• CHETAN RATHOD

When it comes to solving nonlinear differential equations, which are encountered in a wide variety of fields such as applied mathematics, engineering, and physics, algebraic approaches have developed into powerful tools. With the help of these methods, it is possible to find accurate answers in a methodical manner, which is something that is not always possible when relying on more conventional procedures. The Lie group approach, symmetry analysis, and the Hirota bilinear method are all examples of methods that reduce complex problems by transforming nonlinear equations into algebraic forms that can be solved. To be more specific, the Lie group approach is a method that seeks analytical solutions by simplifying differential equations through the investigation of symmetries that exist within them. Through the use of the Hirota method, nonlinear equations are transformed into bilinear form, which makes it simpler to construct solutions for many soliton problems. There are a number of algebraic approaches that can be utilized to determine integrability and to rank equations according to their solvability. One of these methods is the Painlevé analysis. When it comes to dealing with nonlinear phenomena such as wave propagation, fluid dynamics, or population dynamics, these approaches really shine. Algebraic methods are useful for generating numerical or approximate answers in situations where exact solutions are not attainable. These methods give light on the stability and qualitative behavior of solutions, which helps to generate solutions. Algebraic approaches have increased our ability to utilize nonlinear differential equations in a wide variety of engineering and scientific domains. This is because these methods provide a solid foundation for solving and comprehending these problems.

Riccati Equation, Abel Equatns, Cauchy-Kowalewski Theorem, Cauchy-Kowalski System, Unival Cauchy-Kowalski System.

"ALGEBRAIC TECHNIQUES IN NON LINEAR DIFFERENTIAL EQUATIONS", International Journal of Emerging Technologies and Innovative Research (www.jetir.org), ISSN:2349-5162, Vol.11, Issue 10, page no.f666-f673, October-2024, Available :http://www.jetir.org/papers/JETIR2410571.pdf

"ALGEBRAIC TECHNIQUES IN NON LINEAR DIFFERENTIAL EQUATIONS", International Journal of Emerging Technologies and Innovative Research (www.jetir.org | UGC and issn Approved), ISSN:2349-5162, Vol.11, Issue 10, page no. ppf666-f673, October-2024, Available at : http://www.jetir.org/papers/JETIR2410571.pdf