We are concerned with the conventional Chebyshev collocation method (ChC) as well as with the MATLAB-object oriented implementation of this method as Chebfun, in order to solve the following elliptic nonlinear b. v. p.
\begin{equation}
\left\{
\begin{array}{c}
\Delta u+\lambda \exp \left( \frac{au}{a+u}\right) =0,\ x\in \Omega , \\
u=0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in \partial \Omega ,%
\end{array}%
\right. (1)
\end{equation}
where \(\lambda >0,\ a>0\) and \(\Omega\) is a bounded domain from \({{\mathbb R}}^{n},\ n=1,2,3.\) When \(a\rightarrow \infty \) the problem is the so called
Bratu’s problem. For the bifurcation diagram for (1) when \(n:=1\) we refer to [1] and when \(n:=2,3\) we refer to [2].
Of special concern is to simulate numerically the relationship between the multiplicity of solutions to (1) and the dimension \(n\). We also compute the maximal value \(\lambda^*,\) of the parameter \(\lambda,\) for which the problem admits a unique solution!
First of all we compute very accurately these bifurcation values and extend our numerical results reported in [3].
Then we compute a large set of solutions for all three values of $n$ including the radial case when the problem reduces to a mixed b. v. p. attached to a second order nonlinear differential equation.
For \(n:=2\) we additionally solve the problem when the domain \(\Omega\) is the unit square.
In each case we determine the order of Newton’s method in solving the nonlinear algebraic systems of discretization and observe that the convergence of Chebfun outcomes is generally exponential and much accurate (twice as accurate) than those provided by ChC.
References
[1] Huang, S-Y., Wang, S-H.: Proof of a Conjecture for the One-Dimensional Perturbed Gelfand Problem from Combustion Theory. Arch. Rational Mech. Anal. 222, 769–825 (2016).
[2] Jacobsen, J., Schmitt, K., The Liouville-Bratu-Gelfand Problem for Radial Operators, J. Differ. Equations 184, 283–-298 (2002)
[3] Gheorghiu, C. I., Accurate Spectral Collocation Solutions to some Bratu’s Type Boundary Value Problems, arXiv:2011.13212v1