Reliable Iterative Methods Research Articles

Reliable Iterative Method for solving Volterra -Fredholm Integro Differential Equations

The aim of this paper, a reliable iterative method is presented for resolving many types of Volterra - Fredholm Integro - Differential Equations of the second kind with initial conditions. The series solutions of the problems under consideration are obtained by means of the iterative method. Four various problems are resolved with high accuracy to make evident the enforcement of the iterative method on such type of integro differential equations. Results were compared with the exact solution which exhibit that this technique has compatible with the right solutions, simple, effective and easy for solving such problems. To evaluate the results in an iterative process the MATLAB is used as a math program for the calculations.

Solving Cantilever Beam Models by Reliable Iterative Methods

In this article, we implemented two reliable Cantilever beam models of solution methods. The iterative approaches are the Adomain Decomposition Method (ADM) and the Variation Iteration Method (VIM). The ADM and VIM solve several problems at various places, including the precision and reliability of the results. The solution given with an approximate solution is accuracy, since we demonstrate that various figures and tables are available to investigate maximum error reminders. The technology used for the study calculations was MATHEMATICA 11.

Two Reliable Iterative Methods for Solving Chaos synchronization

In this article we proposed two reliable iterative methods for solving Chaos synchronization. The iterative methods are the Adomain Decompustion method (ADM) and Variational Iteration Method (VIM). The ADM and VIM are solved several problems in different areas which accuracy and efficiency in the results. The solution which get is an approximate solution is accuracy as we well show that in figures and tables for the analysis of maximum error reminder various. The software which used in the study for the calculations was MATHEMATICA 11.

Reliable Iterative Methods for Solving the Falkner-Skan Equation

In this work, we suggest reliable iterative methods to solve the Falkner-Skan problem to obtain new approximate solutions. The suggested methods. are Tamimi-Ansari method.(TAM), Daftardair-Jafari. method.(DJM) and Banach countraction method.(BCM). We compare the obtained numerical results with other numerical methods like the Runge-Kutta (RK4) and Euler methods. The fixed point theorm is presented to test the convergence of the suggested methods. Moreover, the results of the remaining maximum error values showing that the suggested methods are reliable and effective. The Software used in our calculations for this work is Mathematica® 10.

Reliable iterative methods for mathematical model of COVID-19 based on data in Anhui, China

In this paper, five reliable iterative methods: Daftardar-Jafari method (DJM), Tamimi-Ansari method (TAM), Banach contraction method (BCM), Adomian decomposition method (ADM) and Variational iteration method (VIM) to obtain approximate solutions for a mathematical model that represented the coronavirus pandemic (COVID -19 pandemic). The accuracy of the obtained results is numerically verified by evaluating the maximum error remainder. In addition, the approximate results are compared with the fourth-order Runge-Kutta method (RK4) and good agreement have achieved. The convergence of the proposed methods is successfully demonstrated and mathematically verified. All calculations were successfully performed with MATHEMATICA®10.

Reliable iterative methods for 1D Swift–Hohenberg equation

In this paper, the nonlinear problem of the 1 D Swift-Hohenberg equation (S-HE) has been solved by using five reliable iterative methods. The first one is the Daftardar-Jafari method namely (DJM), the second method is the Temimi-Ansari method namely (TAM), the third method is the Banach contraction method namely (BCM), the fourth method is the Adomian decomposition method namely (ADM) and finally the fifth method is the Variational iteration method (VIM) to obtain the approximate solutions. In this work, we discussed and applied these iterative methods to solve the S-HE and compared them. In addition, the fixed-point theorem was given to illustrate the convergence of the five methods. To illustrate the accuracy and efficiency of the five methods, the maximum error remainder was calculated since the exact solution is unknown. The results showed that the five iterative methods are accurate, reliable, time saver and effective. All the iterative processes in this paper implemented in MATHEMATICA®11.

A method of analysis of complex hydraulic networks

A simple and reliable iterative solution method of classical hydraulic network flow rate distribution problem is described. The method is based on chord linearization of inverse branch loss function which keeps basic branch properties. It has good speed of convergency which is practically independent of initial values.

Analytic and numerical solutions for linear and nonlinear multidimensional wave equations

We develop three reliable iterative methods for solving the nonlinear 1D, 2D and 3D second-order wave equation and compare the results with a discretization-based solver. The iterative Tamimi–Ansari method (TAM), Daftardar–Jafari method (DJM) and the Banach contraction method (BCM) are used to obtain the exact solution for linear equations. For nonlinear equations and practical problems, however, one obtains the approximate solutions that converge to the exact solution, if one exists. The convergence analysis of the three methods is shown using the fixed-point theorem. The methods prove to be quite efficient and well suited to solve this kind of problems. We present several examples that demonstrate the accuracy and efficiency of the methods. We also compare the methods with a method based on discretization (Boundary Domain Integral Method (BDIM)). The BDIM uses a standard domain grid and discretizes the integral form of the governing equations. The iterative methods were developed with Mathematica® 10, while BDIM is a proprietary development.

Reliable iterative methods for mathematical biology model

In this paper, three iterative methods will be used to solve the nonlinear differential equation systems that represents a model for HIV infection of cells. The iterative methods are the Daftardar-Jafari method namely (DJM), Temimi-Ansari method namely (TAM) and Banach contraction method namely (BCM) to get the approximate solutions. Numerical solutions were also obtained by using the Runge-Kutta (RK4) method and good agreement have achieved. In addition, the convergence of proposed methods is present based on Banach's fixed point theorem. Also, comparisons are made between the proposed methods and the Adomian decomposition method (ADM), and the Variational iteration method (VIM), a good agreement have achieved without required to evaluate the Adomian polynomials to handle the nonlinear terms as in the ADM, does not require to calculate Lagrange multiplier as in the VIM and time saver. Moreover, the results of the maximum error remainder values are obtained and show that the proposed methods are effective and reliable. Our calculations were performed using the MATHEMATICA®10.

Reliable Iterative Methods for Solving Convective Straight and Radial Fins with Temperature-Dependent Thermal Conductivity Problems

In our article, three iterative methods are performed to solve the nonlinear differential equations that represent the straight and radial fins affected by thermal conductivity. The iterative methods are the Daftardar-Jafari method namely (DJM), Temimi-Ansari method namely (TAM) and Banach contraction method namely (BCM) to get the approximate solutions. For comparison purposes, the numerical solutions were further achieved by using the fourth Runge-Kutta (RK4) method, Euler method and previous analytical methods that available in the literature. Moreover, the convergence of the proposed methods was discussed and proved. In addition, the maximum error remainder values are also evaluated which indicates that the proposed methods are efficient and reliable. Our computational works have been done by using the computer algebra system MATHEMATICA®10 to evaluate the terms in the iterative processes.

A semi-analytical iterative method for solving nonlinear thin film flow problems

This paper presents a new implementation of a reliable iterative method proposed by Temimi and Ansari namely (TAM) for approximate solutions of a nonlinear problem that arises in the thin film flow of a third grade fluid on a moving belt. The solution is obtained in the form of a series that converges to the exact solution with easily computed components, without any restrictive assumptions for nonlinear terms. The results are bench-marked against a numerical solution based on the classical Runge–Kutta method (RK4) and an excellent agreement is observed. Error analysis of the approximate solution is performed using the error remainder and the maximal error remainder. An exponential rate for the convergence is achieved. A symbolic manipulator Mathematica ®10 was used to evaluate terms in the iterative process.

A SEMI ANALYTICAL ITERATIVE TECHNIQUE FOR SOLVING DUFFING EQUATIONS

This paper presents new implementation of reliable iterative method proposed by Temimi and Ansari namely (TAM) for analytic and numerical solutions of Duffing equations in two types: damping and undamping equations. The solution is obtained in the series form that converge to the exact solution with easily computed components. Several problems are solved to demonstrate the efficiency of the proposed methods without any restrictive assumptions for nonlinear terms. However, for some examples numerical calculations evidenced by tables and figures for the analysis of the maximal error remainder values.The software used for the calculations in this study was MATHEMATICA r 9.0.

The Laguerre-and-sums-of-powers algorithm for the efficient and reliable approximation of all polynomial roots

We prove the first sufficient convergence criterion for Laguerre's root-finding algorithm, which by empirical evidence is highly efficient. The criterion is applicable to simple roots of polynomials with degree greater than 3. The "Sums of Powers Algorithm" (SPA), which is a reliable iterative root-finding method, can be used to fulfill the condition for each root. Therefore, Laguerre's method together with the SPA is now an efficient and reliable algorithm (LaSPA). In computational mathematics these results solve a central task which was first attacked by L. Euler 266 years ago.

A reliable iterative method for solving Volterra integro-differential equations and some applications for the Lane–Emden equations of the first kind

A reliable iterative method for solving Volterra integro-differential equations and some applications for the Lane–Emden equations of the first kind

A reliable iterative method for solving the epidemic model and the prey and predator problems

In the present article, we implement the new iterative method proposed by Daftardar-Gejji and Jafari (NIM) [V. Daftardar-Gejji, H. Jafari, An iterative method for solving nonlinear functional equations, J. Math. Anal. Appl. 316 (2006) 753-763] to solve two problems; the flrst one is the problem of spread of a non-fatal disease in a population which is assumed to have constant size over the period of the epidemic, and the other one is the problem of the prey and predator. The results demonstrate that the method has many merits such as being derivative-free, overcome the di‐culty arising in calculating Adomian polynomials to handle the nonlinear terms in Adomian Decomposition Method (ADM), does not require to calculate Lagrange multiplier as in Variational Iteration Method (VIM) and no needs to construct a homotopy as in Homotopy Perturbation Method (HPM). The results obtained are compared with the results by existing methods and prove that the presented method is very efiective, simple and does not require any restrictive assumptions for nonlinear terms. The software used for the numerical calculations in this study was MATHEMATICA r 8.0.

A reliable iterative method for solving the time-dependent singular Emden-Fowler equations

Abstract Many problems of physical sciences and engineering are modelled by singular boundary value problems. In this paper, the variational iteration method (VIM) is used to study the time-dependent singular Emden-Fowler equations. This method overcomes the difficulties of singularity behavior. The VIM reveals quite a number of features over numerical methods that make it helpful for singular and nonsingular equations. The work is supported by analyzing few examples where the convergence of the results is observed.

Iterative methods for the force-based quasicontinuum approximation: Analysis of a 1D model problem

Force-based atomistic-continuum hybrid methods are the only known pointwise consistent methods for coupling a general atomistic model to a finite-element continuum model. For this reason, and due to their algorithmic simplicity, force-based coupling methods have become a popular class of atomistic-continuum hybrid models as well as other types of multiphysics models. However, the recently discovered unusual stability properties of the linearized force-based quasicontinuum (QCF) approximation, especially its indefiniteness, present a challenge to the development of efficient and reliable iterative methods. We present analytic and computational results for the generalized minimal residual (GMRES) solution of the linearized QCF equilibrium equations. We show that the GMRES method accurately reproduces the stability of the force-based approximation and conclude that an appropriately preconditioned GMRES method results in a reliable and efficient solution method.

Design of Waveguide Bragg Gratings With Strong Index Corrugations

Waveguide Bragg grating structures with strong index corrugations are analyzed by the complex mode-matching method. As the index corrugation increases, the peak wavelength of the reflection spectrum is shifted towards shorter wavelength from the conventional Bragg condition due to the energy storage effect. It is shown that the peak wavelength corresponds to the center of the stop-band of the Floquet-Bloch mode. An efficient and reliable iterative method is proposed for design of grating structures with strong corrugations.

Efficient and reliable iterative methods for linear systems

The approximate solutions in standard iteration methods for linear systems Ax= b, with A an n by n nonsingular matrix, form a subspace. In this subspace, one may try to construct better approximations for the solution x. This is the idea behind Krylov subspace methods. It has led to very powerful and efficient methods such as conjugate gradients, GMRES, and Bi-CGSTAB. We will give an overview of these methods and we will discuss some relevant properties from the user's perspective view. The convergence of Krylov subspace methods depends strongly on the eigenvalue distribution of A, and on the angles between eigenvectors of A. Preconditioning is a popular technique to obtain a better behaved linear system. We will briefly discuss some modern developments in preconditioning, in particular parallel preconditioners will be highlighted: reordering techniques for incomplete decompositions, domain decomposition approaches, and sparsified Schur complements.

Modeling Multicomponent Reactive Chemical Transport in Nonisothermal Unsaturated/Saturated Soils - Part 2. Numerical Simulations

The set of partial differential equations developed for the simultaneous transfer of moisture, heat, and reactive chemicals under saturated/unsaturated soil conditions (Wu and Chieng, 1995) has been solved by the Galerkin finite element method. The nonlinear algebraic equations of the chemical reactions are solved by the iterative Newton-Raphson method. The chemical transport equations are formulated in terms of the total analytical concentration of each component species, and can be solved sequentially. Three methods of solving the resulting systems of linear equations are implementedthe banded matrix method for one-dimensional and small two-dimensional problems, the iterative method and the sparse matrix method for large two-dimensional and three-dimensional problems. Two efficient and reliable iterative methods are providedthe conjugate gradient method for symmetric matrix equations and the biconjugate gradient squared method for general nonsymmetric matrix equations. A typical subsurface drainage problem was solved using the developed computer program. Two simulations were run to investigate the behavior of the program. Both simulations produced reasonable and satisfactory results.