Objective
To convert the PDE for a density \(u(l,t)\) as a function of the length \(l\) of individuals into a PDE for a density \(N(w,t)\) as a function of the weight \(w\) of individuals, using the transformation \(w = a l^b\).
1. Initial Equations
The PDE for the density \(u(l,t)\) as a function of length is: \[ \frac{\partial u}{\partial t} = - \frac{\partial J}{\partial l} - \frac{m}{l} u \] where the flux \(J\) is \[ J = k (L_\infty - l) u - \frac{\partial (d l u)}{\partial l}. \] Here \(k, L_\infty, c\) and \(m\) are the model parameters.
2. Transformation of Variables and Densities
We want to transform to weight using \(w = a l^b\). We can express \(l\) in terms of \(w\) as \[ l = \left(\frac{w}{a}\right)^{1/b}. \] The Jacobian of this variable transformation is \[ \frac{dw}{dl} = ab l^{b-1}. \] Densities are related by the conservation of probability, \(u(l,t) \, dl = N(w,t) \, dw\), which gives that \[ u = N \frac{dw}{dl} = N (ab l^{b-1}). \]
3. Transformation of the PDE Structure
The original PDE is a conservation law, \(\frac{\partial u}{\partial t} + \frac{\partial J}{\partial l} = - \frac{m}{l} u\). Using the chain rule, this structure transforms to \[ \frac{\partial N}{\partial t} + \frac{\partial J}{\partial w} = - \frac{m}{l} N. \] Our goal is now to express the flux \(J\) and the sink term entirely in terms of \(w, N,\) and their derivatives. The transformed sink term is \[ - \frac{m}{l} N = -m \left(\frac{w}{a}\right)^{-1/b} N = -m a^{1/b} w^{-1/b} N. \]
4. Deriving and Rewriting the Flux J(w)
We substitute \(u = N \frac{dw}{dl}\) into the original expression for \(J\): \[ J = k (L_\infty - l) N \frac{dw}{dl} - \frac{\partial}{\partial l} \left(d l N \frac{dw}{dl} \right). \] Using the identity \(l \frac{dw}{dl} = l(ab l^{b-1}) = ab l^b = bw\), the term in the derivative becomes \(c b w N\). We then apply the chain rule \(\frac{\partial}{\partial l} = \frac{dw}{dl}\frac{\partial}{\partial w}\): \[ J = k (L_\infty - l) N \frac{dw}{dl} - \frac{dw}{dl} \frac{\partial (d b w N)}{\partial w}. \] Factoring out the Jacobian \(\frac{dw}{dl}\) gives \[ J = \frac{dw}{dl} \left[ k (L_\infty - l) N - \frac{\partial (d b w N)}{\partial w} \right]. \] Now, substitute \(l = \left(\frac{w}{a}\right)^{1/b}\) and \(\frac{dw}{dl} = ab l^{b-1} = a^{1/b} b w^{(b-1)/b}\) and let \(n = 1-1/b = (b-1)/b\). This gives \[ J(w) = \left( a^{1/b} b w^n \right) \left[ k \left(L_\infty - \left(\frac{w}{a}\right)^{1/b}\right) N - \frac{\partial (d b w N)}{\partial w} \right]. \]
Next, we fully expand our expression for \(J(w)\): \[ J(w) = \left[ (a^{1/b} b k L_\infty ) w^n N - (b k w) N \right] - \left( a^{1/b} b w^n \right) \frac{\partial (d b w N)}{\partial w}. \] Expanding the derivative and distributing the pre-factor gives \[ J(w) = (a^{1/b} b k L_\infty ) w^n N - (b k w) N - (a^{1/b} d b^2) w^n N - (a^{1/b} d b^2) w^{n+1} \frac{\partial N}{\partial w}. \] Combining terms that multiply \(N\), we get our fully expanded derived flux \[ J(w) = \left[ (a^{1/b} b k L_\infty - a^{1/b} d b^2) w^n - (b k) w \right] N - (a^{1/b} d b^2) w^{n+1} \frac{\partial N}{\partial w}. \]
We would like to write this flux in the simplfied form \[ J_{target} = (A w^n - B w) N - \frac{\partial(D w^{n+1} N)}{\partial w}, \] where the new parameters \(A,B\) and \(D\) should be expressed in terms of the old parameters \(k,L_\infty\) and \(d\) and the new exponent \(n\) should be expressed in terms of the exponent \(b\). We expand this target form using the chain rule on the derivative to get \[ \begin{split} J_{target} &= (A w^n - B w) N - \left( D(n+1)w^n N + D w^{n+1} \frac{\partial N}{\partial w} \right)\\ &= \left[ (A - D(n+1)) w^n - B w \right] N - D w^{n+1} \frac{\partial N}{\partial w}. \end{split} \] By equating the coefficients of the derived form and the target form, we find the constants:
- From the \(\frac{\partial N}{\partial w}\) term: \[D w^{n+1} = (a^{1/b} c b^2) w^{n+1} \implies D = a^{1/b} c b^2.\]
- From the \(wN\) term: \[B w = (b k) w \implies B = bk.\]
- From the \(w^n N\) term: \[A - D(n+1) = a^{1/b} b k L_\infty - a^{1/b} c b^2\] implies \[\begin{split} A &= a^{1/b} b k L_\infty - a^{1/b} c b^2 + D(n+1)\\ &= a^{1/b} b k L_\infty - a^{1/b} c b^2 + (a^{1/b} c b^2)(n+1) \\ &= a^{1/b} b k L_\infty + n(a^{1/b} c b^2)\\ &=a^{1/b}\left(b k L_\infty + n c b^2\right). \end{split}\]
5. Final Result
The transformed PDE is \[ \frac{\partial N}{\partial t} = - \frac{\partial J}{\partial w} - \frac{m}{l} N. \] Substituting the rewritten flux and the transformed sink term yields the final equation: \[ \frac{\partial N}{\partial t} = - \frac{\partial}{\partial w} \left[ (A w^n - B w) N - \frac{\partial(D w^{n+1} N)}{\partial w} \right] - M w^{n-1} N, \] where the constants are defined as: \[\begin{split} n &= 1 - 1/b\\ A &= a^{1/b}\left(b k L_\infty + n c b^2\right)\\ B &= b k\\ D &= a^{1/b} c b^2\\ M &= a^{1/b}m \end{split}\]