A Unified Framework Based on the Conformable Formable Transform for Linear and Nonlinear Fractional Equations

The conformable formable transform is a new integral transform that combines the computational ease of the conformable fractional derivative with the analytical flexibility of the formable transform. The proposed transform is rigorously established in terms of its fundamental features, operational principles, and derivative formulas. Additionally, a unified framework for analytical studies in conformable fractional calculus is provided by deriving a duality relationship among the conformable formable transform, the traditional formable, and Laplace transforms. Inspired by these theoretical advances, the suggested transform is integrated with the Adomian decomposition method to create a novel computational scheme known as the conformable formable decomposition method. Applications to linear and nonlinear conformable fractional differential equations, such as the Newell–Whitehead–Segel equation, demonstrate the applicability and consistency of the suggested method within the tested equation classes. The resulting series solutions show rapid convergence, and exact analytical answers are found in closed form. The results show that a wide class of conformable fractional ordinary and partial differential equations that arise in mathematical physics and applied mathematics may be solved effectively and accurately using the suggested transform and the related decomposition technique.