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Terminal value problems of fractional linear systems with non-homegenous terms are investigated in this paper for the first time. They are equivalent to a second kind weakly singular Volterra–Fredholm integral system. Picard’s method is used to obtain a closed form solution. The exact solution is checked to satisfy the terminal value problem. Numerical solutions are provided in comparison with truncated exact ones.

A new fractional two-dimensional quadric polynomial discrete chaotic map (2D-QPDM) with the discrete fractional difference is proposed. Afterwards, the new dynamical behaviors are observed, so that the bifurcation diagrams, the largest Lyapunov exponent plot and the phase portraits of the proposed map are given, respectively. The new discrete fractional map is exploited into color image encryption algorithm and it is illustrated with several examples. The proposed image encryption algorithm is analyzed in six aspects which indicates that the proposed algorithm is superior to other known algorithms as a conclusion.