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Chebfun V4

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Chebfun V4

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30 Apr 2009 (Updated )

Numerical computation with functions instead of numbers.

Editor's Notes:

This file was selected as MATLAB Central Pick of the Week

Blowup equation (Frank-Kamenetskii)

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)

Reference:

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.

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