Functional fractional calculus

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When a new and extraordinary theory emerges, it often faces skepticism due to its divergence from conventional concepts. Fractional calculus, though not new, has remained underexplored, primarily due to its limited application to real-world problems. It stands out because it transcends ordinary differential calculus, offering solutions where existing theories fall short. This work discusses mathematical abstractions clearly, supplemented by physical and geometrical explanations, alongside practical applications in system identification, description, and control. Traditional physical laws, such as transport theory and electrodynamics, are reinterpreted within the framework of fractional calculus, addressing complexities like heterogeneity in transport, charge distribution, and material properties. The book elaborates on fractional and complex order differentiation and integration, providing mathematical, physical, and geometrical insights with examples. It explores the practical utility of local fractional differentiation in characterizing phase transitions and response irregularities. Additionally, it discusses viscoelastic experiments, fractional control designs, and circuit synthesis. The text connects classical integer order differential equations to fractional contexts, addressing initialization challenges. It presents a modern approach to solving fractional and other differential equations using the principle of action-an