Image encryption scheme based on random fractional discrete cosine transform and dependent scrambling and diffusion

An image encryption scheme is proposed by combining the random fractional discrete cosine transform (RFrDCT) with the dependent scrambling and diffusion (DSD). The application of the randomization, irrational choice and vectorization of fractional orders and the randomization of generating sequences improves the key-sensitivity and thus enlarges the key space greatly. Both the locations and the values of RFrDCT transformed coefficients are changed during the stage of DSD to further enhance the security of image encryption scheme. Numerical simulations demonstrate that the proposed image encryption scheme is feasible, secure and capable of resisting common classical attacks.     

Error estimates of numerical solutions for a cyclic plasticity problem

A cyclic plasticity problem is numerically analyzed in [13], where a sub-optimal order error estimate is shown for a spatially discrete scheme. In this note, we prove an optimal order error estimate for the spatially discrete scheme under the same solution regularity condition. We also derive an error estimate for a fully discrete scheme for solving the plasticity problem.     

Numerical inversions for diffusion coefficients in two-dimensional space fractional diffusion equation

This article deals with an inverse problem of determining the diffusion coefficients in 2D fractional diffusion equation with a Dirichlet boundary condition by the final observations at the final time. The forward problem is solved by the alternating direction implicit finite-difference scheme with the discrete of fractional derivative by shift Grünwald formula and a numerical text which is to prove its numerically stability and convergence is given. Furthermore, the homotopy regularization algorithm with the regularization parameter chosen by a Sigmoid-type function is introduced to solve the inversion problem numerically. Numerical inversions both with accurate data and noisy data are carried out for the unknown diffusion coefficients of constant and variable with polynomials, trigonometric and index functions. The reconstruction results show that the inversion algorithm is efficient for the inverse problem of determining diffusion coefficients in 2D space fractional diffusion equation, and the algorithm is also numerically stable for additional date having random noises.     

On the numerical solution of fractional differential equations under white noise processes

The aim of this paper is to elucidate some relevant aspects concerning the numerical solution of stochastic differential equations in structural and mechanical applications. Specifically, the attention is focused on those differential problems involving fractional operators to model the viscoelastic behavior of the structural/mechanical components and involving a white noise process as stochastic input. Starting from the consideration that the Grünwald–Letnikov based integration scheme, that is a step-by-step procedure often invoked in literature to discretize and integrate the aforementioned differential equations, is not properly employed due to the discontinuous nature of the input, an alternative numerical integration scheme is proposed. The latter is based on the Riemann–Liouville fractional integral and relies on the parabolic piecewise approximation of the response function to be integrated, leading to a more effective and more advantageous solution than that provided by the Grünwald–Letnikov based integration scheme. This is demonstrated analyzing the case study of a fractional Euler–Bernoulli beam and comparing the numerical results with those obtained by an analytical solution available in literature.     

A dual reciprocity boundary element formulation using the fractional step method for the incompressible Navier–Stokes equations

This paper presents a dual reciprocity boundary element solution method for the unsteady Navier–Stokes equations in two-dimensional incompressible flow, where a fractional step algorithm is utilized for the time advancement. A fully explicit, second-order, Adams–Bashforth scheme is used for the nonlinear convective terms. We performed numerical tests for two examples: the Taylor–Green vortex and the lid-driven square cavity flow for Reynolds numbers up to 400. The results in the former case are compared to the analytical solution, and in the latter to numerical results available in the literature. Overall the agreement is excellent demonstrating the applicability and accuracy of the fractional step, dual reciprocity boundary element solution formulations to the Navier–Stokes equations for incompressible flows.     

Navier-Stokes方程的集中质量非协调有限元法

本文通过所谓的速度-压力型公式讨论了Navier-Stokes方程的集中质量非协调有限元法(半离散情形)。首先给出了所讨论方程的集中质量非协调有限元逼近格式,其次对所讨论方程的真解与逼近格式的解之间的误差进行了分析,最后利用Navier-Stokes投影算子及其性质,得到了在确定模意义下的速度、压力误差估计,且某些误差估计能达到最优。     

A Kansa-type MFS scheme for two-dimensional time fractional diffusion equations

In this paper, an efficient Kansa-type method of fundamental solutions (MFS-K) is extended to the solution of two-dimensional time fractional sub-diffusion equations. To solve initial boundary value problems for these equations, the time dependence is removed by time differencing, which converts the original problems into a sequence of boundary value problems for inhomogeneous Helmholtz-type equations. The solution of this type of elliptic boundary value problems can be approximated by fundamental solutions of the Helmholtz operator with different test frequencies. Numerical results are presented for several examples with regular and irregular geometries. The numerical verification shows that the proposed numerical scheme is accurate and computationally efficient for solving two-dimensional fractional sub-diffusion equations.     

Joint image compression–encryption using discrete fractional transforms

Exchange of data in the form of text and image on internet is in fast progression and it is spawning new compression and encryption algorithms for bandwidth and security respectively. We have proposed a new kind of joint algorithm using discrete fractional transforms for compression–encryption of image. In this algorithm, the discrete fractional Fourier transform which is discrete version of fractional Fourier transform, is used to compress the images with variation of its parameter ‘α’ (order of transform). The compressed image is encrypted using discrete fractional cosine transform to provide security. The advantage of this method is its feasible implementation in practice, superior, robustness, security and sensitivity of keys, which has a good prospect and practicability in information security field. Results of computer simulations are presented to verify the validity of the proposed method such as mean square error (MSE) and peak signal to noise ratio between the original image and decrypted image. Sensitivity for right decryption key is proved with respect to MSE.     

Fractional visco‐elastic Timoshenko beam deflection via single equation

This paper deals with the response determination of a visco‐elastic Timoshenko beam under static loading condition and taking into account fractional calculus. In particular, the fractional derivative terms arise from representing constitutive behavior of the visco‐elastic material. Further, taking advantages of the Mellin transform method recently developed for the solution of fractional differential equation, the problem of fractional Timoshenko beam model is assessed in time domain without invoking the Laplace‐transforms as usual. Further, solution provided by the Mellin transform procedure will be compared with classical Central Difference scheme one, based on the Grunwald–Letnikov approximation of the fractional derivative. Moreover, Timoshenko beam response is generally evaluated by solving a couple of differential equations. In this paper, expressing the equation of the elastic curve just through a single relation, a more general procedure, which allows the determination of the beam response for any load condition and type of constraints, is developed. Copyright © 2015 John Wiley & Sons, Ltd.     

A Fully Discrete Spectral Method for Fisher’s Equation on the Whole Line

A generalised Hermite spectral method for Fisher’s equation in genetics with different asymptotic solution behaviour at infinities is proposed, involving a fully discrete scheme using a second order finite difference approximation in the time. The convergence and stability of the scheme are analysed, and some numerical results demonstrate its efficiency and substantiate our theoretical analysis.     

对称正则长波方程的拟紧致守恒差分格式

本文就对称正则长波方程的初边值问题进行了数值研究,提出了一个三层线性拟紧致差分格式,该格式具有较高精度且合理模拟了初边值问题的守恒性质。文章在先验估计基础上运用能量分析方法分析了格式的稳定性及二阶收敛性。数值结果验证了格式的有效性。     

An adaptive hybrid time‐stepping scheme for highly non‐linear strongly coupled problems

This paper deals with the design and implementation of an adaptive hybrid scheme for the solution of highly non‐linear, strongly coupled problems. The term ‘hybrid’ refers to a composite time stepping scheme where a controller decides whether a monolithic scheme or a fractional step (splitting) scheme is appropriate for a given time step. The criteria are based on accuracy and efficiency. The key contribution of this paper is the development of a framework for incorporating error criteria for stepsize selection and a mechanism for choosing from splitting or monolithic possibilities. The resulting framework is applied to silylation, a highly non‐linear, strongly coupled problem of solvent diffusion and reaction in deforming polymers. Numerical examples show the efficacy of our new hybrid scheme on both two‐ and three‐dimensional silylation simulations in the context of microlithography. Copyright © 2005 John Wiley & Sons, Ltd.     

Optical watermark scheme based on singular value decomposition ghost imaging and particle swarm optimization algorithm

An optical image watermarking algorithm, based on singular value decomposition (SVD) ghost imaging and multiple transforms, is designed. The watermark image is first encrypted by applying an SVD ghost imaging system, then the encrypted watermark is embedded into the cover image with the help of multiple transforms, including lifting wavelet transform (LWT), discrete cosine transform (DCT), discrete fractional angular transform (DFAT) and SVD. Four sub-band images are produced from the host image by LWT and DCT. The improved DFAT, whose scaling factors and parameter are optimized by particle swarm optimization algorithm, is operated in the new matrix. Afterwards, SVD is executed in the two-part image and the encrypted watermark is embedded in the host image by mutual operation of different matrices. Simulation results validate that the proposed watermark scheme is superior in the aspects of security, robustness and imperceptibility.     

二维半线性伪抛物方程差分格式及数值模拟

本文讨论了一类具有较强应用背景的二维半线性伪抛物方程,设计了求解此类方程对应的初边值问题的隐式差分格式。基于离散泛函理论和先验估计证明了相应差分格式解的存在唯一性、收敛性和稳定性。最后针对实例给出了数值模拟结果。     

Stochastic response determination of nonlinear oscillators with fractional derivatives elements via the Wiener path integral

A novel approximate analytical technique for determining the non-stationary response probability density function (PDF) of randomly excited linear and nonlinear oscillators endowed with fractional derivatives elements is developed. Specifically, the concept of the Wiener path integral in conjunction with a variational formulation is utilized to derive an approximate closed form solution for the system response non-stationary PDF. Notably, the determination of the non-stationary response PDF is accomplished without the need to advance the solution in short time steps as it is required by the existing alternative numerical path integral solution schemes which rely on a discrete version of the Chapman–Kolmogorov (C–K) equation. This is accomplished by circumventing the solution of the associated Euler–Lagrange equation ordinarily used in the path integral based procedures. The accuracy of the technique is demonstrated by pertinent Monte Carlo simulations.     

抛物问题的质量集中非协调有限元法

主要讨论了一类抛物问题的质量集中非协调有限元方法。首先,我们给出了所讨论问题的质量集中非协调有限元Crank-Nicolson全离散逼近格式。其次,对讨论问题的解与所给出逼近格式的解之间的误差估计进行了分析研究。最后利用椭圆投影算子,我们得到了关于L2模和能量模方面的一些误差估计式。     

A High-Order Difference Scheme for the Generalized Cattaneo Equation

A high-order finite difference scheme for the fractional Cattaneo equation is investigated. The $L_1$ approximation is invoked for the time fractional part, and a compact difference scheme is applied to approximate the second-order space derivative. The stability and convergence rate are discussed in the maximum norm by the energy method. Numerical examples are provided to verify the effectiveness and accuracy of the proposed difference scheme.     

Image encryption based on a reality-preserving fractional discrete cosine transform and a chaos-based generating sequence

A novel image encryption technique based on a reality-preserving fractional discrete cosine transform (FrDCT) and a chaos-based generating sequence is proposed. The FrDCT is a generalization of the discrete cosine transform (DCT). This reality-preserving FrDCT inherits the virtues of both the DCT and fractional transform, providing improved security and flexibility by employing the generating sequence as an extra key in addition to the fractional orders. The most fascinating advantage of the FrDCT is its reality-preserving property, which ensures real ciphertext for real plaintext, which is conducive to display, storage, and transmission. Performance and security analysis demonstrates that this algorithm is valid, secure, sensitive to keys, and robust to noise and occlusion attacks.     

Finite element formulation of viscoelastic sandwich beams using fractional derivative operators

This paper presents a finite element formulation for transient dynamic analysis of sandwich beams with embedded viscoelastic material using fractional derivative constitutive equations. The sandwich configuration is composed of a viscoelastic core (based on Timoshenko theory) sandwiched between elastic faces (based on Euler–Bernoulli assumptions). The viscoelastic model used to describe the behavior of the core is a four-parameter fractional derivative model. Concerning the parameter identification, a strategy to estimate the fractional order of the time derivative and the relaxation time is outlined. Curve-fitting aspects are focused, showing a good agreement with experimental data. In order to implement the viscoelastic model into the finite element formulation, the Grünwald definition of the fractional operator is employed. To solve the equation of motion, a direct time integration method based on the implicit Newmark scheme is used. One of the particularities of the proposed algorithm lies in the storage of displacement history only, reducing considerably the numerical efforts related to the non-locality of fractional operators. After validations, numerical applications are presented in order to analyze truncation effects (fading memory phenomena) and solution convergence aspects.