential equations (ODEs). Subsequently, the obtained system of ODEs is solved using Runge-Kutta (RK4) method. The proposed scheme is named Haar wavelet Runge Kutta (HWRK) iterative solver. The performance of the scheme is investigated by solving several benchmark models. Various error norms are computed to validate the efficiency of our proposed scheme. Moreover, we also check the computational stability and convergence. From numerical experiments, it is observed that proposed method works well for heat-like problems.
In this paper, some decoupled finite element methods based on two-grid discretization are proposed and investigated for the coupled Stokes-Darcy-Darcy model in which the free flow in conduits is coupled with confined flow in fractured porous media. The first algorithm involves solving a coupled problem on a relatively coarse grid and two independent subproblems on a fine grid that could be solved in parallel. The main idea is to use a coarse grid solution to approximate the interaction terms on the interfaces. Numerical analysis indicates that the error estimate is not optimal, then two modified schemes are provided. The first is to change the parallel implementation in the two-grid decoupled method to fine grid with a serial implementation. The second is to introduce a further coarse grid correction. Both algorithms could arrive at the optimal error convergence order. Finally, some numerical results are provided to support the theoretical results.
In this study, we address the problem of locating and approximating solutions of nonlinear Hammerstein-type integral equations with non-separable kernels by employing a higher-order iterative method. To facilitate this process, in the first place, we approximate the non-separable kernel by a separable one and to continue we modify the fifth order iterative method in Arroyo et al. [Approximation of artificial satellites' preliminary orbits: the efficiency challenge. Mathematical and Computer Modelling. 2011;54(7-8):1802-1807] by approximating a solution of a nonlinear Hammerstein-type integral equation. Next, we establish the convergence analysis of a fifth order iterative method, particularly focused on restricted global convergence. After that, we presented the theoretical domains of existence and uniqueness of the solution, by which we are able to find the best ball of location, separation, and uniqueness. Moreover, we examine the effectiveness of approximation of the inverse operator, especially as the number of terms increases in the separable kernel and outline the procedure used to construct the respective operator. We consider a nonlinear Hammerstein-type integral equation to validate the theoretical results.
In this study, a generalized nonlinear diffusion-advection equation of fractional order is defined. We incorporate a concentration-dependent diffusion coefficient and a nonlinear advection term. The defined nonlinear fractional differential equation is based on the Caputo fractional derivative. Using the Banach fixed-point theorem, we obtain the required conditions for the existence and uniqueness of the obtained results. The homotopy perturbation method (HPM) is used to derive an approximate analytical solution in series form. Some examples are provided to illustrate the efficiency and applicability of the proposed method. The results generalize and extend the previous work on fractional diffusion and Burgers-type equations.
Machine learning is an effective method for addressing complicated fluid dynamics issues and evaluating enormous data sets. This work investigates the thermophysical behaviour of Ag-Au/blood hybrid nanofluids on a spinning disk utilizing a two-phase MHD Carreau model and a non-Fourier heat flux technique. The nonlinear governing equations for momentum, temperature, skin friction, and Nusselt number were numerically solved with the MATLAB bvp4c function. Tables and graphs show how important factors affect biomagnetic flow. Temperature increases with increased nanoparticle concentration and radiation, but velocity decreases with increasing porosity and magnetic field intensity. Under favourable conditions, the interaction between dust and fluid improves heat transmission by increasing skin friction and Nusselt number. Radiation increases the Nusselt number by around 19%, while particle contact decreases it by almost 12%. Machine learning integration enables predictive modelling and optimization of biomedical systems such as artificial organs, lab-on-a-chip devices, and medication delivery.
In this study, an innovative strategy integrating the moving least squares (MLS) method with the Genocchi-collocation technique is advanced to approximate the solution of fractal-fractional integro-differential equations. An essential advantage of the proposed technique is that it does not apply meshing and does not depend on the geometry of the computational domain, hence, this method can be considered as a meshless method. Also, accurate results can be achieved with a small number of points and basis functions, thereby significantly reducing computational complexity. By employing the MLS method, Genocchi polynomials, the Gauss-Legendre quadrature rule, and the collocation method, the problem under investigation is transformed into a system of algebraic equations. The convergence analysis of the obtained approximation is established by proving theorems. Several illustrative examples are provided to demonstrate the applicability and efficacy of the proposed strategy.
