Volterra formula

This problem is a bit nasty, but will illustrate the way in which the Volterra formula can be used to derive dislocation displacement fields, (a) Imitate the derivation we performed for the screw dislocation, this time to obtain the displacement fields for an edge dislocation. Use the Volterra formula (as we did for the screw dislocation) to obtain these displacements. Make sure you are careful to explain the logic of your various steps. In particular, make sure at the outset that you make a decent sketch that explains what surface integral you will perform and why. 

When we speak of mathematical models for biology, we usually refer to formulae (such as the Hardy-Weinberg theorem, or the Lotka-Volterra equations) that effectively describe some features of living systems. In our case, embryonic development is not described by integrals and deconvolutions, and the formulae of the reconstruction algorithms cannot be a direct description of what happens in embryos. There is however another type of mathematical model. The formulae of energy, entropy and information, for example, apply to all natural processes, irrespective of their mechanisms, and at this more general level there could indeed be a link between reconstruction methods and embryonic development. For our purposes, in fact, what really matters are not the formulae per se, but... 

Lotka-Volterra cycle in the (u,v) plane, whose intrinsic frequency equals tdo = /. Fig. 15.9c shows that this simple formula gives an excellent estimation for the mean frequency of the chaotic three variable system. 

In some cases, for numerical calculation of nonlinear equations, one can use a fact that fractional derivative is based on a convolution integral, the number of weights used in the numerical approximation to evaluate fractional derivatives. In addition, one can apply predictor-corrector formula for the solution of systems of nonlinear equations of lower order. This approach is based on rewriting the initial value problem (15.68) and (15.69) as an equivalent fractional integral equation (Volterra integral equation of the second kind)... 

---