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

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.