Study of the Differential Transform Method for Solving Volterra Integral Equations of the Third Kind (Abel Type).

Volterra integral equations of the third kind, especially those of the Abel type characterized by singular kernels and fractional-order behavior, frequently arise in the mathematical modeling of dynamic processes in energy systems such as thermal diffusion, phase-change energy storage, and heat transport. These equations pose significant challenges for traditional analytical techniques due to their nonlocal and singular nature. In this paper, we propose the Differential Transform Method (DTM) as an efficient, semi-analytical approach to address such complex integral equations. DTM systematically transforms the integral equation into a set of algebraic recurrence relations, enabling rapid convergence towards a series solution. The proposed method is applied to multiple benchmark problems involving third-kind Volterra integral equations with known exact solutions, particularly those representing heat flux and transient energy phenomena in sustainable energy systems. Detailed error analysis, including absolute and relative error computations, validates the high precision of the DTM results. The approach exhibits excellent agreement with exact solutions while requiring significantly less computational effort compared to fully numerical methods. This study further explores the implications of using DTM in renewable energy modeling, including solar thermal systems, battery thermal management, and Phase-Change Material (PCM) storage. DTM enhances system design and simulation for energy-efficient technologies by enabling accurate modeling of fractional-order systems. The method's suitability for modeling memory effects and energy delays makes it particularly useful in smart grid applications and sustainable thermal energy management. DTM emerges as a robust and computationally sustainable alternative to conventional solvers for complex integral equations. It offers a promising pathway for future research involving coupled, multidimensional, or nonlinear thermal systems and reinforces its role in the advancement of computational sustainability and renewable energy optimization. Major Findings: The study demonstrates that the Differential Transform Method (DTM) is a powerful and efficient tool for solving Volterra integral equations of the third kind, particularly those involving fractional powers and singular kernels. It produces highly accurate series solutions with rapid convergence, as confirmed through comparison with exact analytical results. This makes DTM especially valuable for modeling complex phenomena in renewable energy systems, thermal storage applications, and sustainable energy technologies, where precision and computational efficiency are critical. [ABSTRACT FROM AUTHOR]