Incorporation of initial and boundary conditions

So far, the Lagrangian density for a homogenous problem (no sink or source term in the diffusion equation) has been considered, subject to the requirement that the approximate trial function, ip, can be forced to satisfy the boundary conditions. In this sub-section, these limitations are removed and the Lagrangian density for the Green s function developed. The Green s functions for the forward and backward time process satisfy the equations 

The terms in delta functions will be integrated to get the invariant 5 , which is still space- and time-independent. Hence, these two Green s functions can be combined as they can have the same value at the moment of particle creation (t = 0) and at this time only these terms are non-zero. This expression needs a little further manipulation. Following Lebedev et al. [506] on the variational principle, drop the term V ZJe- 17 VGG and include a new term of the form V (Ae fJUG G — e u4 G — e u4tG )r where f is a unit radial vector and p and A are scalar functions of r which are related directly to the boundary conditions which both G and G satisfy. Multiply the delta function term by 2. 

To show that this is indeed the Lagrangian density, take a small error in G and G, say 5G and SG, which leads to errors in G and G, VG and VG of 5G, 8G, 5VG and 8VG, respectively. Put all these errors into eqn. (261) and take the invariant to first order in the error of G and G  

This is a rather large expression However, because 5G and 5G can be made arbitrarily small, all the first few expressions in the square brackets are zero. The first two brackets are just eqns. (260b) and (260a), respectively. The second two are the boundary conditions on G and G (identical) with A effectively kuct/4nR2 if reaction is to be represented by a boundary condition and 0 = 0. The outer boundary condition is obtained by letting 0 = 0 or for the Green s function or homogenous problem, respectively, with A = 0. Finally, the last term contains (G 6G — G 5G ) , which is zero, since G or G, SG or 5G are zero, respectively at + co and 

Equation (261) for the general Lagrangian density is rather unwieldy and unlikely to be of much direct use. Instead, the steady state version (G = G ) can be written forg = /f Gdf0 

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