How can I solve this integral equation?

Hello everyone, my name is Jose from Sevilla.

I have a function f(T) given by:

f(T)=a+bT+cT^2, where a,b and c are known numbers and T denotes Temperature.

Now I have this equation, where d is yet another constant:

I need to solve for T2, since T1 is also known. In fact, the only unknown here is T2.

I thought about using the trapz function, but I don't know how to include the T2 unknown. Any help will be greatly appreciated! (Important: I don't have the Symbolic Math Toolbox, so I can't do it symbolically. I don't have access to the Optimization Toolbox either, so fsolve and solve are ruled out, too).

Thank you!

Answer 1

[Roger Stafford]

I would suggest utilizing a little calculus here:

integral of a+b*T+c*T^2 w.r. T from T = T1 to T = T2

is equal to:

I = a*(T2-T1)+b/2*(T2^2-T1^2)+c/3*(T2^3-T1^3)

or

c/3*T2^3+b/2*T2^2+a*T2-c/3*T1^3-b/2*T1^2-a*T1-I = 0

You have said everything is known except T2, so you can express T2 as the real solution (or solutions) to:

T2 = roots([c/3,b/2,a,-c/3*T1^3-b/2*T1^2-a*T1-I]);

[Jose Lopez]

Thank you, since f(T) is always a polynomial your solution will work in my case withouth integrating f.

Never heard of root. Thank you again!

[Roger Stafford]

You need to be prepared for multiple roots from 'roots'. Presumably just one of them will be the one you want. An n-th order polynomial always yields n roots even though some of them may be complex-valued. That is not a difficulty introduced by 'roots' or matlab. It is inherent in the statement of your problem.

Answer 2

[Roberto]

One more question. I think the answer wouldn't be as straightforward in case that f(T) was of the form:

f(T) = a + b*T + c*T^2 + d/T

What would be the best approach in this case? Now the roots function wouldn't work since f(T) isn't a polynomial..

[Roger Stafford]

I disagree. ‘roots’ will still work here. The equation

f(T) = a + b*T + c*T^2 + d/T

is equivalent to the equation

c*T^3+b*T^2+(a-f(T))*T+d = 0

which can also be solved with ‘roots’ assuming f(T) is known. You will, however, get three solutions to this third degree equation.

Answer 3

[Zeeshan Salam]

how i implement the integral function of this equation sin square 6 theta dtheta