Asymptotic Stability Analysis Of Numerical Methods For Functional Partial Differential Equations

Partial functional differential equations arise from many biological, chemical, and phys-ical systems which are characterized by both spatial and temporal variables and exhibit var-ious spatio-temporal patterns. The systematic study of such equations from the dynamicalsystems and semigroups point of view began in the 70s, and considerable advance have beenachieved since them. However, partial functional differential equations can not be solved byanalytic techniques, so it is generally necessary to resort to numerical methods.It is well-known that a numerical method which is convergent in a finite interval doesnot necessarily yield the same asymptotic behavior as the underlying partial functional dif-ferential equation.In this thesis, we concentrate on the numerical stability of finite difference methods oftwo classes of partial functional equations. We first study the numerical stability property offirst order forward and backward difference methods when applied to the first class of partialfunctional differential equation and provide some sufficient and necessary conditions suchthat such numerical schemes are asymptotically stable with respect to the trivial solution. Wethen investigate the numerical stability property of Grank-Nicolson difference method whenapplied to the first class of partial functional differential equation and present some sufficientand necessary conditions for asymptotic stability. Finally, we discuss the numerical stabilityproperty of the first order forward and backward difference methods when applied to thesecond class of partial functional differential equation and provide some sufficient conditionssuch that the numerical schemes are asymptotically stable.Some numerical simulations are given to demonstrate our theoretical analysis for as-ymptotical stability of the trivial solution.