| In the dissertation, we analyze a class of Sobolev equation, which is widely used in the field of engineering and technology applications. Specially, it can be used as the encoder of the Heat Flow Cryptosystem, a nontraditional cryptosystem. Therefore, algorithms with higher computational efficiency and higher accuracy are required urgently in practical applications. Recently, the finite difference scheme of the IBVP is extremely studied, and it is difficult to obtain a higher accuracy with this scheme. So, the pseudo-spectral schemes with high accuracy are constructed. In addition, we propose algorithms for encryption and decryption of the cryptosystem.Firstly, some one-dimensional case is considered. By trigonometric interpolation, the governing equation is transformed into a set of ordinary differential equations. The semi-discrete scheme and fully discrete scheme are constructed correspondingly. The errors of these schemes are estimated, respectively. It is evident that the Fourier pseudo-spectral scheme is more accurate than the finite difference scheme commonly. So it is ideal for encryption of the Heat Flow Cryptosystem.Secondly, we focus on the two-dimensional model and also obtain the error estimates of the semi-discrete scheme and fully discrete scheme. This two dimensional model has a high grade in diffusivity. Hence it is more secure than one-dimensional case. In practice, it is natural to treat high dimensional information carrier, such as digital image.Finally, we propose three algorithms for encryption. The first one concentrates on the general case of this one dimensional Sobolev equation. The others are the simplified cases when terms of time derivative are constants. These algorithms have high computational efficiency, and are applicable to encrypting texts and images. In addition, we provide the sensitivity analysis and numerical simulations. From the experimental results, we can see that these algorithms are robust and efficient.
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