Analytical Investigation of Fractional Partial Differential Equations for Complex Dynamical Systems

A mathematical investigation of a time-fractional diffusion equation has been presented. The Caputo derivative was used to introduce temporal memory, while the Mittag–Leffler function provided the natural counterpart of the exponential function found in classical differential equations. Using separation of variables and Fourier expansion, an analytical solution of the fractional diffusion problem was obtained. A theorem establishing the solution representation and uniqueness was proved using the eigenfunction method. Four solved examples demonstrated the principal mathematical ideas and illustrated the transition from classical to fractional behavior. The results show that recovers the classical diffusion model, whereas allows the model to incorporate memory-dependent temporal behavior. This additional flexibility makes fractional PDEs useful for systems where classical models fail to represent observed dynamics adequately. The present work provides an analytical foundation for further investigation. Future research can focus on numerical approximation, variable-order fractional derivatives, multidimensional models, fractional reaction-diffusion equations, and parameter estimation using experimental data.

1.                Podlubny, I. (1999). Fractional Differential Equations. Academic Press. 2.                Kilbas, A. A., Srivastava, H. M., & Trujillo, J. J. (2006). Theory and Applications of Fractional Differential Equations. Elsevier. 3.                Diethelm, K. (2010). The Analysis of Fractional Differential Equations. Springer. 4.                Mainardi, F. (2010). Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press. 5.                Samko, S. G., Kilbas, A. A., & Marichev, O. I. (1993). Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach. 6.                Oldham, K. B., & Spanier, J. (1974). The Fractional Calculus. Academic Press. 7.                Metzler, R., & Klafter, J. (2000). The random walk's guide to anomalous diffusion: A fractional dynamics approach. Physics Reports, 339(1), 1–77. 8.                Gorenflo, R., Mainardi, F., Moretti, D., Pagnini, G., & Paradisi, P. (2002). Discrete random walk models for space-time fractional diffusion. Chemical Physics, 284, 521–544. 9.                Luchko, Y. (2009). Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation. Computers & Mathematics with Applications, 59, 1766–1772. 10.             Sakamoto, K., & Yamamoto, M. (2011). Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. Journal of Mathematical Analysis and Applications, 382, 426–447.