Analytical Solution of the Time-Fractional Schrödinger Equation via Decomposition Methods and Series Expansions - Tayfor, Montasir Salman (Recent Contributions to Physics, 2025)
Understanding quantum systems with intrinsic memory and spatial nonlocality requires mathematical models beyond the limits of classical calculus. In this work, the one-dimensional time-fractional Schrödinger equation is examined through a hybrid analytical framework combining the Aboodh transform with the Adomian Decomposition Method. This formulation enables the reconstruction of the wave function as a rapidly convergent analytical series. The fractional order (α) appears as a physically significant quantity that influences both the energy spectrum and the temporal evolution of quantum states. The theoretical outcomes are compared with optical band-gap variations observed experimentally in ZnO and Al-doped ZnO nanostructures, demonstrating that the fractional model provides a coherent correspondence between theory and measurable quantum behavior. Furthermore, the proposed approach exhibits superior stability and reduced computational effort compared with traditional Laplace and Fourier schemes, making it adaptable to a wide range of fractional quantum models..
Keywords: analytical fractional modeling; fractional Schrödinger equation; nonlocal quantum memory; Adomian decomposition method; analytical fractional modeling nechanics