Search Results (32)

An advanced frequency study in thick-walled functionally graded material (FGM) spherical shells is investigated with advanced shear correction. The values of advanced shear correction can be greater than one, be a negative value, and be affected by a nonlinear term of third-order shear deformation theory (TSDT) of displacements, FGM power law index, and temperature. It is novel and interesting to consider using TSDT and advanced shear correction to derive a simple homogeneous equation with reasonable simplifications into a symmetrical sparse matrix subjected to free vibration. The zero determinant of the symmetrical sparse matrix can be expressed to calculate the natural frequency by Newton’s method. The parameter effects of advanced shear correction, a nonlinear TSDT term, temperature, and the FGM power-law index on the natural frequencies of thick-walled FGM spherical shells are presented. The natural-frequency data for the axial and circumferential mode shapes are obtained. This is a new finding, as the assumed simplification in a sparse matrix causes a numerical truncation error; the natural-frequency values of the presented sparse matrix are much greater than those in a full matrix for thick-walled FGM spherical shells. Full article

We develop a novel Steffensen-type iterative solver to solve nonlinear scalar equations without requiring derivatives. A two-parameter one-step scheme without memory is first introduced and analyzed. Its optimal quadratic convergence is then established. To enhance the convergence rate without additional functional evaluations, we extend the scheme by incorporating memory through adaptively updated accelerator parameters. These parameters are approximated by Newton interpolation polynomials constructed from previously computed values, yielding a derivative-free method with R-rate of convergence of approximately 3.56155. A dynamical system analysis based on attraction basins demonstrates enlarged convergence regions compared to Steffensen-type methods without memory. Numerical experiments further confirm the accuracy of the proposed scheme for solving nonlinear equations. Full article

This paper investigates the design and stability of Traub–Steffensen-type iteration schemes with and without memory for solving nonlinear equations. Steffensen’s method overcomes the drawback of the derivative evaluation of Newton’s scheme, but it has, in general, smaller sets of initial guesses that converge to the desired root. Despite this drawback of Steffensen’s method, several researchers have developed higher-order iterative methods based on Steffensen’s scheme. Traub introduced a free parameter in Steffensen’s scheme to obtain the first parametric iteration method, which provides larger basins of attraction for specific values of the parameter. In this paper, we introduce a two-step derivative free fourth-order optimal iteration scheme based on Traub’s method by employing three free parameters and a weight function. We further extend it into a two-step eighth-order iteration scheme by means of memory with the help of suitable approximations of the involved parameters using Newton’s interpolation. The convergence analysis demonstrates that the proposed iteration scheme without memory has an order of convergence of 4, while its memory-based extension achieves an order of convergence of at least $7.993$, attaining the efficiency index $7.{993}^{1/3}\approx 2$. Two special cases of the proposed iteration scheme are also presented. Notably, the proposed methods compete with any optimal j-point method without memory. We affirm the superiority of the proposed iteration schemes in terms of efficiency index, absolute error, computational order of convergence, basins of attraction, and CPU time using comparisons with several existing iterative methods of similar kinds across diverse nonlinear equations. In general, for the comparison of iterative schemes, the basins of iteration are investigated on simple polynomials of the form $z^n-1$ in the complex plane. However, we investigate the stability and regions of convergence of the proposed iteration methods in comparison with some existing methods on a variety of nonlinear equations in terms of fractals of basins of attraction. The proposed iteration schemes generate the basins of attraction in less time with simple fractals and wider regions of convergence, confirming their stability and superiority in comparison with the existing methods. Full article

In the paper, two nonlinear variants of the Newton method are developed for solving nonlinear equations. The derivative-free nonlinear fractional type of the one-step iterative scheme of a fourth-order convergence contains three parameters, whose optimal values are obtained by a memory-dependent updating method. Then, as the extensions of a one-step linear fractional type method, we explore the fractional types of two- and three-step iterative schemes, which possess sixth- and twelfth-order convergences when the parameters’ values are optimal; the efficiency indexes are $\sqrt{6}$ and $\sqrt[3]{12}$, respectively. An extra variable is supplemented into the second-degree Newton polynomial for the data interpolation of the two-step iterative scheme of fractional type, and a relaxation factor is accelerated by the memory-dependent method. Three memory-dependent updating methods are developed in the three-step iterative schemes of linear fractional type, whose performances are greatly strengthened. In the three-step iterative scheme, when the first step involves using the nonlinear fractional type model, the order of convergence is raised to sixteen. The efficiency index also increases to $\sqrt[3]{16}$, and a third-degree Newton polynomial is taken to update the values of optimal parameters. Full article

A nonlinear equation $f\left(x\right)=0$ is mathematically transformed to a coupled system of quasi-linear equations in the two-dimensional space. Then, a linearized approximation renders a fractional iterative scheme $x_{n+1}=x_n-f\left(x_n\right)/[a+bf\left(x_n\right)]$, which requires one evaluation of the given function per iteration. A local convergence analysis is adopted to determine the optimal values of a and b. Moreover, upon combining the fractional iterative scheme to the generalized quadrature methods, the fourth-order optimal iterative schemes are derived. The finite differences based on three data are used to estimate the optimal values of a and b. We recast the Newton iterative method to two types of derivative-free iterative schemes by using the finite difference technique. A three-point generalized Hermite interpolation technique is developed, which includes the weight functions with certain constraints. Inserting the derived interpolation formulas into the triple Newton method, the eighth-order optimal iterative schemes are constructed, of which four evaluations of functions per iteration are required. Full article

We discuss the weighted complementarity problem, extending the nonlinear complementarity problem on $R^n$. In contrast to the NCP, many equilibrium problems in science, engineering, and economics can be transformed into WCPs for more efficient methods. Smoothing Newton algorithms, known for their at least locally superlinear convergence properties, have been widely applied to solve WCPs. We suggest a two-step Newton approach with a local biquadratic order convergence rate to solve the WCP. The new method needs to calculate two Newton equations at each iteration. We also insert a new term, which is of crucial importance for the local biquadratic convergence properties when solving the Newton equation. We demonstrate that the solution to the WCP is the accumulation point of the iterative sequence produced by the approach. We further demonstrate that the algorithm possesses local biquadratic convergence properties. Numerical results indicate the method to be practical and efficient. Full article

To solve linear and nonlinear eigenvalue problems, we develop a simple method by directly solving a nonhomogeneous system obtained by supplementing a normalization condition on the eigen-equation for the uniqueness of the eigenvector. The novelty of the present paper is that we transform the original homogeneous eigen-equation to a nonhomogeneous eigen-equation by a normalization technique and the introduction of a simple merit function, the minimum of which leads to a precise eigenvalue. For complex eigenvalue problems, two normalization equations are derived utilizing two different normalization conditions. The golden section search algorithms are employed to minimize the merit functions to locate real and complex eigenvalues, and simultaneously, we can obtain precise eigenvectors to satisfy the eigen-equation. Two regularized normalization methods can accelerate the convergence speed for two extensions of the simple method, and a derivative-free fixed-point Newton iterative scheme is developed to compute real eigenvalues, the convergence speed of which is ten times faster than the golden section search algorithm. Newton methods are developed for solving two systems of nonlinear regularized equations, and the efficiency and accuracy are significantly improved. Over ten examples demonstrate the high performance of the proposed methods. Among them, the two regularization methods are better than the simple method. Full article

This paper focuses on the weighted complementarity problem (WCP), which is widely used in the fields of economics, sciences and engineering. Not least because of its local superlinear convergence rate, smoothing Newton methods have widespread application in solving various optimization problems. A two-step smoothing Newton method with strong convergence is proposed. With a smoothing complementary function, the WCP is reformulated as a smoothing set of equations and solved by the proposed two-step smoothing Newton method. In each iteration, the new method computes the Newton equation twice, but using the same Jacobian, which can avoid consuming a lot of time in the calculation. To ensure the global convergence, a derivative-free line search rule is inserted. At the same time, we develop a different term in the solution of the smoothing Newton equation, which guarantees the local strong convergence. Under appropriate conditions, the algorithm has at least quadratic or even cubic local convergence. Numerical experiments indicate the stability and effectiveness of the new method. Moreover, compared to the general smoothing Newton method, the two-step smoothing Newton method can significantly improve the computational efficiency without increasing the computational cost. Full article

A Jacobian-free Newton–Krylov (JFNK) method with effective preconditioning strategies is introduced to solve a diffusion-based tumor growth model, also known as the Fisher–Kolmogorov partial differential equation (PDE). The time discretization of the PDE is based on the backward Euler and the Crank–Nicolson methods. Second-order centered finite differencing is used for the spatial derivatives. We introduce two physics-based preconditioners associated with the first- and second-order temporal discretizations. The theoretical time and spatial accuracies of the numerical scheme are verified through convergence tables and graphs that correspond to different computational settings. We present efficiency studies with and without using the preconditioners. Our numerical findings indicate the excellent performance of the newly proposed preconditioning strategies. In other words, when we turn the preconditioners on, the average number of GMRES and the Newton iterations are significantly reduced. Full article

In the present paper, we introduced a quadratically convergent Newton-like normal S-iteration method free from the second derivative for the solution of nonlinear equations permitting $f^{\prime }\left(x\right)=0$ at some points in the neighborhood of the root. Our proposed method works well when the Newton method fails and performs even better than some higher-order converging methods. Numerical results verified that the Newton-like normal S-iteration method converges faster than Fang et al.’s method. We studied different aspects of the normal S-iteration method regarding the faster convergence to the root. Lastly, the dynamic results support the numerical results and explain the convergence, divergence, and stability of the proposed method. Full article

This study presents a mathematical model of non-integer order through the fractal fractional Caputo operator to determine the development of Ebola virus infections. To construct the model and conduct analysis, all Ebola virus cases are taken as incidence data. A symmetric approach is utilized for qualitative and quantitative analysis of the fractional order model. Additionally, stability is evaluated, along with the local and global effects of the virus that causes Ebola. Using the fractional order model of Ebola virus infections, the existence and uniqueness of solutions, as well the posedness and biological viability and disease free equilibrium points are confirmed. Many applications of fractional operators in modern mathematics exist, including the intricate and important study of symmetrical systems. Symmetry analysis is a powerful tool that enables the creation of numerical solutions for a given fractional differential equation very methodically. For this, we compare the results with the Caputo derivative operator to understand the dynamic behavior of the disease. The simulation demonstrates how all classes have convergent characteristics and maintain their places over time, reflecting the true behavior of Ebola virus infection. Power law kernel with the two step polynomial Newton method were used. This model seems to be quite strong and capable of reproducing the issue’s anticipated theoretical conditions. Full article

In this paper, we construct variants of Bawazir’s iterative methods for solving nonlinear equations having simple roots. The proposed methods are two-step and three-step methods, with and without memory. The Newton method, weight function and divided differences are used to develop the optimal fourth- and eighth-order without-memory methods while the methods with memory are derivative-free and use two accelerating parameters to increase the order of convergence without any additional function evaluations. The methods without memory satisfy the Kung–Traub conjecture. The convergence properties of the proposed methods are thoroughly investigated using the main theorems that demonstrate the convergence order. We demonstrate the convergence speed of the introduced methods as compared with existing methods by applying the methods to various nonlinear functions and engineering problems. Numerical comparisons specify that the proposed methods are efficient and give tough competition to some well known existing methods. Full article

Stress–strain data with a given constitutive model of material can be calculated directly at a single material point. In this work, we propose a framework to perform single-point calculations under large deformations with stress and mixed control, to test and validate sophisticated constitutive models for materials. Inspired by Galerkin–FFT methods, a well-defined mask projector is used for stress and mixed control, and the derived nonlinear equations are solved in Newton iterations with Krylov solvers, simplifying implementation. One application example of the single-point calculator in developing sophisticated models for anisotropic single crystal rate-independent elastoplasticity is given, illustrating that the proposed algorithm can simulate asymmetrical deformation responses under uni-axial loading. Another example for artificial neural network models of the particle reinforced composite is also given, demonstrating that the commonly used machine learning or deep learning modeling frameworks can be directly incorporated into the proposed calculator. The central difference approximation of the tangent is validated so that derivative-free calculations for black-box constitutive models are possible. The proposed Python-coded single-point calculator is shown to be capable of quickly building, testing, and validating constitutive models with sophisticated or implicit structures, thus boosting the development of novel constitutive models for advanced solid materials. Full article

In this paper, static and dynamic stability analyses taking axial excitation into account are presented for a laminated carbon fiber reinforced polymer (CFRP) cylindrical shell under a non-normal boundary condition. The non-normal boundary condition is put forward to signify that both ends of the cylindrical shell are free and one generatrix of the shell is clamped. The partial differential motion governing the equations of the laminated CFRP cylindrical shell with a non-normal boundary condition is derived using the Hamilton principle, nonlinear von-Karman relationships and first-order deformation shell theory. Then, nonlinear, two-freedom, ordinary differential equations on the radial displacement of the cylindrical shell are obtained utilizing Galerkin method. The Newton-Raphson method is applied to numerically solve the equilibrium point. The stability of the equilibrium point is determined by analyzing the eigenvalue of the Jacobian matrix. The solution of the Mathieu equation describes the dynamic unstable behavior of the CFRP laminated cylindrical shells. The unstable regions are determined using the Bolotin method. The influences of the radial line load, the ratio of radius to thickness, the ratio of length to thickness, the number of layers and the temperature field of the laminated CFRP cylindrical shell on static and dynamic stability are investigated. Full article

In the numerical integration of the second-order nonlinear boundary value problem (BVP), the right boundary condition plays the role as a target equation, which is solved either by the half-interval method (HIM) or a new derivative-free Newton method (DFNM) to be presented in the paper. With the help of a boundary shape function, we can transform the BVP to an initial value problem (IVP) for a new variable. The terminal value of the new variable is expressed as a function of the missing initial value of the original variable, which is determined through a few integrations of the IVP to match the target equation. In the new boundary shape function method (NBSFM), we solve the target equation to obtain a highly accurate missing initial value, and then compute a precise solution. The DFNM can find more accurate left boundary values, whose performance is superior than HIM. Apparently, DFNM converges faster than HIM. Then, we modify the Lie-group shooting method and combine it to the BSFM for solving the nonlinear BVP with Robin boundary conditions. Numerical examples are examined, which assure that the proposed methods together with DFNM can successfully solve the nonlinear BVPs with high accuracy. Full article