Verified Computation to a Nonlinear Poisson Equation with Neumann Boundary Condition Derived from the Keller-Segel Model

We present verified computational solutions of a one-dimensional nonlinear Poisson equation with Neumann boundary condition. This equation is derived from the Keller-Segel system known as chemotactic model by steady state, and its solution curve possesses a turning point. Using the idea of the shooting method for the boundary value problems, we transform this equation into two integral equations corresponding to the initial value problems, and use Nakao´s method with local uniqueness to enclose the solutions and a bordering algorithm to treat a turning point. We describe numerical verification conditions and give some numerical results.