Diffusion equation backward

Carasso AS, Sanderson JG, Hyman JM (1978) Digital removal of random media image degradations by solving the diffusion equation backwards in time. SIAM J Numer Anal 15 344-367... 

A major complication exists for constructing the Lagrangian density of a pair of particles diffusing relative to each other. The diffusion (Euler) equation is dissipative and the density of the diffusing species is not conserved. The Euler density, p, would lead to a space—time invariant, Sfr, which would not be constant. This difficulty requires the same approach as that used to handle the Schrodinger equation. Morse and Feshbach [499] define a reverse or backward diffusion equation where time goes backwards compared with that in eqn. (254)... 

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... 

Atoms are diffusing into the boundary laterally from its edges and can diffuse out through its front face into the forward grain. At the same time, atoms will be deposited in the backward grain in the wake of the boundary. In the quasi-steady state in a coordinate system fixed to the moving boundary, the diffusion flux in the forward grain is J = — DXL(dc/dx) — vc and the diffusion equation is... 

Figure8-1 Space-time grid for the one-dimensional diffusion equation, evidencing the explicit forward-difference, implicit backward-difference and C rank-Nicholson discretization schemes.

To obtain an algorithm that is unconditionally stable, we consider an implicit discretization scheme that results from using backward finite-differences for the time derivative. The corresponding difference equation is most conveniently obtained by approximating the diffusion equation at point (Xj,tn+i) ... 

INDICATES RESULTS OF BACKWARDS INTEGRATION OF DIFFUSION EQUATION... 

The anisotropic convection-diffusion equation (3.311) can be written in the form of a backward equation, dpjdt = Lp, if... 

When the backward ET from the acceptor to donor is ignored, i.e. for irreversible reactions, eqn (12.49) and eqn (12.50) become a single reaction-diffusion equation ... 

In this section, the solution to the backward diffusion equation for S2(r, t) using two different types of boundary conditions for both neutral and charged species is presented. These solutions will then be used in the next section to demonstrate the link between the bulk reaction rate and the pair survival probability. Before presenting... 

In order to And the expression for the reaction probability of two neutral particles it is necessary to return to the backward diffusion equation... 

The time-dependent backward diffusion equation for the reaction probability of ions [29, 30] is known to be... 

Solution to the conditioned backward diffusion equation Starting with the time dependent geminate recombination probability for ions and converting to a dimensionless coordinate system such that... 

To generate the required exit times on diffusing from one slice to the next (Fig. 4.6), the backward diffusion equation must be solved for the survival probability S2(r, t) subject to an absorbing inner and outer boundary. The complete derivation is presented in the Sect. A.2 of the Appendix, with the final expression for (r, s) found to be... 

To calculate the probability of moving up or down a slice for a diffusion process one needs to solve the backward diffusion equation subject to the boundary conditions p B ) = 1 and p(A ) = 0 (i.e. the distance of the radical pair moves one slice further apart). The complete derivation is presented in Sect. A.3 of the Appendix, with the final exit probability found to be... 

Unlike diffusion controlled reactions, where reaction takes place as soon as the interparticle separation hits the boundary a partially diffusion controlled reactions involve an extra complexity, such that the probability of reaction must be calculated based on the surface reactivity. This probability can be calculated by solving the backward diffusion equation to And the survival probability (a, B ) on going from boundary a to B (defined as a + 5) subject to a radiation boundary condition at surface a (situation 2 as shown in Fig. 4.7). Using the boundary condition (a) = v/D )p a) and Q. B ) = 1, the survival probability Q(a, B ) is found to be... 

To generate the exit probability between slices, the steady state backward diffusion equation of the form... 

As the reaction scheme contains charged species with a Coulomb potential, generating an analytical recombination time is not possible since the backward diffusion equation for ions cannot be solved in closed form (as discussed in Chap. 4 of this work). However, for high-permittivity solvents such as water, an excellent approximation has been developed [6] which scales the encounter radius and initial separation... 

In an attempt to model the scavenging kinetics, another approach was formulated in which a killing term was introduced into the backward diffusion equation as... 

The backward diffusion equation for the survival probability (S2(f)) to describe this problem can be written as... 

Using the method of separation of variables, the solution to the backward diffusion equation employing the boundary condition as given in Eq. (9.11) is found to be... 

This section extends the outer radiation boundary condition in the previous model to include geminate reaction within the micelle. In this model a single particle diffuses with a sink at the centre and an outer radiation boundary. The required initial condition is S2(( = 0) = 1, inner boundary S2(r = a) = 0 and outer boundary condition — wS2(r = R). Solving the backward diffusion equation with these boundary conditions gives the analytical expression for the survival probability to be... 

In order to decipher whether the mean reaction time corresponds to a reaction with the surface or geminate recombination, it is necessary to calculate the probabilities of each event. Using a similar strategy as before, the steady state backward diffusion equation must be solved subject to the boundary conditions ... 

Figure 4. The Brownian ratchet model of lamellar protrusion (Peskin et al., 1993). According to this hypothesis, the distance between the plasma membrane (PM) and the filament end fluctuates randomly. At a point in time when the PM is most distant from the filament end, a new monomer is able to add on. Consequently, the PM is no longer able to return to its former position since the filament is now longer. The filament cannot be pushed backwards by the returning PM as it is locked into the mass of the cell cortex by actin binding proteins. In this way, the PM is permitted to diffuse only in an outward direction. The maximum force which a single filament can exert (the stalling force) is related to the thermal energy of the actin monomer by kinetic theory according to the following equation ...

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