Blowup equation (Frank-Kamenetskii)
Nick Trefethen, 25 September 2010
(Chebfun example ode/BlowupFK.m)
The Frank-Kamenetskii or "spontaneous combustion" equation is the PDE du/dt = d2u/dx2 + Aexp(u). On the interval [-1,1] with zero initial and boundary conditions, solutions to this equation blow up to infinity in finite time if A is bigger than about 0.878. For smaller A, solutions converge to a steady state.
Here we compute some of these steady-state solutions, which are solutions of the ODE boundary value problem u"+A*exp(u)=0, u(-1)=u(1)=0.
N = chebop([-1 1]); N.bc = 'dirichlet'; FS = 'fontsize'; figure for A = [.2 .4 .6 .8 .87] N.op = @(u) diff(u,2) + A*exp(u); u = N\0; plot(u,'linewidth',2), grid on, hold on text(-.1,max(u)+.04,['A = ' num2str(A)],FS,14) end axis([-1 1 0 1.2]) title('Frank-Kamenetskii blowup equation',FS,16)
H. Fujita, On the nonlinear equations Del u + exp(u) = 0 and dv/dt = Del v + exp(v), Bulletin of the American Mathematical Society 75 (1969), 132-135.