An Exact Analytical Solution to a Certain Class of Nonlinear Ordinary Differential Equations of Membrane Electrochemistry

Various boundary value problems with a small parameter at the highest derivative, i.e. singularly perturbed boundary value problems, arise in the mathematical modeling of transfer processes in electromembrane systems in the form of boundary value problems for systems of Nernst–Planck–Poisson equations. At low current densities, these problems can be solved by various order reduction methods for a certain class of nonlinear ordinary differential equations. In some cases, general solutions to the equations can be found or they can be reduced using certain methods, for example, the boundary layer function method. However, at high current densities, the known methods of asymptotic solution should be modified, as the solution to the degenerate problem does not exist on the entire interval. To address this issue, model problems that admit exact analytical solutions are used to identify the structure of the asymptotic solution, i. e. the asymptotic scale and other parameters. Besides, the exact solution of differential equations is crucial, as it allows for a thorough and complete analysis of the problem. The exact solution also acts as a benchmark for methods of approximate analytical solutions, for example, asymptotic, as well as numerical methods of solutions. The most effective method for solving high-order nonlinear equations is the order reduction method, which allows finding a particular solution. This paper proposes a method of order reduction for a specific class of nonlinear ordinary differential equations and gives examples of specific nonlinear equations and their exact solutions.