Convection-diffusion

Abstract

Considering only diffusion-convection processes, i.e. the equation

$$ u_t = (a(u))_{xx} + (b(u))_{x}, $$

((5.1))

the integral equation (1.9) reduces to the simple identity θ (s) = σ s + b(s). Moreover, by Lemma 2.40 any such ‘solution’θ satisfies the integrability condition in an interval [0, δ) if and only if it is positive in (0, δ ). The search for nonnegative solutions of the integral equation satisfying the integrability condition is therefore reduced to the search for a- such that σ s + b(s) > 0 for all 0 < s < δ for some 0 < δ ≤ ℓ. This leads readily to the next result.