Laplacian equation

Actually, this is not really diffusion-XimiiQd, but rather Laplacian growth, since the macroscopic equation describing the process, apart from fluctuations, is not a diffusion equation but a Laplacian equation. There are some crucial differences, which will become clearer below. In some sense DLA is diffusion-limited aggregation in the limit of zero concentration of the concentration field at infinity. 

When the problem has more than one nonhomogeneous boundary condition, the principle of superposition can be used. For instance, the Laplacian equation in the region 0 < x < a and 0 < y < b is subjected to the following boundary conditions ... 

The following are the steps used in the method of separation of variables in solving the Laplacian equation. 

Substitute XY into the Laplacian equation and select a constant such that the equation with the coordinate that has... 

Put XY into the Laplacian equation So that the following is the resulting condition The variable with homogeneous boundary condition Produces the equation with the eigenfunctions. 

Since we need to obtain the concentration of the reactant as a function of the distance along the pore and in the radial direction, we need to solve a Laplacian equation for the concentration ... 

Likewise, in the M2 case, when Eq. (5.404) represents the gradient equation V 7 =0 the correspondent Laplacian equation At] =0 is also provided through its derivation... 

With no polarization of the electrode and an electrolyte of uniform concentration, we deal only with primary current distribution, which is given by the solution of the Laplacian equation ... 

To obtain the expectation value of the kinetic energy it is usefiil to first apply the Laplacian (Equation (A9.5)) to Remembering that the trial function has no angular dependence, and so any differential with respect to 0 or will be zero, confirm that... 

Equation 2.3-1 expresses the Laplacian meaning of probability. It is applicable when the number of result,s are countable and... 

Taking into account the fact that Ax — Ay — Az — h and substituting Equation (1.62) into the expression for the Laplacian ... 

Equation (6.12) cannot be solved analytically when expressed in the cartesian coordinates x, y, z, but can be solved when expressed in spherical polar coordinates r, 6, cp, by means of the transformation equations (5.29). The laplacian operator in spherical polar coordinates is given by equation (A.61) and may be obtained by substituting equations (5.30) into (6.9b) to yield... 

The laplacian operators in equation (10.23) refer to the spaced-fixed coordinates Qa with components Qxa, Qya, Qza, so that... 

The designations employed in equation (5.6) are as follows D is the EEP diffusion coefficient in its own gas V is the Laplacian operator N is the concentration of EEPs in a gaseous phase N is the concentration of parent gas K is the rate constant of EEP de-excitation by own gas v is the rate constant of EEP radiative de-excitation ro is the cylinder radius v is the heat velocity of EEPs x, r are coordinates traveling along the cylinder axis and radius, respectively. 

The difficulty in solving this equation is that when the Laplacian is written in terms of Cartesian coordinates we find that r is a function of x, y, and z,... 

In order to solve the wave equation for the hydrogen atom, it is necessary to transform the Laplacian into polar coordinates. That transformation allows the distance of the electron from the nucleus to be expressed in terms of r, 9, and (p, which in turn allows the separation of variables technique to be used. Examination of Eq. (2.40) shows that the first and third terms in the Hamiltonian are exactly like the two terms in the operator for the hydrogen atom. Likewise, the second and fourth terms are also equivalent to those for a hydrogen atom. However, the last term, e2/r12, is the troublesome part of the Hamiltonian. In fact, even after polar coordinates are employed, that term prevents the separation of variables from being accomplished. Not being able to separate the variables to obtain three simpler equations prevents an exact solution of Eq. (2.40) from being carried out. 

For an incompressible liquid (i.e. a liquid with an invariant density which implies that the mass balance at any point leads to div v = 0) the time dependency of the concentration is given by the divergence of the flux, as defined by equation (13). Mathematically, the divergence of the gradient is the Laplacian operator V2, also frequently denoted as A. Thus, for a case of diffusion and flow, equation (10) becomes ... 

---