The Fractional-Order Effect Induced by Space–Time Order Mismatch in Operator Differential Equations

This paper reports an interesting phenomenon in which fractional-order effects can be induced by the mismatch of the differential orders of space and time; that is, fractional-order effects can be induced by space–time symmetry breakage. Classical mathematical equations can be transformed into differential equations of undetermined operators. We confirmed that the presence of fractional-order operator solutions in operator differential equations is contingent upon the mismatch of differential orders of space and time, which can induce both fractional operators in the time domain and fractional operators in the space domain. The introduction of symmetry breakage and operators of space and time offers novel insights into understanding nonlocal phenomena within the space–time continuum.