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Solving the non‐linear equation f(x)=0 has nice applications in various branches of physics and engineering. Sometimes the applications of the numerical methods to solve non‐linear equations depending on the second derivatives are restricted in physics and engineering. The purpose of this paper is to propose two new modified Newton's method for solving non‐linear equations. Convergence results show that the order of convergence of the proposed iterative methods for a simple root is four. The iterative methods are free from second derivative and can be used for solving non‐linear equations without computing the second derivative. Finally, several numerical examples are given to illustrate that proposed iterative algorithms are effective.

In this paper, first the authors introduce two new approximations for the definite integral arising from Newton's theorem. Then by considering these approximations, two new iterative methods are provided with fourth‐order convergence which can be used for solving non‐linear equations without computing second derivatives.

In this paper, the authors propose two new iterative methods without second derivatives for solving the non‐linear equation f(x)=0. From numerical results, it is observed that the new methods are comparable with various iterative methods. Also numerical results corroborate the theoretical analysis.

The best property of these schemes is that they are second derivative free. Also from numerical results, it is observed that the new methods are comparable with various iterative methods. The numerical results corroborate the theoretical analysis.

This study implements the fourth-order phase field method (PFM) for modeling fracture in brittle materials. The weak form of the fourth-order PFM requires C¹ basis functions for the crack evolution scalar field in a finite element framework. To address this, non-Sibsonian type shape functions that are nonpolynomial types based on distance measures, are used in the context of natural neighbor shape functions. The capability and efficiency of this method are studied for modeling cracks.

The weak form of the fourth-order PFM is derived from two governing equations for finite element modeling. C⁰ non-Sibsonian shape functions are derived using distance measures on a generalized quad element. Then these shape functions are degree elevated with Bernstein-Bezier (BB) patch to get higher-order continuity (C¹) in the shape function. The quad element is divided into several background triangular elements to apply the Gauss-quadrature rule for numerical integration. Both fourth-order and second-order PFMs are implemented in a finite element framework. The efficiency of the interpolation function is studied in terms of convergence and accuracy for capturing crack topology in the fourth-order PFM.

It is observed that fourth-order PFM has higher accuracy and convergence than second-order PFM using non-Sibsonian type interpolants. The former predicts higher failure loads and failure displacements compared to the second-order model due to the addition of higher-order terms in the energy equation. The fracture pattern is realistic when only the tensile part of the strain energy is taken for fracture evolution. The fracture pattern is also observed in the compressive region when both tensile and compressive energy for crack evolution are taken into account, which is unrealistic. Length scale has a certain specific effect on the failure load of the specimen.

Fourth-order PFM is implemented using C¹ non-Sibsonian type of shape functions. The derivation and implementation are carried out for both the second-order and fourth-order PFM. The length scale effect on both models is shown. The better accuracy and convergence rate of the fourth-order PFM over second-order PFM are studied using the current approach. The critical difference between the isotropic phase field and the hybrid phase field approach is also presented to showcase the importance of strain energy decomposition in PFM.

Modelling accurately the transient behaviour of natural convection flow in enclosures been a challenging task because of a variety of numerical errors which have limited achieving the higher order temporal accuracy. A fourth-order accurate finite difference method in both space and time is proposed to overcome these numerical errors and accurately model the transient behaviour of natural convection flow in enclosures using vorticity–streamfunction formulation.

Fourth-order wide stencil formula with appropriate one-sided difference extrapolation technique near the boundary is used for spatial discretisation, and classical fourth-order Runge–Kutta scheme is applied for transient term discretisation. The proposed method is applied on two transient case studies, i.e. convection–diffusion of a Gaussian Pulse and Taylor Vortex flow having analytical solution.

Error magnitude comparison and rate of convergence analysis of the proposed method with these analytical solutions establish fourth-order accuracy and prove the ability of the proposed method to truly capture the transient behaviour of incompressible flow. Also, to test the transient natural convection flow behaviour, the algorithm is tested on differentially heated square cavity at high Rayleigh number in the range of 10³-10⁸, followed by studying the transient periodic behaviour in a differentially heated vertical cavity of aspect ratio 8:1. An excellent comparison is obtained with standard benchmark results.

The developed method is applied on 2D enclosures; however, the present methodology can be extended to 3D enclosures using velocity–vorticity formulations which shall be explored in future.

The proposed methodology to achieve fourth-order accurate transient simulation of natural convection flows is novel, to the best of the authors’ knowledge. Stable fourth-order vorticity boundary conditions are derived for boundary and external boundary regions. The selected case studies for comparison demonstrate not only the fourth-order accuracy but also the considerable reduction in error magnitude by increasing the temporal accuracy. Also, this study provides novel benchmark results at five different locations within the differentially heated vertical cavity of aspect ratio 8:1 for future comparison studies.

This paper aims to present a high-order algorithm implemented with the modal spectral element method and simulations of three-dimensional thermal convective flows by using the full viscous dissipation function in the energy equation. Three benchmark problems were solved to validate the algorithm with exact or theoretical solutions. The heated rotating sphere at different temperatures inside a cold planar Poiseuille flow was simulated parametrically at varied angular velocities with positive and negative rotations.

The fourth-order stiffly stable schemes were implemented and tested for time integration. To provide the hp-refinement and spatial resolution enhancement, a modal spectral element method using hierarchical basis functions was used to solve governing equations in a three-dimensional space.

It was found that the direction of rotation of the heated sphere has totally different effects on drag, lateral force and torque evaluated on surfaces of the sphere and walls. It was further concluded that the angular velocity of the heated sphere has more influence on the wall normal velocity gradient than on the wall normal temperature gradients and therefore, more influence on the viscous dissipation than on the thermal dissipation.

This paper concerns incompressible fluid flow at constant properties with up to medium temperature variations in the absence of thermal radiation and ignoring the pressure work.

This paper contributes a viable high-order algorithm in time and space for modeling convective heat transfer involving an internal heated rotating sphere with the effect of viscous heating.

Results of this paper could provide reference for related topics such as enhanced heat transfer forced convection involving rotating spheres and viscous thermal effect.

The merits include resolving viscous dissipation and thermal diffusion in stationary and rotating boundary layers with both h- and p-type refinements, visualizing the viscous heating effect with the full viscous dissipation function in the energy equation and modeling the forced advection around a rotating sphere with varied positive and negative angular velocities subject to a shear flow.

This paper aims to develop a new high accuracy numerical method based on off-step non-polynomial spline in tension approximations for the solution of Burgers-Fisher and coupled nonlinear Burgers’ equations on a graded mesh. The spline method reported here is third order accurate in space and second order accurate in time. The proposed spline method involves only two off-step points and a central point on a graded mesh. The method is two-level implicit in nature and directly derived from the continuity condition of the first order space derivative of the non-polynomial tension spline function. The linear stability analysis of the proposed method has been examined and it is shown that the proposed two-level method is unconditionally stable for a linear model problem. The method is directly applicable to problems in polar systems. To demonstrate the strength and utility of the proposed method, the authors have solved the generalized Burgers-Huxley equation, generalized Burgers-Fisher equation, coupled Burgers-equations and parabolic equation in polar coordinates. The authors show that the proposed method enables us to obtain the high accurate solution for high Reynolds number.

In this method, the authors use only two-level in time-direction, and at each time-level, the authors use three grid points for the unknown function u(x,t) and two off-step points for the known variable x in spatial direction. The methodology followed in this paper is the construction of a non-polynomial spline function and using its continuity properties to obtain consistency condition, which is third order accurate on a graded mesh and fourth order accurate on a uniform mesh. From this consistency condition, the authors derive the proposed numerical method. The proposed method, when applied to a linear equation is shown to be unconditionally stable. To assess the validity and accuracy, the method is applied to solve several benchmark problems, and numerical results are provided to demonstrate the usefulness of the proposed method.

The paper provides a third order numerical scheme on a graded mesh and fourth order spline method on a uniform mesh obtained directly from the consistency condition. In earlier methods, consistency conditions were only second order accurate. This brings an edge over other past methods. Also, the method is directly applicable to physical problems involving singular coefficients. So no modification in the method is required at singular points. This saves CPU time and computational costs.

There are no limitations. Obtaining a high accuracy spline method directly from the consistency condition is a new work. Also being an implicit method, this method is unconditionally stable.

Physical problems with singular and non-singular coefficients are directly solved by this method.

The paper develops a new method based on non-polynomial spline approximations of order two in time and three (four) in space, which is original and has lot of value because many benchmark problems of physical significance are solved in this method.

The purpose of this paper is to present a new family of iterative methods with eighth‐order convergence for solving the nonlinear equation.

The paper uses a family of eighth‐order iterative methods for solving nonlinear equation based on Kou's seventh‐order method.

This family of methods is preferable to Ostrowski's, Grau's and Kou's methods in high‐precision computations.

This paper only deals with the nonlinear equations.

This paper is concerned with the iterative methods for finding a simple root of the nonlinear equation f(x)=0. One of the reasons for discussing the solution of nonlinear equation is that many methods for high‐dimensional optimization problems involve solving a sub‐problem which is a one‐dimensional search problem. And the nonlinear finite element problem, the boundary‐value problems appearing in Kinetic theory of gases, elasticity and other applied areas are also reduced to solving such an equation.

New methods of this family require three evaluations of the function and one evaluation of its first derivative and without using the second derivatives per iteration. This new family of methods as a new example agrees with Kung‐Traub's conjecture for n=4 and achieves its optimal convergence order 2ⁿ⁻¹.

The purpose of the article is to construct a new class of higher-order iterative techniques for solving scalar nonlinear problems.

The scheme is generalized by using the power-mean notion. By applying Neville's interpolating technique, the methods are formulated into the derivative-free approaches. Further, to enhance the computational efficiency, the developed iterative methods have been extended to the methods with memory, with the aid of the self-accelerating parameter.

It is found that the presented family is optimal in terms of Kung and Traub conjecture as it evaluates only five functions in each iteration and attains convergence order sixteen. The proposed family is examined on some practical problems by modeling into nonlinear equations, such as chemical equilibrium problems, beam positioning problems, eigenvalue problems and fractional conversion in a chemical reactor. The obtained results confirm that the developed scheme works more adequately as compared to the existing methods from the literature. Furthermore, the basins of attraction of the different methods have been included to check the convergence in the complex plane.

The presented experiments show that the developed schemes are of great benefit to implement on real-life problems.

The paper gives the description of boundary element method (BEM) with subdomains for the solution of convection—diffusion equations with variable coefficients and Burgers’ equations. At first, the whole domain is discretized into K subdomains, in which linearization of equations by representing convective velocity by the sum of constant and variable parts is carried out. Then using fundamental solutions for convection—diffusion linear equations for each subdomain the boundary integral equation (in which the part of the convective term with the constant convective velocity is not included into the pseudo‐body force) is formulated. Only part of the convective term with the variable velocity, which is, as a rule, more than one order less than convective velocity constant part contribution, is left as the pseudo‐source. On the one hand, this does not disturb the numerical BEM—algorithm stability and, on the other hand, this leads to significant improvement in the accuracy of solution. The global matrix, similar to the case of finite element method, has block band structure whereas its width depends only on the numeration order of nodes and subdomains. It is noted, that in comparison with the direct boundary element method the number of global matrix non‐zero elements is not proportional to the square of the number of nodes, but only to the total number of nodal points. This allows us to use the BEM for the solution of problems with very fine space discretization. The proposed BEM with subdomains technique has been used for the numerical solution of one‐dimensional linear steady‐state convective—diffusion problem with variable coefficients and one‐dimensional non‐linear Burgers’ equation for which exact analytical solutions are available. It made it possible to find out the BEM correctness according to both time and space. High precision of the numerical method is noted. The good point of the BEM is the high iteration convergence, which is disturbed neither by high Reynolds numbers nor by the presence of negative velocity zones.

The purpose of this article is to develop and analyze a new derivative-free class of higher-order iterative methods for locating multiple roots numerically.

The scheme is generated by using King-type iterative methods. By employing the Traub-Steffensen technique, the proposed class is designed into the derivative-free family.

The proposed class requires three functional evaluations at each stage of computation to attain fourth-order convergency. Moreover, it can be observed that the theoretical convergency results of family are symmetrical for particular cases of multiplicity of zeros. This further motivates the authors to present the result in general, which confirms the convergency order of the methods. It is also worth mentioning that the authors can obtain already existing methods as particular cases of the family for some suitable choice of free disposable parameters. Finally, the authors include a wide variety of benchmark problems like van der Waals's equation, Planck's radiation law and clustered root problem. The numerical comparisons are included with several existing algorithms to confirm the applicability and effectiveness of the proposed methods.

The numerical results demonstrate that the proposed scheme performs better than the existing methods in terms of CPU timing and absolute residual errors.

The purpose of this paper is to conduct a survey on research in fabric and cloth simulation using mass spring model. Also in this paper some of the common methods in process of fabric simulation in mass spring model are discussed and compared.

This paper reviews and compares presented mesh types in mass spring model, forces applied on model, super elastic effect and ways to settle the super elasticity problem, numerical integration methods for solving equations, collision detection and its response. Some of common methods in fabric simulation are compared to each other. And by using examples of fabric simulation, advantages and limitations of each technique are mentioned.

Mass spring method is a fast and flexible technique with high ability to simulate fabric behavior in real time with different environmental conditions. Mass spring model has more accuracy than geometrical models and also it is faster than other physical modeling.

In the edge of digital, fabric simulation technology has been considered into many fields. 3D fabric simulation is complex and its implementation requires knowledge in different fields such as textile engineering, computer engineering and mechanical engineering. Several methods have been presented for fabric simulation such as physical and geometrical models. Mass spring model, the typical physically based method, is one of the methods for fabric simulation which widely considered by researchers.