Home – Home – Jornals – Numerical Analysis and Applications 2025 number 4

In this paper we consider nonlinear fractional differential equations involving the new Caputo-Fabrizio derivative of order γ ∈]1, 2[. We convert the fractional problem to an equivalent nonlinear Volterra integro-differential equation of the second kind, then we investigate the existence and uniqueness of its solution under certain given conditions by using the Schauder fixed point theorem. Finally, we numerically solve the proposed fractional problem by applying the Nyström method, and we provide some suitable examples to support our study.

Conjugate gradient methods represent a powerful class of optimization algorithms known for their efficiency and versatility. In this research, we delve into the optimization of the Generalized Descent Symmetrical Hestenes-Stiefel (GDSHS) algorithm by refining the parameter c, a critical factor in its performance. We employ both analytical and numerical methodologies to estimate the optimal range for c. Through comprehensive numerical experiments, we investigate the impact of different values of c on the algorithm's convergence behavior and computational efficiency. Comparative analyses are conducted between GDSHS variants with varying c values and established conjugate gradient methods such as Fletcher-Reeves (FR) and Polak-Ribière-Polyak (PRP⁺). Our findings underscore the significance of setting c =1, which significantly enhances the GDSHS algorithm's convergence properties and computational performance, positioning it as a competitive choice among state-of-the-art optimization techniques.

A numerical solution by the finite element method of a homogeneous Dirichlet boundary value problem for an elliptic equation is examined (using a Poisson equation as an example) in a two-dimensional convex polygonal domain Ω with a singular right-hand side given by the Dirac delta function. A theorem on the existence and uniqueness of a generalized solution in the fractional Sobolev space H^s(Ω), 1/2 < s < 1, is proved. An approach to discrete analysis of the problem using the finite element method is proposed and investigated. The results of numerical experiments for a model problem, obtained using the FreeFem++ software, are presented. They confirm the error estimate of the difference between the discrete and exact solutions derived in the paper.

An emerging field of study is the application of fractional calculus to iteratively solve nonlinear equations. Recently, several Newton-type techniques have been proposed that make use of the notion of fractional order derivatives. However, the existence of at least first order derivative is essentially required for the convergence of these methods. On the contrary, we propose a new secant-type method which is inherently derivative-free, although its construction is based on the idea of conformable fractional derivative of order α ∈ (0,1]. The primary objective for the development is to analyze how fractional derivatives have an effect of enlarging the convergence domain. In this regard, the proposed scheme is examined for its convergence characteristics and dynamical features for different values of α in the specified range. Furthermore, the efficacy of the method is demonstrated through solving various applied nonlinear problems including the fractional order Burgers' equation.

In this paper, we will give a new Crank-Nicolson mixed covolume method for parabolic optimal control problems. The state and co-state variables are approximated by the lowest order Raviart-Thomas element and the control variable is approximated by piecewise constant function, while Crank-Nicolson scheme is ultilized for temporal discretization. We derive the priori error estimates for the control variable, the state and the co-state variables.

The aim of this paper is to derive efficient numerical algorithms for the numerical solution of nonstiff ordinary differential equations by applying the Richardson extrapolation technique to a class of explicit two-derivative Runge-Kutta methods. Theoretical results illustrate that the application of this technique has considerable impact on the accuracy and stability properties of the underlying numerical methods. The achieved improvements of the proposed algorithms are also confirmed by the results of some numerical experiments.