Solving Bratu's problem via the Chebyshev collocation method and non-polynomial spline functions with residual error

This paper investigates two numerical techniques for solving Bratu's problem: the Chebyshev polynomial method (CPs) with Gauss-Lobatto points and the non-polynomial spline function method (NPSFs). The main goal of the proposed approaches is to transform Bratu's equation with its associated mixed boundary conditions into matrix forms. Additionally, convergence analysis and error bounds are established for both methods. A residual error correction technique is used to improve the accuracy of the numerical results, and graphical illustrations are provided to examine the behavior of the solutions. The results demonstrate that the proposed approaches are reliable and efficient tools for solving nonlinear boundary value problems, such as Bratu's problem. All computations and graphical representations were performed using Python.