Solving system of nonlinear equation with boundary condition

Hello , I have a sys tem of 3 non linear equation for x, y, z with the boundary conditions. How can I solve them simultaneously please?

a = 1 ; b = 1 ; c = 1 ;
P = 4, Q = 10, R = 3;
% syms x y z

eq1 = ( x^2/(a^2 + P)) + ( y^2/(b^2 + P)) + ( z^2/(c^2 + P)) -1 == 0;  for  P > -c^2
eq2 = ( x^2/(a^2 + Q)) + ( y^2/(b^2 + Q)) + ( z^2/(c^2 + Q)) -1 == 0;  for  -c^2 < Q < -b^2
eq3 = ( x^2/(a^2 + R)) + ( y^2/(b^2 + R)) + ( z^2/(c^2 + R)) -1 == 0 :  for -b^2 < R < -a^2
    

Thanks.

[az]

Sorry the parameter will not be the same

a = 1 ; b = 5 ; c = 30 ;
P = 4; Q = 10; R = 3;
 syms x y z
% function F = root3d(x)

eq1 = ( x^2/(a^2 + P)) + ( y^2/(b^2 + P)) + ( z^2/(c^2 + P)) -1 == 0; % for  P > -c^2
eq2 = ( x^2/(a^2 + Q)) + ( y^2/(b^2 + Q)) + ( z^2/(c^2 + Q)) -1 == 0;  % for  -c^2 < Q < - b^2
eq3 = ( x^2/(a^2 + R)) + ( y^2/(b^2 + R)) + ( z^2/(c^2 + R)) -1 == 0 ;  % for -b^2 < R < - a^2

[az]

the correction in the boundary condition. the 3 equation are basically same it is the boundary condition that will determine what will be the value of x, y and z.

a = 1 ; b = 5 ; c = 30 ;
P = 4; Q = 10; R = 3;
 syms x y z
% function F = root3d(x)

eq1 = ( x^2/(a^2 + P)) + ( y^2/(b^2 + P)) + ( z^2/(c^2 + P)) -1 == 0; % for  P > -c^2
eq2 = ( x^2/(a^2 + Q)) + ( y^2/(b^2 + Q)) + ( z^2/(c^2 + Q)) -1 == 0;  % for  -c^2 > Q > -b^2
eq3 = ( x^2/(a^2 + R)) + ( y^2/(b^2 + R)) + ( z^2/(c^2 + R)) -1 == 0 ;  % for -b^2 > R > -a^2

Answer 1

[darova]

Use isosurface to visualize roots

Use if .. else statement to choose parameter

if P > -c^2
    T = P;
elseif -c^2 < Q && Q < -b^2
    %% -----
    
eq1 = ( x^2/(a^2 + T)) % ... 

[az]

Thank you darova. the problem is treaky.

Can you suggest please what is the 'Solve' command I can use for eq1?

[darova]

What do you want to find? What variable is uknown?

[az]

I have values for a,b,c and P, Q, R.

I want to find values of x, y and z for the boundary condition

for P > -c^2

-c^2 > Q > -b^2

-b^2 > R > -a^2

the P, Q , R are the coordinate values in ellipsoidal coordinate and x, y, z are cartisian coordinate. so I just want to transform coordinate (p, q, R) to (x, y, z).

There are function like 'linsolve' , is there any for nonlinear solve?

Thank you.