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In this paper, we propose an explicit two-grid spectral deferred correction method for solving the nonlinear fractional pantograph differential equations. We design a partition including the global and local grids, which reduces the interaction between the subintervals caused by the delay term. We also analyze the numerical errors of the suggested approach for the prediction step and the correction step, respectively. Numerical experiments confirm the theoretical expectations.

We investigate the application of a posteriori error estimate to a fractional optimal control problem with pointwise control constraints. Specifically, we address a problem in which the state equation is formulated as an integral form of the fractional Laplacian equation, with the control variable embedded within the state equation as a coefficient. We propose two distinct finite element discretization approaches for an optimal control problem. The first approach employs a fully discrete scheme where the control variable is discretized using piecewise constant functions. The second approach, a semi-discrete scheme, does not discretize the control variable. Using the first-order optimality condition, the second-order optimality condition, and a solution regularity analysis for the optimal control problem, we devise a posteriori error estimates. Based on the established error estimates framework, an adaptive refinement strategy is developed to help achieve the optimal convergence rate. Numerical experiments are given to illustrate the theoretical findings.

The periodic nonuniform sampling series, involving periodic samples of both the function and its first r derivatives, was initially introduced by Nathan (Inform Control 22: 172–182, 1973). Since then, various authors have extended this sampling series in different contexts over the past decades. However, the application of the periodic nonuniform derivative sampling series in approximation theory has been limited due to its slow convergence. In this article, we introduce a modification to the periodic nonuniform sampling involving derivatives by incorporating a Gaussian multiplier. This modification results in a significantly improved convergence rate, which now follows an exponential order. This is a significant improvement compared to the original series, which had a convergence rate of (O(N^{-1/p})) where (p>1). The introduced modification relies on a complex-analytic technique and is applicable to a wide range of functions. Specifically, it is suitable for the class of entire functions of exponential type that satisfy a decay condition, as well as for the class of analytic functions defined on a horizontal strip. To validate the presented theoretical analysis, the paper includes rigorous numerical experiments.

Algebraic multigrid (AMG) is one of the most efficient iterative methods for solving large structured systems of equations. However, how to build/check restriction and prolongation operators in practical AMG methods for nonsymmetric structured systems is still an interesting open question in its full generality. The present paper deals with the block-structured dense and Toeplitz-like-plus-cross systems, including nonsymmetric indefinite and symmetric positive definite (SPD) ones, arising from nonlocal diffusion problems. The simple (traditional) restriction operator and prolongation operator are employed in order to handle such block-structured dense and Toeplitz-like-plus-cross systems, which are convenient and efficient when employing a fast AMG. We provide a detailed proof of the two-grid convergence of the method for the considered SPD structures. The numerical experiments are performed in order to verify the convergence with a computational cost of only (mathscr {O}(N text{ log } N)) arithmetic operations, by exploiting the fast Fourier transform, where N is the number of the grid points. To the best of our knowledge, this is the first contribution regarding Toeplitz-like-plus-cross linear systems solved by means of a fast AMG.

A proof that Wronskians of non-zero B-splines are positive is given, using only recursive formulas for B-splines and their derivatives. This could be used to generalize the de Boor–DeVore geometric proof of the Schoenberg–Whitney conditions and total positivity of B-splines to Hermite interpolation. For Wronskians of maximal order with respect to a given degree, positivity follows from a simple formula.

In the present article, we construct a logarithm transformation based Milstein-type method for the stochastic susceptible-infected-susceptible (SIS) epidemic model evolving in the domain (0, N). The new scheme is explicit and unconditionally boundary and dynamics preserving, when used to solve the stochastic SIS epidemic model. Also, it is proved that the scheme has a strong convergence rate of order one. Different from existing time discretization schemes, the newly proposed scheme for any time step size (h>0), not only produces numerical approximations living in the entire domain (0, N), but also unconditionally reproduces the extinction and persistence behavior of the original model, with no additional requirements imposed on the model parameters. Numerical experiments are presented to verify our theoretical findings.

This paper provides a qualitative and quantitative study of a second-order Singularly Perturbed Reaction–Diffusion type System of Integro-differential equations with discontinuous source term. To obtain the numerical solution of the problem, an exponentially-fitted method that can be applied to a Shishkin mesh. This method shows that uniform convergence with respect to the perturbation parameter and necessary examples are given.

This note discusses the computation of the distance function with respect to Finsler metrics. To this end, we show how the Finsler variants of the Eikonal equation can be solved by a primal-dual algorithm exploiting the variational structure. We also discuss the acceleration of the algorithm by preconditioning techniques, and illustrate the flexibility of the proposed method through a series of numerical examples.

This paper presents ConvStabNet, a convolutional neural network designed to predict optimal stabilization parameters for each cell in the Streamline Upwind Petrov Galerkin (SUPG) stabilization scheme. ConvStabNet employs a shared parameter approach, allowing the network to understand the relationships between cell characteristics and their corresponding stabilization parameters while efficiently handling the parameter space. Comparative analyses with state-of-the-art neural network solvers based on variational formulations highlight the superior performance of ConvStabNet. To improve the accuracy of SUPG in solving partial differential equations (PDEs) with interior and boundary layers, ConvStabNet incorporates a loss function that combines a strong residual component with a cross-wind derivative term. The findings confirm ConvStabNet as a promising method for accurately predicting stabilization parameters in SUPG, thereby marking it as an advancement over neural network-based PDE solvers.

We present a simple yet efficient two-stage extended Kaczmarz-type algorithm for solving large least squares problem. During each stage, the current iterate is projected onto a surrogate hyperplane instead of a single one, yielding remarkable reduction in the number of iteration steps and computational time. We prove that the proposed algorithm converges to the unique least-norm least-squares solution with a convergence factor asymptotically smaller than that for some existing randomized extended Kaczmarz-type algorithms. Numerical examples show that the new algorithm outperforms several counterparts for various test problems.