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GHM method for obtaining rationalsolutions of nonlinear differential equations
Hector Vazquez-Leal1, Arturo Sarmiento-Reyes2
1Facultad de Instrumentación Electrónica, Universidad Veracruzana, Cto. Gonzalo Aguirre Beltrán S/N, Xalapa, 91000 Veracruz México.
Springerplus
|June 19, 2015
Summary
The general homotopy method (GHM) provides a powerful way to find accurate rational solutions for nonlinear differential equations. This approach simplifies complex equations, offering high precision with fewer algebraic terms.
Area of Science:
- Applied Mathematics
- Numerical Analysis
- Differential Equations
Background:
- Nonlinear differential equations are fundamental in modeling complex phenomena across science and engineering.
- Existing semi-analytic and numerical methods often face challenges in achieving high precision or efficiency for these equations.
Purpose of the Study:
- To introduce and evaluate the General Homotopy Method (GHM) for obtaining rational solutions to nonlinear differential equations.
- To demonstrate the effectiveness of GHM in providing high-precision representations of nonlinear problems.
Main Methods:
- Application of the General Homotopy Method (GHM).
- Representation of nonlinear differential equations using a minimal number of linear algebraic terms.
- Comparative analysis against established semi-analytic and numerical techniques.
Main Results:
- GHM successfully generated highly accurate rational solutions for three distinct nonlinear problems.
- The method demonstrated superior precision compared to other techniques evaluated.
- GHM offers an efficient approach by simplifying nonlinear equations into linear algebraic forms.
Conclusions:
- The General Homotopy Method (GHM) is a robust and powerful tool for solving nonlinear differential equations.
- GHM provides a significant advantage in achieving high accuracy and efficiency in rational solutions.
- This method holds promise for broader applications in scientific and engineering modeling.
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