Keywords
Abstract
This study investigates the initial and boundary value problems of distributed-order fractional partial differential equations. These equations generalize classical differential equations by incorporating fractional derivatives, which account for memory effects and non-local dynamics in complex systems. The paper analyzes the main properties of fractional derivatives, types of boundary conditions, and solution methods, including analytical and numerical approaches. Applications in heat transfer, population dynamics, viscoelastic materials, and financial systems are discussed. The results demonstrate that distributed-order fractional equations provide more accurate and flexible modeling compared to classical integer-order equations, ensuring stable and unique solutions when proper initial and boundary conditions are applied.
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