Solve a nonlinear equation

Hi, Could you please help me to solve this equation in terms of "q"

1.082 [ (q-1) log((3/4 -3q/4)/(1-3q/4))+ q*log((3q/4) *(3/4 -3q/4)] + 1.4427[-3log(1-q/2) *((q/2)  -1)+(q/2+1/2)^2 *log(q/2+1/2)- (3q/4 -1)*log(1-3q/4)- (3q/4+1/4)*log(3q/4+1/4)] + 0.7213[q*log(q/2)* ( -q-3/2 +1/q)+ (-q-1)*log(q/2 +1/2) + q*(-q/2+1/2)*log(1/2-q/2)-3*log(1-q/2)-(q-1)*log(1/2 - q/2)] -0.36 *[(q-1)^2*log(1/2 - q/2)]=0

[Walter Roberson]

log 10 or natural logarithms?

[cm]

Natural Logarithm

Let me re write it as it is shown in MATLAB command window:

1.082 *( (q-1) *log((3/4 -3*q/4)/(1-3*q/4))+ q*log((3*q/4)* (3/4 -3*q/4)) )+ 1.4427*(-3*log(1-q/2) *((q/2) -1)+((q/2+1/2)^2) *log(q/2+1/2)- (3*q/4 -1)*log(1-3*q/4)- (3*q/4+1/4)*log(3*q/4+1/4)) + 0.7213*(q*log(q/2)*( -q-3/2 +1/q)+ (-q-1)*log(q/2 +1/2) + q*(-q/2+1/2)*log(1/2-q/2)-3*log(1-q/2)-(q-1)*log(1/2 - q/2)) -0.36 *((q-1)^2*log(1/2 - q/2))

thanks,

Answer 1

[Walter Roberson]

If you assume natural logs, then there are no non-complex solutions.

The value of the expression (assuming natural logs) is negative between 0 and 1 (going to negative infinity at those two bounds.) Outside of that range, the expression is real-valued at exactly one point near -3.000138648 and at exactly one point near 1.851008154. At those two exceptional locations, the expression has positive non-zero values. The fact that there are negative values and positive values does not, however, imply that there are points at which the function is real-valued and 0, as the function is discontinuous.

The results have the same shape if log10 is used instead of ln.

[Walter Roberson]

Note to myself for my future reference:

(541/500) * (q-1) * ln((3/4-(3/4) * q) / (1-(3/4) * q))+(541 / 500) * q * ln((3/4) * q * (3/4-(3/4) * q))-(43281/10000) * ln(1-(1/2) * q) * ((1/2) * q-1)+(14427/10000) * ((1/2) * q+1/2)^2 * ln((1/2) * q+1/2)-(14427/10000) * ((3/4) * q-1) * ln(1-(3/4) * q)-(14427/10000) * ((3/4) * q+1/4) * ln((3/4) * q+1/4)+(7213/10000) * q * ln((1/2) * q) * (-q-3/2+1/q)+(7213/10000) * (-q-1) * ln((1/2) * q+1/2)+(7213/10000) * q *(-(1/2) * q+1/2) * ln(-(1/2) * q+1/2)-(21639/10000) * ln(1-(1/2) * q)-(7213/10000) * (q-1) * ln(-(1/2) * q+1/2)-(9/25) * (q-1)^2 * ln(-(1/2) * q+1/2)

[cm]

Thanks, but the point is that we have another constraint that q only should get the values between 0 and 1 since its the value of probability density! does this equation have any answer in this range?

[Matt J]

Walter has already told you that it does not.

[Walter Roberson]

Negative probability densities ??

[Matt J]

Thanks, but the point is that we have another constraint that q only should get the values between 0 and 1 since its the value of probability density!

If you really mean "probability density" then q should be required to be positive, but not less than 1. Only probability mass functions and CDFs have that requirement.

[cm]

yes I was looking for a value between 0 and 1 but it seems that the equation can not satisfy this constraint

Answer 2

[Matt J]

solve, fzero, fsolve?

[cm]

I used "solve" but the process took so long without any appropriate result! i will be so pleased if you introduce some thing that works for this equation, some stronger function

Answer 3

[Andrei Bobrov]

syms q

>> ex = 1.082*( (q-1)*log((3/4 -3*q/4)/(1-3*q/4))+ q*log((3*q/4) *(3/4 -3*q/4)) + 1.4427*(-3*log(1-q/2) *((q/2)  -1)+(q/2+1/2)^2 *log(q/2+1/2)- (3*q/4 -1)*log(1-3*q/4)- (3*q/4+1/4)*log(3*q/4+1/4)) + 0.7213*(q*log(q/2)* ( -q-3/2 +1/q)+ (-q-1)*log(q/2 +1/2) + q*(-q/2+1/2)*log(1/2-q/2)-3*log(1-q/2)-(q-1)*log(1/2 - q/2)) -0.36 *((q-1)^2*log(1/2 - q/2)));

>>solve(ex,q)
ans = 
- 3.7601288392107154402269169733539 + 0.46896675884329957486140629064384*i

Answer 4

[cagatay yilmaz]

Hello,

Could you guys help me the solve following problem. This is the dispersion equation for a symmetric lamb wave. I am looking for the z values for different f (frequencies)? I have to find something similar to Fig 1 a.

 close all

 clear all

 clc

 %cl longitudunal wave speed

 %ct transversal wave speed

 %cp phase velocity of wave

 %cg group velocity of wave

 % z=ct/cp;

 pi=3.14;

 nu=0.33;% poisson ratio

 ro=2700;%density kg/m3

 E=70e9;% elastic modulus Pa

 mu=E/(2*(1+nu)); %shear modulus

 cl=((E*(1-nu))/(ro*(1+nu)*(1-2*nu)))^(0.5);

 ct=(mu/ro).^(0.5);

 k=ct/cl;

 f=10:10:3e6;

 w=2*pi*f;

 d=(w*0.8e-3)/ct;

 fzero (@(z) (2*z.^2-1).^2*(sin(sqrt((1-z.^2))*d))*cos(sqrt((k.^2-z.^2.*d)))-(sin(sqrt  (k.^2-z.^2.*d)))*cos(sqrt((1-z.^2))*d)*(4*z.^2)*sqrt(1-z.^2)*sqrt(k.^2-z.^2),1)

[Walter Roberson]

Please start a new Question for this topic.

[cagatay yilmaz]

I have started a new question

https://www.mathworks.com/matlabcentral/answers/336873-nonlinear-tangent-trigonemetric-equation