Galerkin-ritz procedures for approximate solutions to systems of reaction-diffusion equations

✍ Scribed by Gerald Rosen

• Publisher: Springer
• Year: 1978
• Tongue: English
• Weight: 469 KB
• Volume: 40
• Category: Article
• ISSN: 1522-9602
• DOI: 10.1016/s0092-8240(78)80013-5

✦ Synopsis

For any essentially nonlinear system of reaction diffusion equations of the generic form Oci/?t = DiV~c~ + Q~(e, x, t) supplemented with Robin type boundary conditions over the surface of a closed bounded three-dimensional region, it is demonstrated that all solutions for the concentration distribution n-tuple function e=(Cl(X,t) .... ,c,(x,t)) satisfy a differential variational condition. Approximate solutions to the reaction-diffusion initial-value boundaryvalue problem are obtainable by employing this variational condition in conjunction with a Galerkin-Ritz procedure. It is shown that the dynamical evolution from a prescribed initial concentration n-tuple function to a final steady-state solution can be determined to desired accuracy by such an approximation method. The variational condition also admits a systematic Galerkin-Ritz procedure for obtaining approximate solutions to the multi-equation elliptic boundary-value problem for steady-state distributions e=c(x). Other systems of phenomenological (non-Lagrangian) field equations can be treated by Galerkin Ritz procedures based on analogues of the differential variational condition presented here. The method is applied to derive approximate nonconstant steady-state solutions for an n-species symbiosis model.