Iteration Methods with an Auxiliary Function for Nonlinear Equations

Abstract

Various iterative methods have been introduced by involving Taylor’s series on the auxiliary function <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mi>g</a:mi> <a:mfenced open="(" close=")" separators="|"> <a:mrow> <a:mi>x</a:mi> </a:mrow> </a:mfenced> </a:math> to solve the nonlinear equation <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" id="M2"> <f:mi>f</f:mi> <f:mfenced open="(" close=")" separators="|"> <f:mrow> <f:mi>x</f:mi> </f:mrow> </f:mfenced> <f:mo>=</f:mo> <f:mn>0</f:mn> </f:math> . In this paper, we introduce the expansion of <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" id="M3"> <k:mi>g</k:mi> <k:mfenced open="(" close=")" separators="|"> <k:mrow> <k:mi>x</k:mi> </k:mrow> </k:mfenced> </k:math> with the inclusion of weights <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" id="M4"> <p:msub> <p:mrow> <p:mi>w</p:mi> </p:mrow> <p:mrow> <p:mi>i</p:mi> </p:mrow> </p:msub> </p:math> such that <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" id="M5"> <r:mstyle displaystyle="true"> <r:msubsup> <r:mrow> <r:mo stretchy="false">∑</r:mo> </r:mrow> <r:mrow> <r:mi>i</r:mi> <r:mo>=</r:mo> <r:mn>1</r:mn> </r:mrow> <r:mi>p</r:mi> </r:msubsup> <r:mrow> <r:msub> <r:mrow> <r:mi>w</r:mi> </r:mrow> <r:mrow> <r:mi>i</r:mi> </r:mrow> </r:msub> </r:mrow> </r:mstyle> <r:mo>=</r:mo> <r:mn>1</r:mn> </r:math> and knots <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" id="M6"> <v:msub> <v:mrow> <v:mi>τ</v:mi> </v:mrow> <v:mrow> <v:mi>i</v:mi> </v:mrow> </v:msub> <v:mo>∈</v:mo> <v:mfenced open="[" close="]" separators="|"> <v:mrow> <v:mn>0,1</v:mn> </v:mrow> </v:mfenced> </v:math> in order to develop a new family of iterative methods. The methods proposed in the present paper are applicable for different choices of auxiliary function <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" id="M7"> <ab:mi>g</ab:mi> <ab:mfenced open="(" close=")" separators="|"> <ab:mrow> <ab:mi>x</ab:mi> </ab:mrow> </ab:mfenced> </ab:math> , and some already known methods can be viewed as the special cases of these methods. We consider the diverse scientific/engineering models to demonstrate the efficiency of the proposed methods.