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 0αα0 and λ>0, and has three solutions when α>α0 and λ*λλ*. 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 4.0686722336α04.0686722344. 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.

Keywords

• differential equation
• application
• multiple solutions with symmetry
• scientific computation