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As part of my 4th year individual project I am programming a series of equations into Matlab. My project heavily relies on shallow water hydrodynamics and more specifically on the article for slender-body resistance published by Mitchell in 1898 and later expanded upon by Tuck in 1966.

My problem arises with the fact that are aren't any explanations to how an integral is converted into it's final form, which I must use. The issue is that I do not even know what to call the final result. It looks like some kind of convolution, but I need help shedding some light on what I am dealing with more specifically. I have the following equation:

Cf

I have values but no functions for any of the things appearing in the equation. What I can't figure out is: what is $s'(ξ)$? I have an array for $s(x)$ and can differentiate that to get $s'(x)$ . I also have a $k(x)$ array, but I am completely clueless on how I get the ξ variable. I spoke with my supervisor and was advised that this is a dummy variable, but that doesn't help me deal with the integrals numerically. To deal with the integrals, I simply use trapezoidal rule since each of the arrays has 100 elements the error is of no consequence in this case. Any help would be highly appreciated

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    In one integral you are keeping x fixed and integrating over $\xi$. As your supervisor says, this is a dummy variable and you just need to know its range of values to sum over that range (your Trapezoid method). This will leave you with a function of x which you then integrate (sum) over its range. – 2017-02-15
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    Hi Paul, thank you very much for answering. If I understand what you're saying, I need to figure out what the limits of the inner integral are? since I am looking at a ship hull I would assume that would be along the length? But would this mean that I have to integrate the inner integral first analytically and set the limits, or substitute the ξ and then integrate. In the latter case, would ξ be an array? – 2017-02-15
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    You need to use latex. And do you know how to compute something like $f' \ast g(t) = \int_{-\infty}^\infty f'(u) g(t-u)du$ with $f(t) = e^{-t^2} \sin(t)$ and $g(t) = \frac{1}{1+t^2}$ ? – 2017-02-15
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    Unfortunately this is beyond the maths I can do. How would I do this if I don't have a function for my \f or \g? – 2017-02-15
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    I now realise that I might be able to use $s'(x)$ and $k(x)$ and input them in Matlab's built in convolution function. Do you think this will do the trick? – 2017-02-15

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