Exact solution of time-fractional differential-difference equations: invariant subspace, partially invariant subspace, generalized separation of variables

We present how the invariant subspace method of differential equations can be extended to scalar and coupled fractional differential-difference equations, and illustrate its applicability through fractional discrete Burger’s equation, fractional discrete heat equation, fractional Toda lattice system, coupled fractional discrete Volterra equation. Then, we show how to find the partially invariant subspaces for the nonlinear difference operator that possesses at least one linear difference factor and illustrate its applicability through the fractional Lotka Volterra equation, fractional discrete KdV equation, and fractional Toda lattice equation. Also, we extend the generalized separation of variables technique to time-fractional differential-difference equation. We demonstrate its efficiency through the fractional discrete KdV equation, fractional discrete Modified KdV equation, fractional discrete Burgers equation, fractional Lotka Volterra equation, and fractional discrete heat equation. Finally, we generalize the invariant subspace method to a time-space fractional differential-difference equation, and its usefulness is illustrated through examples. The above-proposed methods provide efficient analytical techniques to derive the exact solution of time-fractional differential-difference equations in the Caputo sense or Riemann Liouville sense.