A new Analytical Method of General Conformable Differential Equations with Applications to Chemical Kinetics, Fractal Calculus and Vibrations

This paper introduces a novel method for transforming ordinary nth-order linear differential equations into equivalent forms using generalized conformable derivatives and integrals, with an extension to fractal derivatives. The proposed approach generalizes the concept of the fractal derivative, revealing that it can also be viewed as a θ-Derivative, as formulated by Todorov. This transformation method demonstrates that both conformable and fractal differential equations can be expressed as ordinary differential equations with variable coefficients, establishing a significant link between these generalized derivatives and traditional differential equations. By utilizing recurrence expressions derived from the Faa Di Bruno formula, this technique proves its applicability to a range of scenarios, such as generalized Bernoulli equations and separable first-order equations under specific conditions. Additionally, a novel approach for calculating Laurent polynomials based on Bell polynomials is presented. The method is further applicable to autonomous systems of linear differential equations and diagonalizable systems. Practical examples, including chemical systems, viscoelastic fluid oscillators, Fractal growth model and fractional oscillators, illustrate the utility of this technique, demonstrating its value in expanding the scope of differential equation modeling while preserving foundational connections to classical methods.