Constructing Frozen Jacobian Iterative Methods for Solving Systems of Nonlinear Equations, Associated with ODEs and PDEs Using the Homotopy Method (original) (raw)
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Mathematics
This research aims to propose a new family of one-parameter multi-step iterative methods that combine the homotopy perturbation method with a quadrature formula for solving nonlinear equations. The proposed methods are based on a higher-order convergence scheme that allows for faster and more efficient convergence compared to existing methods. It aims also to demonstrate that the efficiency index of the proposed iterative methods can reach up to 43≈1.587 and 84≈1.681, respectively, indicating a high degree of accuracy and efficiency in solving nonlinear equations. To evaluate the effectiveness of the suggested methods, several numerical examples including their performance are provided and compared with existing methods.
Frozen Jacobian iterative methods are of practical interest to solve the system of nonlinear equations. A frozen Jacobian multi-step iterative method is presented. We divide the multi-step iterative method into two parts namely base method and multi-step part. The convergence order of the constructed frozen Jacobian iterative method is three, and we design the base method in a way that we can maximize the convergence order in the multi-step part. In the multi-step part, we utilize a single evaluation of the function, solve four systems of lower and upper triangular systems and a second frozen Jacobian. The attained convergence order per multi-step is four. Hence, the general formula for the convergence order is 3 + 4(m − 2) for m ≥ 2 and m is the number of multi-steps. In a single instance of the iterative method, we employ only single inversion of the Jacobian in the form of LU factors that makes the method computationally cheaper because the LU factors are used to solve four system of lower and upper triangular systems repeatedly. The claimed convergence order is verified by computing the computational order of convergence for a system of nonlinear equations. The efficiency and validity of the proposed iterative method are narrated by solving many nonlinear initial and boundary value problems. c 2016 All rights reserved. Keywords: Frozen Jacobian iterative methods, multi-step iterative methods, systems of nonlinear equations, nonlinear initial value problems, nonlinear boundary value problems. 2010 MSC: 65H10, 65N22.
Journal of Computational and Theoretical Nanoscience, 2015
We presented multi-step iterative method for solving systems of nonlinear equations, associated with ordinary differential equations (ODEs). In particular we considered the ODEs of the form L x t + f x t = g t : here L • is a linear differential operator and f • is a nonlinear smooth function. The computational efficiency of our proposed iterative method is hidden in frozen Jacobian at a single point. The direct inversion of frozen Jacobian is avoided by using either LU-decomposition or GMRES-type methods. The higher-order Frechet derivatives of systems of nonlinear equations associated with under-discussion ODEs are diagonal matrices, and the computational cost of higher-order Frechet derivatives is same as Jacobian. The use of second-order Frechet-derivative enhances the convergence-order. The inclusion of multi-step increase the convergence-rate and the design of multi-step consist on the product of a matrix polynomial and frozen Jacobian inverse. The convergence-order (CO) obeys the formula CO = 3 m − 1 , where m ≥ 2 is the number of steps per full-cycle of the iterative scheme.
New Higher Order Iterative Methods for Solving Nonlinear Equations
Hacettepe Journal of Mathematics and Statistics, 2017
In this paper, using the system of coupled equations involving an auxiliary function, we introduce some new efficient higher order iterative methods based on modified homotopy perturbation method. We study the convergence analysis and also present various numerical examples to demonstrate the validity and efficiency of our methods.
Research Article Convergent Homotopy Analysis Method for Solving Linear Systems
2016
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. By using homotopy analysis method (HAM), we introduce an iterative method for solving linear systems.This method (HAM) can be used to accelerate the convergence of the basic iterative methods. We also show that by applying HAM to a divergent iterative scheme, it is possible to construct a convergent homotopy-series solution when the iteration matrix G of the iterative scheme has particular properties such as being symmetric, having real eigenvalues. Numerical experiments are given to show the efficiency of the new method. 1.
Newton-like iterative methods for solving system of non-linear equations
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Homotopy perturbation method (HPM) is applied to construct a new family of Newton-like iterative methods for solving system of non-linear equations. Comparison of the result obtained by the present method with Newton-Raphson method reveals that the accuracy and fast convergence of the new method.
New iterative methods for solving nonlinear equation by using homotopy perturbation method
Applied Mathematics and Computation, 2012
In this paper, we suggest and analyze a new class of iterative methods for solving nonlinear equations by using the homotopy perturbation method. Convergence of their method is also considered. Here we also discuss the efficiency index and computational order of convergence of new methods. Several numerical examples are given to illustrate the efficiency and performance of these new methods. These new iterative methods may be viewed as an extension and generalization of the existing methods for solving nonlinear equations.
A higher order frozen Jacobian iterative method for solving Hamilton-Jacobi equations
It is well-known that the solution of Hamilton-Jacobi equation may have singularity i.e., the solution is non-smooth or nearly non-smooth. We construct a frozen Jacobian multi-step iterative method for solving Hamilton-Jacobi equation under the assumption that the solution is nearly singular. The frozen Jacobian iterative methods are computationally very efficient because a single instance of the iterative method uses a single inversion (in the scene of LU factorization) of the frozen Jacobian. The multi-step part enhances the convergence order by solving lower and upper triangular systems. The convergence order of our proposed iterative method is 3(m − 1) for m ≥ 3. For attaining good numerical accuracy in the solution, we use Chebyshev pseudo-spectral collocation method. Some Hamilton-Jacobi equations are solved, and numerically obtained results show high accuracy.
High order iterative methods without derivatives for solving nonlinear equations
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The new second-order and third-order iterative methods without derivatives are presented for solving nonlinear equations; the iterative formulae based on the homotopy perturbation method are deduced and their convergences are provided. Finally, some numerical experiments show the efficiency of the theoretical results for the above methods.