Existence and uniqueness of the p-generalized modified error function

TL;DR

This paper establishes the existence and uniqueness of the p-generalized modified error function as a solution to a nonlinear differential equation with Robin boundary conditions, and explores its convergence and special cases.

Contribution

It introduces the p-generalized modified error function, proves its existence and uniqueness, and analyzes its convergence to the p-modified error function under different boundary conditions.

Findings

Proved existence and uniqueness of the p-generalized modified error function.

Showed convergence to the p-modified error function as p varies.

Extended analysis to problems with Neumann boundary conditions.

Abstract

In this paper, the p-generalized modified error function is defined as the solution to a non-linear ordinary differential problem of second order with a Robin type condition at x=0. Existence and uniqueness of a non-negative C^\infty solution is proved by using a fixed point strategy. It is shown that the p-generalized modified error function converges to the p-modified error function defined as the solution to a similar problem with a Dirichlet condition at x=0. In both problems, for p=1, the generalized modified error function and the modified error function, studied recently in literature, are recovered. In addition, existence and uniqueness of solution to a problem with a Neumann condition is also analysed.