GiNaCDE: the high-performance F-expansion and First Integral Methods with C++ library for solving Nonlinear Differential Equations

TL;DR

GiNaCDE is a C++ library that automates the F-expansion, modified F-expansion, and first integral methods to efficiently find exact traveling-wave solutions of nonlinear differential equations, including complex and parameter-dependent cases.

Contribution

The paper introduces a generalized, improved set of algorithms for solving nonlinear differential equations, implemented in a high-performance C++ library called GiNaCDE, capable of handling complex functions and parameter conditions.

Findings

Successfully solves a wide range of NLODEs and NLPDEs.

Reproduces known solutions and discovers new exact solutions.

Demonstrates high efficiency and effectiveness through examples.

Abstract

We present the algorithms for three popular methods: F-expansion, modified F-expansion, and first integral methods to automatically get closed-form traveling-wave solutions of nonlinear partial differential equations (NLPDEs). We generalize and improve the methods. The proposed algorithms are manageable, straightforward, and powerful tools providing a high-performance evaluation of the exact solutions of nonlinear ordinary differential equations (NLODEs) and NLPDEs. For differential equations with parameters, the new algorithms determine the conditions on the parameters to obtain exact solutions. The algorithms show solutions to a wide variety of NLODEs and NLPDEs, both integrable and non-integrable. It can solve NLODEs and NLPDEs that contain complex functions. The algorithms are implemented in a C++ library named GiNaCDE. The efficiency and effectiveness of the algorithms are demonstrated by some examples with the help of GiNaCDE. The output results tally with the previously known results, and in some cases, new exact traveling-wave solutions are explicitly obtained. Use of the library, implementation issues, scope, limitations, and future extensions of the software are addressed.