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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.

The analysis of certain structures must be performed with due consideration to non‐linear behavior, such as material and geometric non‐linearities. The existing methods for treating non‐linear structural behavior generally make use of repeated linearization, such as load increment methods. This paper demonstrates that there is an alternative type of linearization that appears to have significant advantages when applied to the analysis of non‐linear structural systems. Briefly stated, this alternative linearization can be thought of as a “monomialization”. This monomial (single‐termed power function) approximation more faithfully models the power function behavior inherent in typical structural systems. Conveniently, it becomes a linear form when transformed into log space. Thus, computational tools based on linear algebra remain useful and effective. Preliminary results indicate that the monomial approximation provides a higher quality approximation to non‐linear phenomena exhibited in structural applications. Consequently, incremental and iterative methods become more effective because larger steps can be taken. The net result is an increase in reliability of the solution process and a significant reduction in computational effort. Two examples are presented to demonstrate the method.

This paper examines several modifications to Newton's method for the numerical solution of the nonlinear Poisson equation which describes the electrostatic potential distribution in a semiconductor device. Two methods for a more efficient solution of the equation when the device is a Metal‐Oxide‐Semiconductor Field Effect Transistor are proposed. Their extension to the solution of the fully coupled system of equations is also discussed. The modifications to Newton's method are also compared numerically.

The purpose of this paper is to present a fast nonlinear solver for the prediction of transients in network flows.

Broyden method‐based nonlinear solvers are developed to solve the system of conservation equation for fluids by judiciously exploiting physical coupling among the equations.

To demonstrate the feasibility and robustness of the solvers, two test cases of practical engineering interest were solved. The results obtained by the solvers were verified against analytical results for a simplified case. The performance of the solvers was found to be comparable or better than existing solvers.

The proposed solver enables predictions of fluid and thermal transients in complex flow networks feasible in reduced computational time.

The purpose of this paper is to present a numerical model of coupled heat, moisture transfer and their effects on the mechanical deformations of wood during the drying process.

Coupling among heat, moisture, and mechanical deformations is solved consecutively by use of sparse solver of MATLAB. The weighted residual of the equilibrium equations of drying process of wood, based on finite element method, is investigated. The stress and plastic strain increments can be solved with Newton's method.

The numerical model is applied to a plain strain problem of a long wood board taken from the outer region of the wood log. Numerical simulation reveals the stress reversal during the drying process. The mechanical deformations and the principle stresses of a three‐dimensional wood board in consideration of the orthotropic properties are presented.

Plane strain and plane stress are analysed. The tangential modulus is derived. The transformation of the stress and strain tensors between the local coordinate system resulting from the cylindrical properties of wood and the global one is evaluated. Selection of element type for temperature, moisture content and displacement is discussed.

Discrete eigenvalues occur in eddy current problems in which the solution domain was truncated on its edge. In case of conductive material with a hole, the eigenvalues are complex numbers. Their computation consists of finding complex roots of a complex function that satisfies the electromagnetic interface conditions. The purpose of this paper is to present a method of computing complex eigenvalues that are roots of such a function.

The proposed approach involves precise determination of regions in which the roots are found and applying sets of initial points, as well as the Cauchy argument principle to calculate them.

The elaborated algorithm was implemented in Matlab and the obtained results were verified using Newton’s method and the fsolve procedure. Both in the case of magnetic and nonmagnetic materials, such a solution was the only one that did not skip any of the eigenvalues, obtaining the results in the shortest time.

The paper presents a new effective method of locating complex eigenvalues for analytical solutions of eddy current problems containing a conductive material with a hole.

Adaptive slicing is a key step in 3D printing as it is closely related to the building time and the surface quality. This study aims to develop an adaptive layering algorithm that can coordinate the optimization of printing quality and efficiency to meet different printing needs.

A multiobjective optimization model is established for printing quality, printing time and layer height based on the variation of surface features, profile slope and curvature of the model. The optimal solution is found by an improved method combining Newton's method and gradient method and adapts to different printing requirements by adjusting the parameter thresholds.

Several benchmarks are applied to verify this new method. The proposed method has also been compared with the uniform layering method, it reduces the volume error by 46.4% and shortens the printing time by 28.1% and is compared with five existing adaptive layering methods to demonstrate its superior performance.

Compared with other methods with only one layered result, this method is a demand-oriented algorithm that can obtain different results according to different needs and it can reach a trade-off between the building time and the surface quality.

This paper presents a hybrid algorithm used, in conjunction with the Finite Integration Technique (FIT), for solving static and quasistatic electromagnetic field problems in nonlinear media. The hybrid technique is based on new theoretical results regarding the similarities between the Picard‐Banach fixed‐point (polarization) method and the Newton method. At each iteration, the solution is obtained as a linear combination of the old solution, and the new Picard‐Banach and Newton solutions. The numerical solutions are calculated through a “triangle” (bidimensional) minimization of the residual or of the energy functional. The goal of this combination is to increase the robustness of the iterative method, without losing the quadratic speed of convergence in the vicinity of the solution. The proposed method generalizes and unifies in a single algorithm the overrelaxed Picard‐Banach and the underrelaxed Newton methods.

The purpose of this paper is to solve unsteady compressible Navier–Stokes equations without the commonly used dual-time loop. The authors would like to use an adaptive time-stepping (ATS)-based local error control instead of CFL-based time-stepping technique. Also, an all-speed flow algorithm is implemented with simple low dissipation AUSM convective scheme, which can be computed without preconditioning which in general destroys the time accuracy.

In transient flow computations, the time-step is generally determined from the CFL condition. In this paper, the authors demonstrate the usefulness of ATS based on local time-stepping previously used extensively in ordinary differential equations (ODE) integration. This method is implemented in an implicit framework to ensure the numerical domain of dependence always contains the physical domain of dependence.

In this paper, the authors limit their focus to capture the unsteady physics for three cases: Sod’s shock-tube problem, Stokes’ second problem and a circular cylinder. The use of ATS with local truncation error control enables the solver to use the maximum allowable time-step, for the prescribed tolerance of error. The algorithm is also capable of converging very rapidly to the steady state (if there is any) after the initial transient phase. The authors present here only the first-order time-stepping scheme. An algorithmic comparison is made between the proposed adaptive time-stepping method and the commonly used dual time-stepping approach that indicates the former will be more efficient.

The original method of ATS based on local error control is used extensively in ODE integration, whereas, this method is not so popular in the computational fluid dynamics (CFD) community. In this paper, the authors investigate its use in the unsteady CFD computations. The authors hope that it would provide CFD researchers with an algorithm based on an adaptive time-stepping approach for unsteady calculations.

The purpose of this paper is to obtain the nonlinear Schrodinger equation (NLSE) numerical solutions in the presence of the first-order chromatic dispersion using a second-order, unconditionally stable, implicit finite difference method. In addition, stability and accuracy are proved for the resulting scheme.

The conserved quantities such as mass, momentum and energy are calculated for the system governed by the NLSE. Moreover, the robustness of the scheme is confirmed by conducting various numerical tests using the Crank-Nicolson method on different cases of solitons to discuss the effects of the factor considered on solitons properties and on conserved quantities.

The Crank-Nicolson scheme has been derived to solve the NLSE for optical fibers in the presence of the wave packet drift effects. It has been founded that the numerical scheme is second-order in time and space and unconditionally stable by using von-Neumann stability analysis. The effect of the parameters considered in the study is displayed in the case of one, two and three solitons. It was noted that the reliance of NLSE numeric solutions properties on coefficients of wave packets drift, dispersions and Kerr nonlinearity play an important control not only the stable and unstable regime but also the energy, momentum conservation laws. Accordingly, by comparing our numerical results in this study with the previous work, it was recognized that the obtained results are the generalized formularization of these work. Also, it was distinguished that our new data are regarding to the new communications modes that depend on the dispersion, wave packets drift and nonlinearity coefficients.

The present study uses the first-order chromatic. Also, it highlights the relationship between the parameters of dispersion, nonlinearity and optical wave properties. The study further reports the effect of wave packet drift, dispersions and Kerr nonlinearity play an important control not only the stable and unstable regime but also the energy, momentum conservation laws.