how to integrate the function?

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Thomas 2021-6-13

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评论： Walter Roberson 2021-6-13

can someone help me with the integration.I tried int(f,t',o,t), but this does not seem to be working. Are there any possible ways to solve the integration. Here theta, tc, vx,w etc are constants.

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Answer 1

Walter Roberson 2021-6-13

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syms theta t__c t_prime x v__x y omega t

part1 = 1/(1+2*t_prime/t__c)

part1 =

part2 = exp(-2*((x-v__x*t_prime).^2 + y.^2)/((1+2*t_prime/t__c)*omega^2))

part2 =

part3 = exp(-2*(v__x*t_prime)^2 / ((1+2*t_prime/t__c)*omega^2))

part3 =

del_theta = theta/(2*t__c) * int(part1 * (part2 - part3), t_prime, 0, t)

del_theta =

I suspect that you are hoping for a closed form solution. It is unlikely that a closed form solution exists.

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Thomas 2021-6-13

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Hi walter,

thank you for your reponse.

I tried integrating the function as mentioned above,

syms x y t_prime t
w=0.00035;
K=0.17;
c_p=2440;
rho=789;
P=40*10^-3; 
lambda=532*10^-9;
alpha=40;
dn_dt=4*10^-4;
L=10*10^-3;
v_x=5*10^-3;
D=K/(rho*c_p);
t_c=w^2/(4*D);
theta=(dn_dt*alpha*P*L)/(lambda*K);
part1 = 1/(1+2*t_prime/t_c);
part2 = exp(-2*((x-v_x*t_prime).^2 + y.^2)/((1+2*t_prime/t_c)*w^2));
part3 = exp(-2*(v_x*t_prime)^2 / ((1+2*t_prime/t_c)*w^2));
del_theta = theta/(2*t_c) * int(part1 * (part2 - part3), t_prime, 0, t)

i guess, still matlab could not integrate the function,

the result i obtained was

3589593118749817*int(-(exp(-t_prime^2/(20000*((1156979788303063*t_prime)/1637772947977591586816 + 1156979788303063/9444732965739290427392))) - exp(-(2*(t_prime/200 - x)^2 + 2*y^2)/((1156979788303063*t_prime)/1637772947977591586816 + 1156979788303063/9444732965739290427392)))/((18014398509481984*t_prime)/3123803993182357 + 1), t_prime, 0, t))/35184372088832

Walter Roberson 2021-6-13

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You are not likely to be able to find a closed form solution.

If you split the integration into two integration, then the first one can be rewritten to have the form

syms b c t T

int(exp(t^2/(b*t + c)), t,0,T)

ans =

MATLAB cannot resolve it. Maple cannot resolve it. Mathematica cannot resolve it.

You could experiment with approximations, such as taking a taylor series of the inside of the integral and integrating that. However, in practice it leads to some really large numbers -- order 10 expansion leads to a lot of values in the 10^510 range for example.

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