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To further enhance the security of image encryption, a new chaos-based image encryption algorithm (IEA) based on discrete Fourier transform and Joseph traversal is proposed to encrypt the plain image in both the frequency domain and space domain simultaneously. In the proposed IEA, the logistic map is used to generate the appropriate chaotic sequence, and the improved Joseph traversal is used to scramble the image in different starting positions and variable step sizes. Then, block diffusion is performed at the end. The main finding concerning the proposed IEA is that the combination of discrete Fourier transform and Joseph traversal can enhance the security of the image information, which has been validated by measuring the performance in resisting the common types of attacks. [ABSTRACT FROM AUTHOR]

New classes of evolution differential equations with the Liouville derivative in Banach spaces are studied. Equations are considered on the whole real line and are not endowed by the initial conditions. Using the methods of the Fourier transform theory, we prove the unique solvability in the sense of classical solutions for the equation solved with respect to the Liouville fractional derivative with a bounded operator at the unknown function. This allows us to obtain the analogous result for the equation with a linear degenerate operator at the fractional derivative and with a spectrally bounded pair of operators. Abstract results are applied to obtain new results on the unique solvability of systems of ordinary differential equations, boundary problems to partial differential equations, and systems of equations. [ABSTRACT FROM AUTHOR]