Initialized Fractional Calculus

The fractional calculus (which admits of integrals and derivatives of non-integer order) dates back almost to the origin of the better-known ordinary (integer-order) calculus, but thus far has been treated more as a mathematical curiosity than as a scientific and engineering tool. Increasingly many physical processes are found to be best described using fractional differential equations. These processes include: viscoelasticity, rheology, electrochemistry, fractal processes, and many diffusion processes. The application of the fractional calculus to scientific and engineering problems has been inhibited by difficulties that arise from the basic definitions given heretofore for integrals and derivatives of arbitrary order. These difficulties are associated with the initialization problem, which is explained below.