Solving a integral with a unknown value with known limits

2020 6 15
2 个回答
更新时间：2020 6 15
5 次查看（30 天）

0 个投票

Hello guys,
I am trying the solve the following equations in matlab to find Bg. All other variables are known.

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回答（2 个）

Walter Roberson
Walter Roberson 2020-6-15

0 个投票

The below is Maple notation, easily change to MATLAB notation with the Symbolic Toolbox
simplify([solve](LHS = 2*B*int(-B^2*((1 - xi)^2 - 2*(1 - rho__v/rho__1)*xi - 1), xi = 0 .. 1), B));
[ (1/3)
[ (1/3) / 2\
[12 \LHS rho__1 (5 rho__1 - 3 rho__v) /
[------------------------------------------------,
[ 10 rho__1 - 6 rho__v
(1/3)
(1/3) / 2\ / (1/2) \
12 \LHS rho__1 (5 rho__1 - 3 rho__v) / \I 3 - 1/
---------------------------------------------------------------, -
20 rho__1 - 12 rho__v
(1/3)
(1/3) / 2\ / (1/2) \
12 \LHS rho__1 (5 rho__1 - 3 rho__v) / \I 3 + 1/
---------------------------------------------------------------
20 rho__1 - 12 rho__v
]
]
]
]
]
The first is certain to be real-valued if the coefficients are real-valued. The other two might be real-valued for particular combinations of values, if you use the definition that x^(1/3) is exp(log(x)/3) and x is negative, then the complex part can vanish.

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Sam Joseph
Sam Joseph 2020-6-15
Okay I will test this out to see if it will work.
Thanks for the quick reply.
Sam Joseph
Sam Joseph 2020-6-15
Apologies I do not have access to symbolic toolbox. Is there a way to convert this to matlab code more easily?
Walter Roberson
Walter Roberson 2020-6-15
You can use the results:
[ 12^(1/3)*(LHS*rho__1*(5*rho__1-3*rho__v)^2)^(1/3)/(10*rho__1-6*rho__v), ...
12^(1/3)*(LHS*rho__1*(5*rho__1-3*rho__v)^2)^(1/3)*(1i*3^(1/2)-1)/(20*rho__1-12*rho__v), ...
-12^(1/3)*(LHS*rho__1*(5*rho__1-3*rho__v)^2)^(1/3)*(1i*3^(1/2)+1)/(20*rho__1-12*rho__v) ]
where LHS is the expression that appears to the left of your =

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Steven Lord
Steven Lord 2020-6-15

0 个投票

Subtract the left side of your equation from both sides to give an equation of the form 0 = someFunctionOf(B_g). Then use fzero to find a root of someFunctionOf.

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Sam Joseph
Sam Joseph 2020-6-15
Will the integral and exponential term work with someFunctionOf?
Steven Lord
Steven Lord 2020-6-15
Inside the someFunctionOf function you write you can certainly call exp and integral.

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