Journal of Numerical Analysis and Approximation Theory

Book reviews

jnaat — 2024-12-18

Book reviews for:

Brad G. Osgood, Lectures on the Fourier Transform and Its Applications, AMS, 2019, 693 pp., ISBN 978-1-4704-4191-3 (paperback), ISBN 978-1-4704-4976-6 (ebook). Reviewed by Călin Gheorghiu.

Jeffrey Humpherys, Tyler J. Jarvis, Emily J. Evans, Foundations of Applied Mathematics. Volume 1: Mathematical Analysis, SIAM, Philadelphia, 2017, XX + 689 pp., ISBN 978-1-61197-489-8 (paperback), ISBN 978-1-61197-490-4 (ebook). Part of the Other Titles in Applied Mathematics series. Reviewed by Eduard Grigoriciuc.

Jeffrey Humpherys, Tyler J. Jarvis, Foundations of Applied Mathematics. Volume 2: Algorithms, Approximation, Optimization, SIAM, Philadelphia, 2020, XVIII + 788 pp., ISBN 978-1-61197-605-2 (paperback), ISBN 978-1-61197-606-9 (ebook). Reviewed by Andra Malina.

Extended convergence of two-step iterative methods for solving equations with applications

Ioannis K. Argyros — 2024-12-18

The convergence of two-step iterative methods of third and fourth order of convergence are studied under weaker hypotheses than in earlier works using our new idea of the restricted convergence region. This way, we obtain a finer semilocal and local convergence analysis, and under the same or weaker hypotheses. Hence, we extend the applicability of these methods in cases not covered before. Numerical examples are used to compare our results favorably to earlier ones.

A numerical study of an infeasible interior-point algorithm for convex quadratic semi-definite optimization

Yasmina Bendaas — 2024-12-18

The focus of this research is to apply primal-dual interior-point pathfollowing methods, specifically those derived from Newton’s method for solving convex quadratic semidefinite optimization (CQSDO) problems. In this paper, we present a numerical study of an infeasible primal-dual interior-point method for tackling this class of optimization problems. Unlike the feasible interior-point algorithms, the proposed algorithm can be start with any initial positive definite matrix and does not require the strictly feasible initial points. Under certain conditions, the Newton system is well defined and its Jacobian is nonsingular at the solution. For computing an iteration throughout the algorithm, a Newton direction and a step-size are determined. Here, our search direction is based on Alizadeh-Haeberly-Overton (AHO) symmetrization. However, for the step size along this direction an efficient procedure is suggested. Preliminary numerical results demonstrate the efficiency of our algorithm.

Additive operator splitting scheme for a general mean curvature flow and application in edges enhancement

Rafaa Chouder — 2024-12-18

Many models that use non-linear partial differential equations (PDEs) have been extensively applied for different tasks in image processing. Among these PDE-based approaches, the mean curvature flow filtering has impressive results, for which feature directions in the image are important. In this paper, we explore a general model of mean curvature flow, as proposed in [4, 5]. The model
can be re-arranged to a reaction-diffusion form, facilitating the creation of an unconditionally stable semi-implicit scheme for image filtering. The method employs the Additive Operator Split (AOS) technique. Experiments demonstrated that the modified general model of mean curvature flow is highly effective for reducing noise and has a superior job of preserving edges.

Chebfun approximation to structure of positive radial solutions for a class of supercritical semi-linear Dirichlet problems

Călin I. Gheorghiu — 2024-12-18

We use the Chebfun programming package to approximate numerically the structure of the set of positive radial solutions for a class of supercritical semilinear elliptic Dirichlet boundary value problems. This structure (bifurcation diagram) is provided only at the heuristic level in many important works. In this paper, we investigate this structure, as accurately as possible, for the class
of problems mentioned above taking into account the dimension of Euclidean space as well as the physical parameter involved.

A new preconditioned Richardson iterative method

Hassan Jamali — 2024-12-18

In this paper, we propose a new iterative technique for solving an operator equation \(Ax=y\) based on the Richardson iterative method. Then, by using the Chebyshev polynomials, we modify the proposed method to accelerate the convergence rate. Also, we present the results of some numerical experiments that demonstrate the efficiency and effectiveness
of the proposed methods compared to the existing, state-of-the-art methods.

On generation and properties of triple sequence-induced frames in Hilbert spaces

Asif H. Jan — 2024-12-18

In this paper, we present the innovative idea of ”t-frames,” frames produced by triple sequences within Hilbert spaces. The paper explores various properties of these t-frames, delving into topics like frame operators, alternative dual frames, and the stability
inherent in t-frames.

Exponential B-spline collocation method for singularly perturbed time-fractional delay parabolic reaction-diffusion equations

Feyisa E. Merga — 2024-12-18

The singularly perturbed time-fractional delay parabolic reaction-diffusion of initial boundary value problem is provided by the present study. Employing implicit Euler's method along with the Caputo fractional derivative, the time-fractional is discretized. Spatial domain is handled by implementing the exponential B-spline collocation technique. The converge of the method is varified and has an accuracy of \(O(N^{-2}(lnN)^{2})\). Two model examples are examined in order to examine the extent to which the scheme is effective. The findings generated by tables and figures indicate the scheme has dual layers at the end spatial domain and is uniformly convergent.

Convergence of the θ-Euler-Maruyama method for a class of stochastic Volterra integro-diﬀerential equations

Samiha Mouchir — 2024-12-18

This paper addresses the convergence analysis of the θ-Euler-Maruyama method for a class of stochastic Volterra integro-diﬀerential equations (SVIDEs). At ﬁrst, we discuss the existence, uniqueness, boundedness and H¨older continuity of the theoretical solution. Subsequently, the strong convergence order of the θ-Euler-Maruyama approach for SVIDEs is shown. Finally, we provided numerical examples to illustrate the theoretical results.

Falkner hybrid block methods for second-order IVPs: A novel approach to enhancing accuracy and stability properties

Robert I. Okuonghae — 2024-12-18

Second-order initial value problems (IVPs) in ordinary differential equations (ODEs) are ubiquitous in various fields, including physics, engineering, and economics. However, their numerical integration poses significant challenges, particularly when dealing with oscillatory or stiff problems. This article introduces a novel Falkner hybrid block method for the numerical integration of second-order IVPs in ODEs. The newly developed method is of order six with a large interval of absolute stability and is implemented using a fixed step size technique. The numerical experiments show the accuracy of our methods when compared with Falkner linear multistep methods, block methods, and other hybrid codes proposed in the scientific literature. This innovative approach demonstrates improved accuracy and stability in solving second-order IVPs, making it a valuable tool for researchers and practitioners.

Numerical analysis and stability of the Moore-Gibson-Thompson-Fourier model

Ali Smouk — 2024-12-18

This work is concerned the Moore-Gibson-Thompson-Fourier Model. Our contribution will consist in studying the numerical stability of the Moore-Gibson-Thompson-Fourier system. First we introduce a finite element approximation after the discretization, then we prove that the associated discrete energy decreases and later we establish a priori error estimates. Finally, we obtain some numerical simulations.

Higher-order approximations for space-fractional diffusion equation

Anura Gunarathna Wickramarachchi — 2024-12-18

Second-order and third-order finite difference approximations for fractional derivatives are derived from a recently proposed unified explicit form. The Crank-Nicholson schemes based on these approximations are applied to discretize the space-fractional diffusion equation. We theoretically analyse the convergence and stability of the Crank-Nicholson schemes, proving that they are unconditionally stable. These schemes exhibit unconditional stability and convergence for fractional derivatives of order in the range . Numerical examples further confirm the convergence order and unconditional stability of the approximations, demonstrating their effectiveness in practice.