system of non linear equations

Question

sepideh 2023-10-16

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评论： sepideh 2023-10-24

Hello

I am dealing with a system of two non-linear equations and I tried to solve it by 'fsolve', but the solver stopped and could not fine the answers.

Would you please help me?

I attached the files

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Answer 1

Torsten 2023-10-16

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https://ww2.mathworks.cn/matlabcentral/answers/2034179-system-of-non-linear-equations#answer_1334159

在 MATLAB Online 中打开

%solving stsyem of nonlinear equation 3.31 and 3.32 Woods paper with fsolve
% eta_l= x1 , eta_u= x2
r=2;
v=20;
F=13/9;
fun= @(x)rooted(x,r,v,F);
x0 = [0 0];
x = fsolve(fun,x0)
No solution found.

fsolve stopped because the problem appears regular as measured by the gradient,
but the vector of function values is not near zero as measured by the
value of the function tolerance.
x = 1×2
     0     0
function f = rooted (x,r,v,F)
  %UNTITLED3 Summary of this function goes here
  %   Detailed explanation goes here
  % F<Fd Fd= 2.6 I put F=1.5
  f(1)= (r*v-1)/(v*(r-1)) * ((r*v*x(1)^2-x(2)^2)/(r*v-1))^(3/2) - ...
        (v-1)/(v*(r-1)) * ((v*x(1)^2-x(2)^2)/(v-1))^(3/2) - 9 ;
  f(2)= x(2)^3/(r*v) + (v-1)/(v*(r-1)) * ((v*x(1)^2-x(2)^2)/(v-1))^(3/2) - ... 
        (r*v-1)/(r*v*(r-1)) * ((r*v*x(1)^2-x(2)^2)/(r*v-1))^(3/2) - 9*F;
end

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sepideh 2023-10-24

编辑：Torsten 2023-10-24

在 MATLAB Online 中打开

Dear Torsten

I have the similar problem. this time the non linear system is a bit different.

would you please check this too?

again the solver can not find the roots

I put different initial guess but still no results

thank you in advance

% upper layer detacheded

% plane geometry

clc

clear all

close all

%solving stsyem of nonlinear equation 3.38 and 3.39 Woods paper with fsolve

% eta_l= x1 , eta_u= x2

%%%% the condition r*v *xil^2>xiu^2> xil^2 must be held %%%

r=2;

v=0.1;

Fu= 1/( ( (1-v)/(v*(r-1)))^(1/2)-1) %eq 3.37 is used here as indicator

Fu = 0.5000

% gam > Fu

F=1.44;

fun= @(x)rooted(x,r,v,F);

x0 = [0 5];

x = fsolve(fun,x0)

No solution found. fsolve stopped because the last step was ineffective. However, the vector of function values is not near zero, as measured by the value of the function tolerance.

x = 1×2

0 1.4104

fun(x)

ans = 1×2

-7.2060 0.9307

xil= x(1)

xil = 0

xiu=x(2)

xiu = 1.4104

xi_star=((xiu^2-v*xil^2)/(1-v))^(1/2) %here is the point that lower layer goes to zero, or upper layer starts

xi_star = 1.4867

xi_plus= ((xiu^2-r*v*(xil^2))/(1-r*v))^(1/2)%here is the point that upper layer goes to zero.

xi_plus = 1.5769

% 0<xi<xi+ is the region that only lower layer is present

xi = linspace(0,xi_plus,1000);

fll= @(xi) ((xiu.^2-xi.^2)./r*v*6);

plot(xi,fll(xi),'r')

hold on

xii = linspace(xi_plus,xi_star,1000);

fl=@(xii) ((xiu^2-(v*xil^2))/(6*v*(r-1)))+(((v-1)*xii.^2)/(6*v*(r-1)));

fu=@(xii) (((r*v*xil^2)-xiu^2)/(6*v*(r-1)))+(((1-r*v)*xii.^2)/(6*v*(r-1)))+((xiu^2-(v*xil^2))/(6*v*(r-1)))+(((v-1)*xii.^2)/(6*v*(r-1)));

plot (xii,fl(xii),'r')

hold on

plot (xii,fu(xii),'b')

hold on

xiii=linspace(xi_star,xil,1000);

fuu=@(xiii) ((xil^2-xiii.^2)./(6));

plot (xiii,fuu(xiii),'b')

hold off

B =area(xi, fll(xi));

B.FaceColor = [1 0 0]; % red

B.LineWidth = 1;

hold on

C =area(xii, fu(xii));

C.FaceColor = [0 0 1];

C.EdgeColor = [0 0 1];

C.LineWidth = 1;

hold on

d =area(xii, fl(xii));

d.FaceColor = [1 0 0]; % red

d.EdgeColor = [1 0 0];

d.LineWidth = 1;

hold on

A =area(xiii, fuu(xiii));

A.FaceColor = [0 0 1];

A.EdgeColor = [0 0 1];

A.LineWidth = 1;

xlabel('Î¾')

ylabel('f(Î¾)')

legend([C B],'Upper Layer','Lower Layer')

function f = rooted (x,r,v,F)

%eq 3.38 and 3.39

% F>Fu Fu= 0.500 I put F=1.5

f(1)= (1-r*v)/(v*(r-1)) * ((x(2)^2-r*v*x(1)^2)/(1-r*v))^(3/2) - ...

(1-v)/(v*(r-1)) * ((x(2)^2-v*x(1)^2)/(1-v))^(3/2) + x(1)^3 - 9 ;

f(2)= (1-v)/(v*(r-1)) * ((x(2)^2-v*x(1)^2)/(1-v))^(3/2) - ...

(1-r*v)/(r*v*(r-1)) * ((x(2)^2-r*v*x(1)^2)/(1-r*v))^(3/2) - 9*F;

end

Torsten 2023-10-24

I don't know what the reason for non-convergence is. The equations seem to be correct - and concerning parameters r, v and F and initial values for x - I don't know if they make sense.

sepideh 2023-10-24