Analysis and approximation of a nonlocal equation
Ali A. Al-Shomrani · 2014
In this thesis we study a nonlocal problem for u(x,t), 0 ≤ x ≤ 1 ∂u/∂t (x,t) = f (ˆ u(x,t))−u(x,t). (*) Here f is a nonlinear function and ˆ u is an average of u given by K(x)ˆ u(x,t) =Z 1 0 J(x−y)u(y,t)dy where K(x) =Z 1 0 J(x−y)dy. We consider three different versions of the averaging function J(x). Chapter 2 and 3 contain an analytical study of equilibrium solutions to (*). Chapter 4 concentrates on a numerical approximation of the full problem given by (*). Finally, Chapter 5 studies and approximates slowly varying solutions to (*).