Physically, integral and fractional order terms in equations are associated to nonlocal processes. I can think of two types of physical models associated to this type of equation. The first would be diffusion processes with random motion with long-range memory (Hurst phenomena) and jumps (Lévy flights). In this type of diffusion process the equation describing the evolution of probabilities (Fokker-Planck of forward Kolmogorov equation) would be described by a partial differential equation with fractional and integral terms.
The second class of models would be associated to nonlinear asymptotic models for dispersive waves, in particular when the dispersion relation of the system is not described by a transcendental function. See for instance the Benjamin-Ono equation that describe the dynamics of internal waves.
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