Non-linear system.

I have 5 equation with 5 unknown system that I am performing a steady state analysis on. It has boiled down to two equations with two unknowns but it is not pretty and I am not sure how to solve it without getting the trivial solution. It is of the following form, where the upper case denotes a variable, and lower case denotes constants:

0 = a*Y - b*A - c*Y^3 + d*Y^2*A - f*Y*A^2

0 = -g*A^3 - h*Y^2*A + k*Y*A^2 - l*Y^3 - m*A - n*Y

Any idea how I would go about doing this? Thanks

Answer 1

[Walter Roberson]

There is the trivial solution of Y=0, A=0 that is what you are probably referring to.

Y is the roots of an octic (order 8) polynomial that only uses the even powers (8,6,4,2,0), and thus could be solved as a quartic.

Z is an expression that has several occurrences of the roots of an octic that uses only the even powers. It appears to me in glancing through the expression of Z that the roots of the octic are places where Y has been substituted back in to one of the two equations. If one examines the first equation then one can see that once Y was known, A could be solved as a quadratic in Y.

It thus appears there are 8 non-trivial values for Y and 2 values of A for each of the 8 values of Y. It could be that some of the solutions will turn out not to be unique. It is not possible to tell ahead of time just looking at the form of the equation how many of the solution pairs would be real-valued.

Y = roots(

[f^3*l^2 + (l*h*d + c*(2*l*k + h^2))*f^2 + ((-3*c*d*l - 2*c^2*h)*g + k*(c^2*k - d*h*c - l*d^2))*f + g*(-k*d*c^2 + d^3*l + g*c^3 + c*h*d^2),

0,

2*n*l*f^3 + (-l*h*b + (l*m + n*h)*d + (-2*l*k - h^2)*a + 2*c*(n*k + m*h))*f^2 + ((3*c*b*l + (-3*n*c + 3*a*l)*d + 4*h*c*a - 2*m*c^2)*g - 2*((-d*l - (1/2)*c*h)*b + (1/2)*n*d^2 + ((1/2)*c*m - (1/2)*h*a)*d + c*a*k)*k)*f - 3*(c^2*a*g + ((2/3)*d*h*c - (1/3)*c^2*k + l*d^2)*b - (2/3)*((1/2)*n*d^2 + ((1/2)*c*m - (1/2)*h*a)*d + c*a*k)*d)*g,

0,

n^2*f^3 + ((-n*h - l*m)*b + n*m*d + (-2*m*h - 2*n*k)*a + m^2*c)*f^2 + (((-3*a*l + 3*n*c)*b - 2*(-(3/2)*d*n - 2*c*m + h*a)*a)*g + (-l*b^2 + (c*m - h*a + 2*d*n)*b + a^2*k + a*m*d)*k)*f + 3*g*(c*a^2*g + ((1/3)*c*h + d*l)*b^2 + (-n*d^2 + (-(2/3)*c*m + (2/3)*h*a)*d - (2/3)*c*a*k)*b - (1/3)*a*d*(k*a + d*m)),

0,

(-n*m*b - m^2*a)*f^2 + ((-3*a*b*n - 2*m*a^2)*g - k*b*(b*n + m*a))*f - (a^3*g - (-l*b^2 + (c*m + 3*d*n - h*a)*b + a^2*k + 2*a*m*d)*b)*g,

0,

-g*b^2*(b*n + m*a)])