摘要：In this work, we propose a novel analytical solution approach for solving a general homogeneous time-invariant fractional initial value problem in the normal form D t α [ u ( x ‾ , t ) ] = F ( u ( x ‾ , t ) ) , 0 ≤ t < R , u ( x ‾ , 0 ) = f ( x ‾ ) , $$\begin{aligned}& D_{t}^{\alpha } \bigl[u(\overline{x},t) \bigr] = F \bigl(u( \overline{x},t) \bigr),\quad 0\leq t < R, \\& u(\overline{x},0) = f(\overline{x}), \end{aligned}$$ where D t α $D_{t}^{\alpha }$ is the Caputo fractional operator with 0 < α ≤ 1 $0<\alpha \leq 1$ . The solution is given analytically in the form of a convergent multi-fractional power series without using any particular treatments for the nonlinear terms. The new approach is taken to search patterns for compacton solutions of several nonlinear time-fractional dispersive equations, namely K α ( 2 , 2 ) $K_{\alpha }(2,2)$ , Z K α ( 2 , 2 ) $ZK_{\alpha }(2,2)$ , D D α ( 1 , 2 , 2 ) $DD_{\alpha }(1,2,2)$ , and K α ( 2 , 2 , 1 ) $K_{\alpha }(2,2,1)$ . Remarkably, the graphical analysis showed that the n-term approximate memory solutions, labeled by the memory parameter 0 < α ≤ 1 $0<\alpha \leq 1$ , are continuously homotopic as they reflect, in some sense, some memory and heredity properties.
关键词：Caputo derivative ; Fractional power series ; Fractional PDE ; Power series method
