Various problems of pure and applied sciences can be studied in the unified framework of nonlinear equations. In this paper, a new family of iterative methods for solving nonlinear equations is developed by using a new decomposition technique. The convergence of the new methods is proven. For the implementation and performance of the new methods, some examples are solved and the results are compared with some existing methods.
The conceptualization and creation of diverse iterative methods for finding efficient and precisely the approximate solution of nonlinear equation,
In this section, we are going to present some new iterative methods by the help of quadrature method and basic fundamental law of calculus. Consider the following nonlinear equation:
Now, rearrange (
From (
From (
The iterative scheme which will compute the approximate root
From (
The iterative scheme which will compute the approximate root
The iterative scheme which will compute the approximate root
The iterative scheme which will compute the approximate root
The iterative scheme which will compute the approximate root
The iterative scheme which will compute the approximate root
The iterative scheme which will compute the approximate root
Take
The iterative scheme which will compute the approximate root
Algorithms
After putting the value of
The iterative scheme which will compute the approximate root
The iterative scheme which will compute the approximate root
The iterative scheme which will compute the approximate root
The iterative scheme which will compute the approximate root
The iterative scheme which will compute the approximate root
The iterative scheme which will compute the approximate root
Shah and Noor [
This section helps us to find out the convergence of Algorithm
Let
Let
We use Intel Core i5 computer with 4 GB RAM and operating system is Windows 10 (32bit). We use Maple 13 for computation and Matlab to plot the graphs. In our computation, we use the following stopping criteria:
Consider the growth of population over short periods of time by assuming that population grows continuously with time at a rate proportional to the number present at that time. Let
In this example, we consider a model in which a particle is moved on an inclined plane making angle
In this paper, the coupled system of equations with the new decomposition technique has been used to develop a family of iterative methods for solving nonlinear equations, which includes several wellknown and new methods. Technique of derivation of the iterative methods is very simple as compared to the Adomian decomposition method. This is another aspect of the simplicity. The convergence analysis of the new iterative methods has been proven. We have solved some examples and the methods are compared, which are exhibited in Tables
Numerical results for Example
Method 






NM  —  13  0.100997929685740  0.00 

Algorithm 
0.5  4  0.100997929685740  0.00 

Algorithm 
0.4  6  0.100997929685740  0.00 

Algorithm 
0.2  3  0.100997929685740  0.00 

Algorithm 
0.3  4  0.100997929685740  0.00 

Algorithm 
0.2  5  0.100997929685740  0.00 

Algorithm 
0.13  3  0.100997929685740  0.00 

Numerical results for Example
Method 






NM  —  10 



Algorithm 

4 



Algorithm 

3 



Algorithm 

4 



Algorithm 

4 



Algorithm 

2 



Algorithm 

4 



The authors declare that they have no conflicts of interest.