Lower and Upper Bounds of the Solution of Bratu-Like Equation Arising in Electrospinning

The Bratu equation, which is a basic one-dimensional model in combustion theory was introduced and solved by Bratu in 1914. The closed-form solution of this generalized equation when λ(t) is a function of t was also provided under certain conditions on the function λ(t). Since then, this equation along with its variations and extensions, has garnered significance and found diverse applications. One such equation is Bratu-like equation, which arises in electrospinning and vibration-electrospinning processes when λ(u) is represented as a function of the dependent variable u. This development complicates this new model and making it impossible to obtain explicit or implicit solutions. So, it may be useful to address this challenge by deriving the closed-forms of lower and upper bounds of the solutions, which we present in this work. This approach can serve as a practical method for verifying the accuracy of numerical solutions obtained through various numerical methods.