Convergence and stability analysis of the θ-method for delayed diffusion mathematical models

摘要

A number of nonlinear phenomena in many branches of the applied sciences and engineering are described in terms of delay differential equations, which arise when the evolution of a system depends both on its present and past time. In this work a θ-method is proposed to treat mixed problems for delay reaction–diffusion equations. The conditions so that the proposed reaction–diffusion model is asymptotically stable is studied. The numerical stability of the proposed scheme in study is analysed via the spectral radius condition and then a necessary and sufficient conditions so that our scheme is asymptotically stable in both cases, when θ∈[0,1/2) and when θ∈[1/2,1] is presented. The consistence and convergence are also studied. Numerical examples to validate the effectiveness of the method are included.