Nonnegative Source Research Articles

Approximate general solutions to nonlinear reaction-diffusion equations

Upper and lower bounds c ̂ + = c ̂ +( x,t) and c ̂ − = c ̂ −( x,t are obtained on the solution to the generic initial-value problem for nonlinear reaction-diffusion equations of the form ∂c ∂t = D∇ 2c − gf(c) + f , where c = c( x,t) is the concentration of a reactant molecular species, φ(c) is a prescribed monotone increasing positive-definite function of c, and f = f( x,t) is a prescribed nonnegative source distribution. It follows from the bounding relation c ̂ − ⩽ c ⩽ c ̂ + that a suitably accurate approximate general solution may often be given immediately by c ̄ ≡ 1 2 ( c ̂ + + c ̂ −) . This is illustrated here for a second-order process [ i. e., φ( c) ∝ c 2] in the infinite (unbounded) x domain.