To decompose the solution space, we divide the space into pieces called groups.
To get a better order of convergence, we decompose the regular part of the solution, thus: mathbf{u}_{r}=tilde{mathbf{u}}_{r} + lambda_{2} mathbf{S}_{2}+ lambda_{3} mathbf{S}_{3}+cdots+lambda_{k} mathbf{S}_{k}.
Now, based on Lemmas (3.2) and (3.3), one can also decompose the solution (u(t)) as follows.
The key idea is to decompose the solution into two parts, a singular part and a regular part.
Instead of solving the equation in a single solution step we decompose the problem into two consecutive problems based on an asymptotic analysis of the physical problem where each problem is asymptotically independent of the ratio of coefficients ε="β−/β+.
Then we decompose further the solution (z x,t)) of (3.5) as (z x,t)=v x,t)+phi(x)), where (phi(x)) is the unique solution of (2.8) and (v x,t)) solves the following equation: textstylebegin{cases} v_{tt}-Delta v_{t}-Delta v+h z,z_{t} -h phi,0)=0 quadtext{in } Omega timesmathbb{R}^, v+h z,z_{t} -h phi,0 (v(x,0),v_{t}(x,0))=xi_{u}(0)- (phi(x),0 ).
By simply applying a nonoxidizing ac voltage to a platinum electrode in the same solution we decomposed the whole bulk wire in minutes, while the solution turned into a black suspension of nanoparticles.[ 24] To understand these observations, we outline the most important experimental findings.
To assess the contribution of the constraints to the solution, we decomposed the gradient component of equation 2 into two parts, one describing the data (A T W d r d ) and one describing the constraints (μ D T W c r c ).
Transmission characteristics away from the source region are deduced by decomposing the solution into creeping rays via a Watson transformation.
By decomposing the solution of a general eikonal equation as the product of two factors: the first factor is the solution to a simple eikonal equation (such as distance) or a previously computed solution to an approximate eikonal equation.
