On a Conjecture for the One-Dimensional Perturbed Gelfand Problem for the Combustion Theory

Abstract

We investigate the well-known one-dimensional perturbed Gelfand boundary value problem and approximate the values of α0,λ* and λ* such that this problem has a unique solution when 0<α<α0 and λ>0, and has three solutions when α>α0 and λ*<λ<λ*. The solutions of this problem are always even functions due to its symmetric boundary values and autonomous characteristics. We use numerical computation to show that 4.0686722336<α0<4.0686722344. This result improves the existing result for α0≈4.069 and increases the accuracy of α0 to 10−8. We developed an algorithm that reduces errors and increases efficiency in our computation. The interval of λ for this problem to have three solutions for given values of α is also computed with accuracy up to 10−14.

The Bifurcation Curves of a Category of Dirichlet Boundary Value Problems

“…In [19], we get that DBVP (25) has exactly three solutions when α > α 0 and 4.0686722336 < α 0 < 4.068722344. Using the transformation v � u/α, one may convert problem (25) to…”