1. What are the contributions in "Blues function method in computational physics" ?
The authors introduce a computational method in physics that goes “ beyond linear use of equation superposition ” ( BLUES ).. Their aim in this contribution is to provide a method that can assist in constructing a solution to a nonlinear differential equation ( DE ) with a source term using superposition.. Here, the authors propose a different approach.. In order to set a concrete stage for illustrating the method the authors give an example in the field of population dynamics [ 2 ] and the physics of active matter and recall the Fisher equation [ 3 ], ∂u ∂t = αu ( 1− u ) +D u ∂x2, ( 1 ) which describes the growth of a ( dimensionless ) concentration u ( x, t ) up to a certain level ( here normalized to 1 ) and its diffusion.. After the transformation: t← αt, x← √ α/D x, the authors arrive at a dimensionless representation, ∂u ∂t = u ( 1− u ) + ∂ u ∂x2 ( 2 ) The following generalization of the Fisher equation allows for convective motion [ 4 ], ∂u ∂t + ∂h ( u ) ∂x = u ( 1− u ) + ∂ u ∂x2 ( 3 ) For the function h ( u ) they consider the form h ( u ) = a0 + a1u+ a2u.. It was recently observed that when a delta-source term in the co-moving frame is added, ∂u ∂t + ku ∂u ∂x − u ( 1− u ) − ∂ u ∂x2 = 1 k δ ( x− c ( k ) t ), ( 4 ) where the function c ( k ) is chosen to match the propagation velocity self-consistently, the DE possesses the following traveling wave solution with an exponential tail [ 7 ], u ( x, t ) ≡ U ( z ) = 1, z < 0 e− z k, z ≥ 0, ( 5 ). Areas for further applications are suggested.
read more
