Iterative Methods And Adomian's Decomposition Technique For Solving Nonlinear Equations

Nonlinear problem play a great role in modern scientific computing disciplines. Many equations derived from practical problem are always the nonlinear forms.So how to solve these nonlinear problem appropriately becomes more and more hot in research disciplines.The main content of the article is constructing iterative methods for solving nonlinear equations.The whole article contains of four chapters.In the first chapter,the background and history of nonlinear problem and iterative methods are presented.The concept which occur in many places of the article is introduced firstly.Because the main method constructing iterative methods are geometric ways and multi-step ways,some important methods constructed through these ways are given.In the second chapter,present the iterative methods which constructed using the function and derivative about only one point.Firstly,using quadratic curve approximate substitute function f,and take the next iterate through the graph of quadratic equation intersecting with the x-axis.So a irrational iterative family included a parameterλ_n is given.After using a well known approximation,a rational iterative family is proposed correspondingly.The convergence theorem prove the two iterative families are all third-order.Because quadratic curve substituting f to construct iterative method will compute the square root at every iterate step.Sometimes it is not convenient for practical use,especially such operation is no meaning in some sense and the iterative form cannot be extended to the multi-dimensional space.So a well known approximation is used to avoid the square root computing.The convergence theorem prove the iterative family is also third-order.Because the rational iterative formulae are all have f″'s computing,in order to avoid the second-derivative computing two technique are presented.Numerical examples are given at the end of the chapter.When the parameterλ_n is constant or automatic variable of iterative families compare with compare with some well-known methods.It is interesting that the iterative families with parameter automatic variable have better performance than classic third-order methods.And it don't consume additional computing of function or derivative.In the third section,present the iterative methods which constructed using the function and derivative about two points.The parabola and rational function are taken to substitute nonlinear function f.In other word,using parabola and rational function approximate substitute function f,and take the next iterate through the graph of quadratic equation intersecting with the x-axis.When using the parabola method,irrational and rational iterative families are presented. When using rational function method,a rational iterative family is proposed. Through the convergence theorem prove the two iterative families are all can have fifth-order.Because the two rational formulae all have the computing of f′(z_n), which point z_n is the Newton step.In order to avoid the computing of f′(z_n) for less computing consumption,a technique of difference quotient substitute derivative is taken.The convergence theorem prove the rational iterative family is fourth-order.Numerical examples are given at the end of the chapter and compare with some well-known methods.In the fourth chapter,Adomian decomposition method which proposed in 1980's by American mathematician Adomian discussed for solving nonlinear problem.In 2003,Abbasbandy using Adomian decomposition technique constructed a iterative family for solving nonlinear equations.In his article,he declare the family have super-cubic convergence or more higher.In this chapter, we prove the family proposed by Abbasbandy is at most third-order.And a fourth-order iterative family is presented using Abbasbandy's idea.Numerical examples are given and compare with some well-known methods.