MATLAB: How to solve this differential equation

differential equationsequationiteration

I want to solve the equation iteratively for rcfv. I have the value of gama = 1.399 and beta = 0.675. How can I solve this equation to get the value of rcfv.

Best Answer

This will get you two real values for ‘r_CFV’:

gama = 1.399;
beta = 0.675;
f = @(rcfv) rcfv.^(1-gama)/gama + ((gama-1)/2)*beta.^4 * rcfv.^(2/gama) - (gama+1)/2;
r_cfv(1) = fzero(f,  1)
r_cfv(2) = fzero(f, 10)
x = linspace(-20, 20);
figure(1)
plot(x, f(x))
grid

Use the Optimization Toolbox fsolve function with real and complex initial estimates to get complex values. Another option is the Symbolic Math Toolbox solve function.

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MATLAB: How to write this equation

I used y and B to make it easier to read:

num = 2*y*(r^((y-1)/y)-1)
den = (y-1)*(B^4 - r^(-2/y))
out = sqrt(num/den)

MATLAB: Getting different results between Matlab integration and Maple

Hi Zeyad,

Here is an alternative function quad() with its set tolerance to compute integration that changes the solutution a bit.

clearvars
length_shaft = 0.4;
% Determining 4 values of beta
f = 'cos(x)*cosh(x)-1';            
OPTs = optimset('tolX',1e-17);
Roots = [fzero(f,4.0, OPTs) fzero(f,7.0, OPTs) fzero(f,10.0, OPTs) fzero(f,14.0, OPTs)];   
beta = Roots./length_shaft;
% Determining Constant (T) (a value for each beta)
for ii=1:numel(beta)
B(ii) = quad(@(y) ((cos(beta(ii)*y) + cosh(beta(ii)*y)) - (((cos(beta(ii)*length_shaft) - ...
     cosh(beta(ii)*length_shaft))/(sin(beta(ii)*length_shaft) - ...
     sinh(beta(ii)*length_shaft)))*(sin(beta(ii)*y) + sinh(beta(ii)*y)))).^2, 0, length_shaft, 1e-16);
T(ii) = (B(ii)*2.5)^(-0.5);
end
format long
B = double(B)
T = double(T)

An alternative way would be for the integration part to write your own code based on Simpson's rule that gives much higher accuracy.

Good luck.

Best Answer

Related Solutions

MATLAB: How to write this equation

MATLAB: Getting different results between Matlab integration and Maple

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