SECOND INITIAL–BOUNDARY VALUE PROBLEM FOR DISTRIBUTED-ORDER FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS
Abstract
In this article, second-type initial–boundary value problems for fractional-order partial differential equations are studied from a mathematical perspective. The main focus is placed on diffusion and heat conduction equations formulated with the Caputo fractional derivative, including their analytical and numerical solutions, as well as their applications in physical, engineering, and biological processes. In particular, the application of fractional-order equations to modeling real-world phenomena such as heat transfer along a rod, nanofluid flow, and substance transport in biological tissues is discussed. Second-type (Neumann-type) boundary conditions, which prescribe heat or mass flux at the boundary, are analyzed in terms of their analytical solutions, numerical methods, and physical interpretation. The presented examples and practical applications demonstrate the significant practical relevance of the theory of fractional-order differential equations.
Keywords
fractional-order partial differential equations; Caputo fractional derivative; second initial–boundary value problem; Neumann boundary conditions; heat conduction; diffusion processes; numerical methods
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