Journal of Numerical Analysis and Approximation Theory
Founded in 1972, Journal of Numerical Analysis and Approximation Theory is an open access, single-blind peer-reviewed journal which publishes original and survey papers in all areas of Numerical Analysis and Approximation Theory. The journal is edited by Tiberiu Popoviciu Institute of Numerical Analysis (Romanian Academy) and published by the Publishing House of the Romanian Academy (Editura Academiei Române). Its former name is Revue d'analyse numérique et de théorie de l'approximation (see the history section). ISSN 2457-6794, ISSN-E 2501-059X
Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
A simplified homotopy perturbation method for solving nonlinear ill-posed operator equations in Hilbert spaces
One popular regularization technique for handling both linear and nonlinear ill-posed problems is homotopy perturbation. In order to solve nonlinear ill-posed problems, we investigate an iteratively-regularized simplified version of the Homotopy perturbation approach in this study. We examine the method's thorough convergence analysis under typical circumstances, focusing on the nonlinearity and the convergence rate under a H\"{o}lder-type source condition. Lastly, numerical simulations are run to confirm the method's effectiveness.
Copyright (c) 2025 Sharad Kumar Dixit https://creativecommons.org/licenses/by/4.0
Efficient global optimization of multivariate functions with unknown H\"{o}lder constants via \(\alpha\)-dense curves
In this paper, we consider the global optimization problem of non-smooth functions over an \(n\)-dimensional box, satisfying the H\"{o}lder condition. We focus our study on the case when the H\"{o}lder constant is not a priori known. We develop and analyze two algorithms. The first one is an extended version of the Piyavskii's method that adaptively construct a linear secants of the parabolic support functions, avoiding the need for a known H\"{o}lder constant. The second algorithm employs a reducing transformation approach which consists of generating, in the feasible box, an $\alpha$-dense curves, effectively converting the multivariate initial problem to a problem of a single variable. We prove the convergence of both algorithms. Their practical efficiency is evaluated through numerical experiments on some test functions and comparison with existing techniques.
Copyright (c) 2025 Nor Sahnoune, Djaouida Guettal, Mohamed Rahal https://creativecommons.org/licenses/by/4.0
A Stancu type extension of the Campiti-Metafune operator
We consider an extension of the Campiti-Metafune operator using a Stancu type technique. We study some properties of the new obtained operator.
Copyright (c) 2025 Teodora Cătinaș https://creativecommons.org/licenses/by/4.0
Modified adaptive quadrature by expansion for Laplace and Helmholtz layer potentials in 2D
An adaptive algorithm based on quadrature by expansion (QBX) is proposed for computing layer potentials at target points near or on a smooth boundary in \(\mathbb{R}^2\). The algorithm can be viewed as major modifications to the two-phase algorithm AQBX, proposed recently by Klinteberg et al. [SIAM Journal on Scientific Computing, 40(3), 2018]. In the modified AQBX (MAQBX), we consider sharper bounds for the involved truncation error. As a result, the involved stopping criteria are met earlier, and the total computational cost is reduced. Moreover, MAQBX is a single-phase algorithm and its structure is far simpler than that of AQBX. It is recommended that QBX (or any version of it) should be applied on a small part of the boundary that is near the target point, and a classical quadrature is applied on the rest of the boundary (this is often referred to as local QBX). We partially show that for Laplace and Helmholtz potentials, parametric symmetricity of the target point with respect to the near part, can improve the convergence of QBX. Based on this observation, we suggest the local MAQBX that is very efficient in practice both for computing layer potentials and for solving boundary integral equations via the Nystr\"{o}m scheme.
Copyright (c) 2025 Hassan Majidian https://creativecommons.org/licenses/by/4.0
A working set algorithm for support vector regression problems
Due to its dual quadratic structure and the existence of complex structural constraints, support vector regression (SVR) is still a computationally intensive operation. This paper presents a new working set technique called WSA-SVR, which extends the primal simplex method, which was just suggested for SVM classification, to the regression scenario. The suggested approach successfully overcomes the inherent difficulties of SVR, such as the coupling of dual variables, the ϵ- insensitive loss function, and the equality constraint on coefficient differences. WSA-SVR maintains primal feasibility throughout the iterations and uses effective rank-one updates, eliminating the need for null space and reduced Hessian projections. These characteristics guarantee numerical stability as well as theoretical scalability. Extensive experiments on widely used benchmark datasets show that WSA–SVR not only converges rapidly, but also provides competitive predictive accuracy. In many cases, it achieves comparable or better results, while requiring substantially fewer iterations than traditional solvers. By addressing the structural constraints of SVR in a principled way and demonstrating strong empirical performance, WSA–SVR emerges as an efficient and adaptable solution for kernel-based regression problems.
Support Vector Regression; Dual Quadratic Programming; Primal Simplex Algorithm; Working Set Methods; Convex Optimization; Iterative Algorithms.
Mathematics Subject Classification: 90C20. 90C25. 65K05. 68T05. 90C90.
Copyright (c) 2025 Saida Azib, Belkacem Brahmi https://creativecommons.org/licenses/by/4.0
Power series for the half width of the Voigt function, rederived
The Voigt function is the convolution of a Gaussian and a Lorentzian. We rederive power series for its half width at half maximum for the limiting cases of near-Gaussian and near-Lorentzian line shapes. We thereby provide independent verification and slight corrections of the expansion coefficients reported by Wang et al (2022). Results are used in our implementation of function voigt_hwhm in the open-source library libcerf.
Copyright (c) 2025 Joachim Wuttke https://creativecommons.org/licenses/by/4.0
Two step Steffensen-type methods with or without memory for nondifferentiable equations
A wide range of applications emerging from diverse disciplines are formulated as an equation or system of equations in abstract spaces such as Euclidean multidimensional, Hilbert or Banach to mention a few, a plethora of which are nondifferentiable. Different methodologies are developed worldwide to handle the solutions of such equations. Such methodologies can also be applied to solve the differentiable equations. In the current study, two step Steffensen-Type methods with and without memory free from derivatives are considered for the nondifferentiable equations and the local as well as semilocal convergence analysis is proved under more generalized conditions. Numerical applications are provided which demonstrate the theoretical results. Better results in terms of radii of convergence balls and number of iterations are obtained using the proposed approach as compared to the existing ones.
Copyright (c) 2025 Ioannis K. Argyros, Gagan Deep https://creativecommons.org/licenses/by/4.0
Reconstruction inversion formulas for the Laguerre Gabor transform
In this paper, we define and study the Gabor transform in the context of the Laguerre hypergroup. We prove some of its basic properties, such as Plancherel theorem, inversion formula and Calder´on’s reproducing inversion formula. Next, using the harmonic analysis related to Laguerre hypergroup, we examine spaces of Sobolev type for which we make explicit kernels reproducing. Exploiting the aforesaid theory, we introduce and study the extremal function associated with the Gabor transform. Finally, by utilizing the reproducing kernels we establish important estimates for this extremal function.
Copyright (c) 2025 Khaled Hleili, Manel Hleili https://creativecommons.org/licenses/by/4.0
On the coordinate system-dependence of the accuracy of symplectic numerical methods
Symplectic numerical methods have become a widely-used choice for the accurate simulation of Hamiltonian systems in various fields, including celestial mechanics, molecular dynamics and robotics. Even though their characteristics are well-understood mathematically, relatively little attention has been paid in general to the practical aspect of how the choice of coordinates affects the accuracy of the numerical results, even though the consequences can be computationally significant. The present article aims to fill this gap by giving a systematic overview of how coordinate transformations can influence the results of simulations performed using symplectic methods. We give a derivation for the non-invariance of the modified Hamiltonian of symplectic methods under coordinate transformations, as well as a sufficient condition for the non-preservation of a first integral corresponding to a cyclic coordinate for the symplectic Euler method. We also consider the possibility of finding order-compensating coordinate transformations that improve the order of accuracy of a numerical method. Various numerical examples are presented throughout.
Copyright (c) 2025 Donát M. Takács, Tamás Fülöp https://creativecommons.org/licenses/by/4.0
Laurent operator-based representation of discrete solutions in the Newmark scheme with non-homogeneous terms
This paper investigates representation results for second-order evolution equations arising in structural dynamics, discretized using the Newmark time integration scheme. More precisely, the discrete solution is expressed in terms of bi-infinite Toeplitz or Laurent operators. A spectral analysis of the associated discrete operators is discussed, and a convergence analysis is performed under relaxed regularity assumptions on the source term. Furthermore, we examine the errors introduced by some truncation strategies, including one that is commonly used in engineering practice.
Copyright (c) 2025 Eliass Zafati https://creativecommons.org/licenses/by/4.0
Applications of the theory of generalized Fourier transforms to Tikhonov problems
We define and study multiplier operators associated with the Sturm-Liouville operator involving a nonnegative function satisfying certain conditions. We introduce and study the extremal function, and we deduce approximate inversion formulas for the multipliers operators in a Paley-Wiener type space.
SL-multiplier operators; Paley-Wiener type space; approximate inversion formulas
Copyright (c) 2025 Fethi Soltani, Akram Nemri https://creativecommons.org/licenses/by/4.0
Some simple full-range inverse-normal approximations
Two approximations are given for numerically inverting the full range of the standard normal cumulative distribution function. The first approximation has two fitted constants and only modest accuracy but is very simple and well suited for hand calculators. The second approximation has four fitted constants and is more accurate. Alternate versions of the approximations are also given, with the constants chosen to minimize the maximum relative error in the implied approximate q rather than the maximum absolute error in the approximate z.
Copyright (c) 2025 Raymond Koopman https://creativecommons.org/licenses/by/4.0
Best approximation of Hartley-Bessel multiplier operators on weighted Sobolev spaces
The main goal of this paper is to introduce the Hartley-Bessel \(L^2_\alpha\)-multiplier operators and to give for them some new results as Plancherel’s, Calderon’s reproducing formulas and Heisenberg’s, Donoho-Stark’s uncertainty principles. Next, using the theory of reproducing kernels we give best approximation and an integral representation of the extremal functions related to these operators on weighted Sobolev spaces.
Copyright (c) 2025 Ahmed Chana, Abdellatif Akhlidj https://creativecommons.org/licenses/by/4.0
Upper and lower solution method for control of second-order Kolmogorov type systems
In this paper, an upper and lower solution method for the control of second-order Kolmogorov systems is introduced. Two iterative algorithms, one exact and one approximate, are proposed and their convergence is studied. The technique is based on Perov's fixed point theorem, matrices convergent to zero, and the use of Bielecki's norm.
Copyright (c) 2025 Alexandru Hofman https://creativecommons.org/licenses/by/4.0
Stability of piecewise flat Ricci flow in three dimensions
For a recently developed piecewise flat approximation of the Ricci flow, numerical instabilities are seen to arise for a particularly useful class of mesh-types. Here, a geometrically motivated adaptation to these meshes is introduced, and a linear stability analysis and numerical simulations used to show that the instability is then suppressed. These adapted meshes have also been successfully used in a recently published paper to show the convergence of the piecewise flat Ricci flow to known smooth Ricci flow solutions for a variety of manifolds.
Numerical Ricci flow, piecewise flat, linear stability
Copyright (c) 2025 Rory Conboye https://creativecommons.org/licenses/by/4.0
Andrei Stan: David L. Chopp, Introduction to High Performance Scientific Computing, SIAM, Philadelphia, 2019, ISSN: 9781611975642, ISSN (print): 9781611975635. Eduard Stefan Grigoriciuc: Samuel D. Conte, Carl De Boor, Elementary numerical analysis: an algorithmic approach, SIAM, Philadelphia, 2018, XXIV + 454 pp., ISBN 978-1-61197-519-2 (paperback), eISBN 978-1-61197-520-8 (ebook). Part of the Classics in Applied Mathematics series.
Copyright (c) 2025 JNAAT https://creativecommons.org/licenses/by/4.0
Products of parametric extensions: refined estimates
We present point wise estimates on approximation by bounded linear operators of real-valued continuous functions defined on the cartesian product of d compact intervals. The main purpose is to provide a unified theory to deal with pointwise estimates on approximation processes of the above type which are generated by the tensor product method. This will constitute an extension and a refinement of earlier work of Haussmann and Pottinger. As an example a new estimate for approximation by multivariate positive linear operators is given.
Copyright (c) 2025 Heiner Gonska https://creativecommons.org/licenses/by/4.0
A numerical approach for singularly perturbed parabolic reaction-diffusion problem on a modified graded mesh
This paper addresses the numerical approximations of solutions for one dimensional parabolic singularly perturbed problems of reaction-diffusion type. The solution of this class of problems exhibit boundary layers on both sides of the domain. The proposed numerical method involves combining the backward Euler method on a uniform mesh for temporal discretization and an upwind finite difference scheme for spatial discretization on a modified graded mesh. The numerical solutions presented here are calculated using a modified graded mesh and the error bounds are rigorously assessed within the discrete maximum norm. The primary focus of this study is to underscore the crucial importance of utilizing a modified graded mesh to enhance the order of convergence in numerical solutions. The method demonstrates uniform convergence, with first-order accuracy in time and nearly second-order accuracy in space concerning the perturbation parameter. Theoretical findings are supported by numerical results presented in the paper.
Copyright (c) 2025 Kishun Kumar Sah, Subramaniam Gowrisankar https://creativecommons.org/licenses/by/4.0
A comparative study on balanced truncation and projection onto the dominant eigenspace of the Gramian using Laguerre functions
This paper presents a comparative analysis of two model reduction techniques: balanced truncation and projection onto the dominant eigenspace of the Gramian. Both methods are evaluated through the lens of Laguerre functions which is a reduction technique based on Laguerre function expansion, that approximates the low-rank factors of the Controllability and Observability Gramians to produce approximately balanced systems without solving Lyapunov equations. We demonstrate the effectiveness of each approach in reducing the complexity of linear time-invariant (LTI) systems while preserving system dynamics. Numerical experiments highlight the strengths and weaknesses of both methods, showcasing their applicability to large-scale systems. By thoroughly analyzing both methods in conjunction with Laguerre functions, we contribute to the ongoing discourse in the field and lay the groundwork for future advancements in model reduction techniques.
Copyright (c) 2025 Shafiqul Islam , Saiduzzaman Sayed, Monir Uddin, Osman Gani https://creativecommons.org/licenses/by/4.0
A stable finite difference scheme for fractional viscoelastic wave propagation in space-time domain
This work focuses on the numerical approximation of the one dimensional wave propagation equation in a viscoelastic medium modelled by the fractional Zener model. First, we propose an explicit centred finite difference scheme. Then, we prove a sufficient stability condition for this scheme using an energy technique. Furthermore, we demonstrate that this condition is numerically necessary. Finally, we present various numerical simulations to validate the proposed scheme.
Copyright (c) 2025 Mohamed Ait ichou, Abdelaaziz Ezziani https://creativecommons.org/licenses/by/4.0
Extended convergence of two-step iterative methods for solving equations with applications
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.
Banach space, restricted convergence region, convergence of iterative method.
Copyright (c) 2024 Santhosh George, Ioannis K Argyros https://creativecommons.org/licenses/by/4.0
On generation and properties of triple sequence-induced frames in Hilbert spaces
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.
Copyright (c) 2024 Asif Hussain Jan, Tanweer Jalal, Younis Ahmad Bhat https://creativecommons.org/licenses/by/4.0
A new preconditioned Richardson iterative method
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.
Copyright (c) 2024 Hassan Jamali, Reza Pourkani, Mohammad Abdi Arablou https://creativecommons.org/licenses/by/4.0
Convergence of the θ-Euler-Maruyama method for a class of stochastic Volterra integro-diﬀerential equations
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.
Stochastic Volterra integro-diﬀerential equations, Θ-Euler-Maruyama method, strong convergence, H¨older continuity.
Numerical methods for solving stochastic differential equations and their application
Copyright (c) 2024 Samiha Mouchir https://creativecommons.org/licenses/by/4.0
A numerical study of an infeasible interior-point algorithm for convex quadratic semi-definite optimization
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.
Convex quadratic semidefinite optimization, Interior-point methods, Primal-dual algorithm, Newton method.
Copyright (c) 2024 Yasmina Bendaas, Mohamed Achache https://creativecommons.org/licenses/by/4.0
Falkner hybrid block methods for second-order IVPs: A novel approach to enhancing accuracy and stability properties
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.
Copyright (c) 2024 Robert Okuonghae, Kaidi https://creativecommons.org/licenses/by/4.0
Exponential B-spline collocation method for singularly perturbed time-fractional delay parabolic reaction-diffusion equations
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.
Delay Parabolic reaction-diffusion, Time-fractional, Exponential B-spline Collocation Method
Copyright (c) 2024 Feyisa Edosa Merga, G.F. Duressa https://creativecommons.org/licenses/by/4.0
Numerical analysis and stability of the Moore-Gibson-Thompson-Fourier model
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.
Copyright (c) 2024 ALI SMOUK https://creativecommons.org/licenses/by/4.0
Higher-order approximations for space-fractional diffusion equation
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.
Fractional derivatives, Grünwald approximation, Unified explicit form, Space Fractional diffusion equation
Copyright (c) 2024 W Anura Gunarathna, Haniffa Mohamed Nasir https://creativecommons.org/licenses/by/4.0
Chebfun approximation to structure of positive radial solutions for a class of supercritical semi-linear Dirichlet problems
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.
Copyright (c) 2024 Calin-Ioan Gheorghiu https://creativecommons.org/licenses/by/4.0