Sunday, October 16, 2016

Can We Solve a Nonlinear Equation With Many Variables?

Typically, we are able to solve system of linear or nonlinear equations which are a set of simultaneous equations (SE). Definitely, solving of a linear SE is very easy while we have to use Newton's method to solve nonlinear SEs. The common case for both of them is, to generate an equation for each variable. It means, we can solve an equation with three variables, if we have three simultaneous equations or solving of four variables needs to find a system of four simultaneous equations and so on.

Can we solve a nonlinear equation with many variables? Yes. In the special conditions, the answer is 
positive.

The purpose of this article is to present some examples which show us possibility to solve a nonlinear equation with many variables where we have a good estimation for limited domain and range of variables. The method applied is the same method stated in article of "The Generating New Probability Theorems" posted on link:


The experienced physicists and engineers have usually the true speculation of domain and range for the variables while they need to obtain precise amounts for the variables. Therefore, this method can be useful for them.
At the first, I start by a nonlinear equation with three variables then four variables and finally five variables.

Example (1): Solve Circle Equation

  When we open a calculus book, we can see the signs and footprints of Pythagoras (582 B.C – 496 B.C) everywhere. Therefore, let me start by solving of circle equation for a limited domain and range as follows:

Consider the circle equation with below domain:     

If 

           x^2 + y^2 = r^2

x, y ϵ N,            x, y ≤ 100

Then the range will be

r ϵ N,            r ≤ 141


 Now, I apply above method and get all results of "x, y" for "r" in given range. All number of results to "r" has been presented on below graph







In this case, total sum of possible answers is equal to 126. 
For instance, above graph shows us, if r = 25, 50, 75, 100 then the number of results for "x, y" are equal to 4 and if r = 65, 85 then the number of results for "x, y" are equal to 8. The results are as follows 






 Now, consider the circle equation with below domain

    

If 

           x^2 + y^2 = r^2

    0 0 0 x, y ϵ N,            x, y ≤ 1

Then the range will be

 4 r ϵ N,            r ≤ 14 1

 If I apply above method, I will generate all results of "x, y" for "r" in given range. All number of results to "r" has been presented on below graph:






In this case, total sum of possible answers is equal to 2068. 

For instance, above graph shows us, if r = 325, 425, 650,725,850,925,975 then the number of results for "x, y" are equal to 14. The results are as follows: 




Example (2): Solve Circle Equation