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In this work, we propose a novel analytical solution approach for solving a general homogeneous time-invariant fractional initial value problem in the normal formwhere D α t is the Caputo fractional operator with 0 < α ≤ 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), ZK α (2, 2), DD α (1, 2, 2), and K α (2, 2, 1). Remarkably, the graphical analysis showed that the n-term approximate memory solutions, labeled by the memory parameter 0 < α ≤ 1, are continuously homotopic as they reflect, in some sense, some memory and heredity properties. MSC: 26A33; 35R11; 35F25; 35C10; 40C15