Hi,

You can use the following links:

http://www.polymath-software.com/PolymathHelp/nle.htm

https://in.mathworks.com/matlabcentral/answers/251576-how-to-solve-system-of-3-non-linear-equations-using-fsolve-provided-that-all-the-three-equations-ha

http://math.stackexchange.com/questions/191257/3-nonlinear-equations-in-3-variables-with-the-same-variable-raised-to-differen

waiting for links

I have attached them, in my previous answer.

Dear Maurya Gyanprakash 

If you are looking for the constants and the variables you need to reformulate the problem as:-

k1= a1*x^2+b1*x*y+c1*z=d1

k2= a2*x+b2*x*y^2+c2*z*x=d2

k3= a3*x+b3*x*y+c3*z=d3;

Thus the objective function becomes:-

f= k1+ k2 + k3 -d1-d2-d3

the Optimizer will change both the constants and the variables to reach the value of f to zero

Good luck

Prof. S. EL-Rabaie

Thanks Prof El-Sayed Mahmoud El-Rabaie..

You can eliminate z from and get two equations for x and y. Then you can eliminate y from these two equations and get a 4th degree polynomial equation for x. Then use a program like Mathematica to get the roots of this 4th degree polynomial equation. Then from these roots get the corresponding solutions for y and z.

Use "fsolve" in Matlab (a built-in optimization based on the Newton-Rhapson iteration method) to optimize the roots. Visit MathWorks and Matlab central for more details. Hope this help.

Assuming you know the values of all your constants a1,a2,b1,b2,etc... d2, d3.

Using Maple, gives the solution for x analytically. However if you don't have access to Maple but have access to MatLab or similar software the procedure below will give you multiple solutions. I have kept it general so you can insert any values for any of the coefficients.

First solve the following 4th Order polynomial equation for the dummy variable R. This is easily done in Matlab:

(a1^2*b2*c3^2+a1*b1*b3*c2*c3-a1*b3^2*c1*c2)*R^4

+

(-2*a1*a3*b2*c1*c3-a3*b1^2*c2*c3+a3*b1*b3*c1*c2)*R^3

+

(2*a1*b2*c1*c3*d3-2*a1*b2*c3^2*d1+a2*b1^2*c3^2-2*a2*b1*b3*c1*c3+a2*b3^2*c1^2+a3^2*b2*c1^2+b1^2*c2*c3*d3-b1*b3*c1*c2*d3-b1*b3*c2*c3*d1+b3^2*c1*c2*d1)*R^2

+

(-2*a3*b2*c1^2*d3+2*a3*b2*c1*c3*d1-b1^2*c3^2*d2+2*b1*b3*c1*c3*d2-b3^2*c1^2*d2)*R

+

b2*c1^2*d3^2-2*b2*c1*c3*d1*d3+b2*c3^2*d1^2)

=0

For Example, using a1=1, a2=2,etc... b1=1, etc... c1=1,etc... gives:

3*R^2 - 3*R - 3*d1 + d3=0

This is a simple quadratic which can be solved for R if d1,d3 are known. Once R is obtained you know "x" as follows:

x=R

Then y becomes:

(1/2)*(b3*c2*x^2+sqrt(b3^2*c2^2*x^4+4*a3*b2*c2*c3*x^3-4*a2*b2*c3^2*x^2-4*b2*c2*c3*d3*x^2+4*b2*c3^2*d2*x))/(b2*c3*x)

Then z becomes:

-(b3*x*y+a3*x-d3)/c3

Hope that was helpful.

you could refer to this source code using matlab. It solves similar equations using many methods (load control method - displacement control method - arc length method - work control method)

Code Solving Nonlinear Equations Using Matlab

Code Solving Nonlinear Equations Using Matlab

Thanks all, the equations have been solved

No idea Abolfazl Zaraki

You can get 4 exact solutions for x, by eliminating z and get 2 equations for x and y. Then you can eliminate y from these 2 equations and get a SINGLE 4th degree equation for x!

Then use the MATHEMATICA function

Solve[a x^4 + b x^3 + c x^2 + d x + e == 0, x] here a,b,c,d,e are constants in terms of a1,a2,a3,b1,b2,b3,ci,c2,c3,d1,d2,d2.

This will give you 4 EXACT solutions for x!