The target of this work is to present a multiplication-based iterative method for two Hermitian positive definite matrices to find the geometric mean. The method is constructed via the application of the matrix sign function. It is theoretically inve...
In this article, we suggest the local analysis of a uni-parametric third and fourth order class of iterative algorithms for addressing nonlinear equations in Banach spaces. The proposed local convergence is established using an
In this work, using the weight function technique, we introduce a new family of fourth-order iterative methods optimal in the sense of Kung and Traub for scalar equations, generalizing Jarratt’s method. Through Taylor series expansions, we conf...
A number of optimal order multiple root techniques that require derivative evaluations in the formulas have been proposed in literature. However, derivative-free optimal techniques for multiple roots are seldom obtained. By considering this factor as...
Nonlinear equations are frequently encountered in many areas of applied science and engineering, and they require efficient numerical methods to solve. To ensure quick and precise root approximation, this study presents derivative-free iterative meth...
A new parametric family of iterative schemes for solving nonlinear systems is presented. Fourth-order convergence is demonstrated and its stability is analyzed as a function of the parameter values. This study allows us to detect the most stable elem...
High-order iterative techniques without derivatives for multiple roots have wide-ranging applications in the following: optimization tasks, where the objective function lacks explicit derivatives or is computationally expensive to evaluate; engineeri...
In this article, we develop a compact finite difference scheme for a variable-order-time fractional-sub-diffusion equation of a fourth-order derivative term via order reduction. The proposed scheme exhibits fourth-order convergence in space and secon...
The present paper includes the local and semilocal convergence analysis of a fourth-order method based on the quadrature formula in Banach spaces. The weaker hypotheses used are based only on the first Fréchet derivative. The new approach provides th...
In this paper, a three-stage fourth-order numerical scheme is proposed. The first and second stages of the proposed scheme are explicit, whereas the third stage is implicit. A fourth-order compact scheme is considered to discretize space-involved ter...
Singular singularly-perturbed problems (SSPPs) are a powerful mathematical tool for modelling a variety of real phenomena, such as nuclear reactions, heat explosions, mechanics, and hydrodynamics. In this paper, the numerical solutions to fourth-orde...
In this work, the homotopy analysis method (HAM) is applied to solve the fourth-order parabolic partial differential equations. This equation practically arises in the transverse vibration problems. The proposed iterative scheme finds the solution wi...
This paper presents a finite difference approach for solving the time-fractional Burgers’ equation, which is a model for nonlinear flow with memory effects. The method leverages the
In this manuscript, by using the weight-function technique, a new class of iterative methods for solving nonlinear problems is constructed, which includes many known schemes that can be obtained by choosing different weight functions. This weight fun...
In this paper, an implicit difference scheme is proposed and analyzed for a class of nonlinear fourth-order equations with the multi-term Riemann–Liouvile (R–L) fractional integral kernels. For the nonlinear convection term, we handle imp...
We provide a ball comparison between some 4-order methods to solve nonlinear equations involving Banach space valued operators. We only use hypotheses on the first derivative, as compared to the earlier works where they considered conditions reaching...
The purpose of this study is to improve the computational efficiency of solvers for nonlinear algebraic problems with simple roots. To this end, a multi-step solver based on Newton’s method is utilized. Divided difference operators are applied...
King’s method applies to solve scalar equations. The local analysis is established under conditions including the fifth derivative. However, the only derivative in this method is the first. Earlier studies apply to equations containing at least...
In this paper, a few single-step iterative methods, including classical Newton’s method and Halley’s method, are suggested by applying
In this paper, we address a key issue in Numerical Functional Analysis: to perform the local convergence analysis of a fourth order of convergence iterative method in Banach spaces, examining conditions on the operator and its derivatives near the so...
There is no doubt that there is plethora of optimal fourth-order iterative approaches available to estimate the simple zeros of nonlinear functions. We can extend these method/methods for multiple zeros but the main issue is to preserve the same conv...
There is no doubt that the fourth-order King’s family is one of the important ones among its counterparts. However, it has two major problems: the first one is the calculation of the first-order derivative; secondly, it has a linear order of converge...
This study presents an efficient high-order radius function Hermite finite difference (RBF-HFD) scheme for the numerical solution of Caputo time-fractional sub-diffusion equations with integral boundary conditions. The spatial derivatives are approxi...
Based on the Steffensen-type method, we develop fourth-, eighth-, and sixteenth-order algorithms for solving one-variable equations. The new methods are fourth-, eighth-, and sixteenth-order converging and require at each iteration three, four, and f...
In this work, we have developed a fourth order Newton-like method based on harmonic mean and its multi-step version for solving system of nonlinear equations. The new fourth order method requires evaluation of one function and two first order Fréchet...
Many optimal order multiple root techniques involving derivatives have been proposed in literature. On the contrary, optimal order multiple root techniques without derivatives are almost nonexistent. With this as a motivational factor, here we develo...
A plethora of higher order iterative methods, involving derivatives in algorithms, are available in the literature for finding multiple roots. Contrary to this fact, the higher order methods without derivatives in the iteration are difficult to const...
A generalized high-order class for approximating the solution of nonlinear systems of equations is introduced. First, from a fourth-order iterative family for solving nonlinear equations, we propose an extension to nonlinear systems of equations hold...
This study introduces a novel, iterative algorithm that achieves fourth-order convergence for solving nonlinear equations. Satisfying the Kung–Traub conjecture, the proposed technique achieves an optimal order of four with an efficiency index
In this paper, high-order compact difference methods (HOCDMs) are proposed to solve the semi-linear Sobolev equations (SLSEs), which arise in various physical models, such as porous media flow and heat conduction. First, a two-level numerical method...
This paper studies the existence and uniqueness of the classical solution of inverse problems for a fourth-order hyperbolic equation with a complex-valued coefficient with Dirichlet and Neumann boundary conditions. Using the method of separation of v...
This paper discusses the Crank–Nicolson compact difference method for the time-fractional damped plate vibration problems. For the time-fractional damped plate vibration equations, we introduce the second-order space derivative and the first-or...
Singularly perturbed integro-partial differential equations with reaction–diffusion behavior present significant challenges due to boundary layers arising from small perturbation parameters, which complicate the development of accu...
The block-centered finite-difference method has many advantages, and the time-fractional fourth-order equation is widely used in physics and engineering science. In this paper, we consider variable-coefficient fourth-order parabolic equations of frac...
In this paper, some one-step iterative schemes with memory-accelerating methods are proposed to update three critical values
Here, we propose optimal fourth-order iterative methods for approximating multiple zeros of univariate functions. The proposed family is composed of two stages and requires 3 functional values at each iteration. We also suggest an extensive convergen...
Here, we suggest a high-order optimal variant/modification of Schröder’s method for obtaining the multiple zeros of nonlinear uni-variate functions. Based on quadratically convergent Schröder’s method, we derive the new family o...
The purpose of this paper is to develop a numerical scheme for the two-dimensional fourth-order fractional subdiffusion equation with variable coefficients and delay. Using the
The main goal of this paper is to study Jarratt-like iterative methods to obtain their order of convergence under weaker conditions. Generally, obtaining the
The study of the fuzzy differential equation is a topic that researchers are interested in these days. By modelling, this fuzzy differential equation can be used to resolve issues in the real world. However, finding an analytical solution to this fuz...
We propose a novel family of seventh-order iterative methods for computing multiple zeros of a nonlinear function. The algorithm consists of three steps, of which the first two are the steps of recently developed Liu–Zhou fourth-order method, w...
The principal objective of this work is to propose a fourth, eighth and sixteenth order scheme for solving a nonlinear equation. In terms of computational cost, per iteration, the fourth order method uses two evaluations of the function and one evalu...
In this paper, a high-accuracy conservative implicit algorithm for computing the space fractional coupled Schrödinger–Boussinesq system is constructed. Meanwhile, the conservative nature, a priori boundedness, and solvability of the numeri...
We suggest a new and cost-effective iterative scheme for nonlinear equations. The main features of the presented scheme are that it does not involve any derivative in the structure, achieves an optimal convergence of fourth-order factors, has more fl...
We present a new two-parameter family of fourth-order iterative methods for solving systems of nonlinear equations. The scheme is composed of two Newton–Jarratt steps and requires the evaluation of one function and two first derivatives in each...
Non-linear systems of equations play a fundamental role in various engineering and data science models, where accurate solutions are essential for both theoretical research and practical applications. However, solving such systems is highly challengi...
In this work, we propose a new fourth-order Jarratt-type method for solving systems of nonlinear equations. The local convergence order of the method is proven analytically. Finally, we validate our results via some numerical experiments including an...
We present a local convergence analysis of an eighth order three step methodin order to approximate a locally unique solution of nonlinear equation in a Banach spacesetting. In an earlier study by Sharma and Arora (2015), the order of convergence was...
This article considers the fourth-order family of weighted-Newton methods. It provides the range of initial guesses that ensure the convergence. The analysis is given for Banach space-valued mappings, and the hypotheses involve the derivative of orde...
We present a new Jarratt-type family of optimal fourth- and sixth-order iterative methods for solving nonlinear equations, along with their convergence properties. The schemes are extended to nonlinear systems of equations with equal convergence orde...
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