How to solve a system of equations with integrals

x1, x2 and x3 are variables. Could anyone tell me how to solve the below system of equations? Thanks!

[darova]

Try fsolve

Answer 1

[Walter Roberson]

syms x1 x2 x3 t real
assume(x1>=0 & x2>=0 & x3>=0 & t>=0)
eqn1 = simplify(int(x1*x2*t, t, x3, 0.8)) == 0.5

eqn1 = 

eqn2_part1 = simplify(int(x1*x2*t, t, x3, t))

eqn2_part1 = 

eqn2_part2 = simplify(x1+x2+t)

eqn2_part2 = 

eqn2_part3 = simplify(int(eqn2_part1 / eqn2_part2, t, 0.1, x3))

eqn2_part3 = 

eqn2 = simplify(eqn2_part3 == 0.4)

eqn2 = 

eqns = (subs([eqn1; eqn2], x3, 1-x1-x2))

eqns = 

sol = solve(eqns, 'returnconditions', true)

sol = struct with fields:
            x1: [1×1 sym]
            x2: [1×1 sym]
    parameters: [1×2 sym]
    conditions: [1×1 sym]

sol.x1

ans = 

z

sol.x2

ans = 

sol.conditions

ans = 

eqnexpr = lhs(eqns)-rhs(eqns);
char(eqnexpr)

ans = '[- (x1*x2*(25*(x1 + x2 - 1)^2 - 16))/50 - 1/2; 5*x1*x2*(x1/10 + x2/10 + (x1 + x2 - 1)^2/2 - log(x1 + x2 + 1/10)*(x1 + x2)^2 + (x1 + x2)*(x1 + x2 - 1) - 1/200) + 5*x1*x2*log(x1 + x2 + 1/10)*(x1 + x2 - 1)^2 - 4]'

Feqns = matlabFunction(eqnexpr, 'vars', {[x1, x2]})

Feqns = function_handle with value:
    @(in1)[in1(:,1).*in1(:,2).*((in1(:,1)+in1(:,2)-1.0).^2.*2.5e+1-1.6e+1).*(-1.0./5.0e+1)-1.0./2.0;in1(:,1).*in1(:,2).*(in1(:,1)./1.0e+1+in1(:,2)./1.0e+1+(in1(:,1)+in1(:,2)-1.0).^2./2.0-log(in1(:,1)+in1(:,2)+1.0./1.0e+1).*(in1(:,1)+in1(:,2)).^2+(in1(:,1)+in1(:,2)).*(in1(:,1)+in1(:,2)-1.0)-1.0./2.0e+2).*5.0+in1(:,1).*in1(:,2).*log(in1(:,1)+in1(:,2)+1.0./1.0e+1).*(in1(:,1)+in1(:,2)-1.0).^2.*5.0-4.0]

nsol = fsolve(Feqns, [1/2 1/3])

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.

nsol = 1×2

1.0756 1.0757

[Walter Roberson]

To improve on this substantially requires adding some constraints, such as constraints that x1, x2, x3 are non-negative and that x3 >= 1/10 and x3 <= 8/10

[Jingjiao Guan]

Thank you so much for your thoughtful and kind reply! I actually don’t know whether my question has a solution or not. I am trying to numerically solve a system of equations that are similar but slightly more complex than the ones I posted. I am new to Matlab, so I am still digesting your codes slowly.

[Walter Roberson]

There are other approaches as well, but their success depends upon knowing the constraints.

[Jingjiao Guan]

Thanks again! Below is my real system of equations. x3 is between 0 and 1. x4 is between 0 and 2. x1 and x2 should be real numbers. I modified your code (I only kept things I understand). Please see the code under the equations. It has been running for almost an hour without giving out result. I would really appreciate if you could tell me if it is correct or if can be accelerated.

syms x1 x2 x3 x4 t 
eqn1 = int(((x1*t+x2/t)^(-2)-1)^(-0.5), t, x3, 0.8) - x4 == 0
eqn2_part1 = 1 - (int(((x1*t + x2/t)^(-2)-1)^(-0.5), t, x3, t))^2 - t^2
eqn2_part2 = ((x1*t + x2/t)^(-2)-1)^(-0.5)
eqn2_part3 = int(eqn2_part1 / eqn2_part2, t, 0.8, x3 ) 
eqn2 = 0.75 * eqn2_part3 - 0.1 == 0
eqn3 = (x1 - 1) * x3^2 + x2 == 0
eqn4 = x3^2 + (1-x4)^2 - 1 == 0
eqns = [eqn1, eqn2, eqn3, eqn4];
sol = solve(eqns, [x1 x2 x3 x4])
sol.x1
sol.x2
sol.x3
sol.x4