This paper studies the use of the Laplace transform as a key tool for solving fractional differential equations which involve non-integer derivatives and are used to model various physical phenomena such as viscoelastic materials and control systems Fractional differential equations pose significant challenges due to the complexity of fractional derivatives and integral terms making classical solution methods inefficient The methodology in the paper relies on the Laplace transform to convert fractional equations shifting to a frequency domain to the temporal domain simplifying the handling of these complex equations This approach enables precise and efficient solutions and transforms complex equations into more manageable forms The study also explores practical applications such as solving equations related to viscoelastic materials which exhibit dynamic behavior governed by fractional equations This contributes to a deeper understanding of these materials and their mathematical modeling The paper concludes that the Laplace transform offers a robust framework for solving a wide range of fractional differential equations more efficiently with significant benefits in mathematical modeling and analysis Additionally the study highlights the importance of integrating digital methods with the Laplace transform for solving complex boundary problems thereby enhancing practical applications in fields like applied mathematics and engineering