Fractional Differential Equations: Methods and Applications

Fractional calculus, first introduced by G. W. Leibniz in 1695, has evolved from a purely theoretical concept to a powerful tool for modeling complex natural phenomena, including polymer mechanics, anomalous diffusion, dielectric relaxation, and fractional rheology. This handbook presents a new method for solving linear and nonlinear fractional differential equations, combining the Adomian decomposition method with Laplace transforms. Applicable to both ordinary and partial differential equations, the method provides analytic and numerical solutions. To validate the approach, it is first applied to integer-order differential equations with known solutions and then extended to fractional cases. Maple and Mathematica procedures are included to facilitate practical computations. Designed for researchers, engineers, and advanced students, this book bridges theory and application, offering a comprehensive guide to modern fractional calculus and its solutions.