Uniformly Stable Explicitly Solvable Finite Difference Method for Fractional Diffusion Equations

A finite difference scheme for the one-dimensional space fractional diffusion equation is presented and analysed. The scheme is constructed by modifying the shifted Grünwald approximation to the spatial fractional derivative and using an asymmetric discretisation technique. By calculating the unknowns in differential nodal point sequences at the odd and even time levels, the discrete solution of the scheme can be obtained explicitly. We prove that the scheme is uniformly stable. The error between the discrete solution and the analytical solution in the discrete $l^2$ norm is optimal in some cases. Numerical results for several examples are consistent with the theoretical analysis.     

High Order Difference Schemes for a Time Fractional Differential Equation with Neumann Boundary Conditions

A compact finite difference scheme is derived for a time fractional differential equation subject to Neumann boundary conditions. The proposed scheme is second-order accurate in time and fourth-order accurate in space. In addition, a high order alternating direction implicit (ADI) scheme is also constructed for the two-dimensional case. The stability and convergence of the schemes are analysed using their matrix forms.     

Simultaneous inversion for a diffusion coefficient and a spatially dependent source term in the SFADE

This paper deals with an inverse problem of determining a diffusion coefficient and a spatially dependent source term simultaneously in one-dimensional (1-D) space fractional advection–diffusion equation with final observations using the optimal perturbation regularization algorithm. An implicit finite difference scheme for solving the forward problem is set forth, and a fine estimation to the spectrum radius of the coefficient matrix of the difference scheme is given with which unconditional stability and convergence are proved. The simultaneous inversion problem is transformed to a minimization problem, and existence of solution to the minimum problem is proved by continuity of the input–output mapping. The optimal perturbation algorithm is introduced to solve the inverse problem, and numerical inversions are performed with the source function taking on different forms and the diffusion coefficient taking on different values, respectively. The inversion solutions give good approximations to the exact solutions demonstrating that the optimal perturbation algorithm with the Sigmoid-type regularization parameter is efficient for the simultaneous inversion problem in the space fractional diffusion equation.     

Structure Preserving Algorithm for Fractional Order Mathematical Model of COVID-19

In this article, a brief biological structure and some basic properties of COVID-19 are described. A classical integer order model is modified and converted into a fractional order model with as order of the fractional derivative. Moreover, a valued structure preserving the numerical design, coined as Grunwald–Letnikov non-standard finite difference scheme, is developed for the fractional COVID-19 model. Taking into account the importance of the positivity and boundedness of the state variables, some productive results have been proved to ensure these essential features. Stability of the model at a corona free and a corona existing equilibrium points is investigated on the basis of Eigen values. The Routh–Hurwitz criterion is applied for the local stability analysis. An appropriate example with fitted and estimated set of parametric values is presented for the simulations. Graphical solutions are displayed for the chosen values of (fractional order of the derivatives). The role of quarantined policy is also determined gradually to highlight its significance and relevancy in controlling infectious diseases. In the end, outcomes of the study are presented.     

Asymmetric hybrid encryption scheme based on modified equal modulus decomposition in hybrid multi-resolution wavelet domain

In this paper, an asymmetric hybrid encryption scheme, using coherent superposition and modified equal modulus decomposition in a hybrid multi-resolution wavelet, is proposed. The hybrid multi-resolution wavelet is generated using fractional Fourier transform of multiple orders and Walsh transform. The fractional orders of the fractional Fourier transform increase the key space and hence provide additional strength to the cryptosystem. The designed scheme has a large key space to avoid brute-force attack and is non-linear in nature. The scheme is validated on grey-scale images. Computer-based simulations have been performed to verify the validity and performance of the proposed scheme against various attacks. Scheme's robustness to the special attack is also checked. Results show that single equal modulus decomposition with fractional Fourier transform and hybrid transform are vulnerable to the special attack, whereas the proposed scheme endures the special attack.     

Flow past an impulsively started circular cylinder at high Reynolds number

The unsteady viscous incompressible flow around a circular cylinder is studied numerically for high Reynolds number. The two dimensional Navier-Stokes equation in stream function-vorticity formulation is solved. A fractional step method is used to solve the stream function equation while the vorticity transport equation is solved using a third order upwinding scheme. The computed solutions by the finite difference method agree reasonably well with the available experimental and other computational results at a Reynolds number of 9500 and 4500. These comparisons are for the initial stages of flow evolution when the wake bubble is symmetric.     

Fractional M-band dual tree complex wavelet transform for digital watermarking

In this paper, a novel digital watermarking scheme using fractional M-band dual tree complex wavelet transform (Fr-M-band-DT-CWT) is proposed. High frequency channels have wide bandwidth and low frequency channels have narrow bandwidth. These characteristics are suitable for analysing low frequency signal, but not for relatively high frequency signal. The images often contain many edges, which may cause rich middle and high frequency components in the 2-band wavelet domain. Therefore, the ordinary 2-band dual tree complex wavelet transform (DT-CWT) is not well-suited for analysing the image. So, the M-band DT-CWT with the FrFT called Fr-M-band-DT-CWT is proposed in this paper to address this problem. Further, we integrate the Fr-M-Band-DT-CWT with singular value decomposition (SVD) in order to enhance the performance. Experimental results of the proposed watermarking scheme are compared with the previously available watermarking algorithms, fractional Fourier transform (FrFT), fractional wavelet transform (FrWT). Further, the proposed watermark extraction scheme is also tested on different attacks. The results of the present investigations show that the proposed watermarking scheme is superior as compared to other existing watermarking schemes.     

Asymmetric encryption algorithm for colour images based on fractional Hartley transform

This paper presents a new scheme for encryption of single-channel colour images. The scheme uses amplitude- and phase-truncation approach to introduce non-linearity for enhanced security. Further, the colour image encryption is performed in the fractional Hartley domain, which is relatively less investigated. The encryption starts with an affine transform of each channel of the input colour image. Thereafter, one of the channels is considered as the input amplitude image while the other two are used as phase masks, one in the spatial and the other in the frequency domain. A detailed analysis of the scheme’s sensitivity to various encryption parameters has been carried out. In addition, security analysis of the scheme against attacks establishes the scheme’s robustness. The combined use of the affine transform and phase-truncation approach for colour image encryption in the fractional Hartley domain is attempted for the first time in this study. It is shown that the proposed scheme resists the special attack.     

带有非奇异核分数阶差分方程的比较原理及解的不确定性分布

比较原理对于研究分数阶方程的性质具有重要的作用。首先，证明了 ABR 型分数阶差分方程的比较原理。其次，利用比较原理，建立了不确定性分数阶差分方程的解与其 $\alpha$-路径之间的联系。接着，给出了 ABR 型不确定性分数阶差分方程解的不确定性分布。最后，具体的实例验证了主要结论的正确性。     

Prediction of bed level variations in nonuniform sediment bed channel

In the present study, a numerical model, based on one-dimensional de Saint-Venant equations along with sediment continuity equation, is developed for prediction of bed levels in non-cohesive sediments in aggrading alluvial channels. One-dimensional, unsteady flow equations and sediment continuity equations are solved using ‘shock-capturing’, second order accurate, explicit MacCormack finite difference scheme while considering upstream and downstream boundary conditions in the channel. Series of experimental investigations have been undertaken for measurements of bed and water levels in an aggrading channel due to overloading of nonuniform sediments, extracted from the bed of Tapi River at Surat City, in a flume installed in Advanced Hydraulics Laboratory of SVNIT, Surat, India. A satisfactory coupling between the water flow and sediment flow has been achieved. The sediment continuity equation is used for the each size class to compute the volume of each size class after each time step at any computational node in the computational grid. The fractional bed and suspended load transport capacities for different size fractions have been computed using fractional transport laws for nonuniform sediments. The active bed layer concept has been implemented in finite difference scheme to consider the interaction and exchange of sediment and water flow near the mixing layer. The performance of developed numerical model has been satisfactorily verified with independent experimental data of nonuniform sediment bed. Also, consideration of sediment nonuniformity in computation of bed level variation has been demonstrated by comparing the results based on sediment transport functions of uniform and nonuniform sediments.     

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.     

Optical solitons of some fractional differential equations in nonlinear optics

This paper deals with two different forms of the conformable fractional Benjamin–Bona–Mahony (BBM) equations by an analytical method. These physical models have important applications for describing the propagation of optical pulses in non-linear media. The conformable fractional symmetric BBM equation and the conformable time fractional Equal-width (EW) equation are considered. The extended Jacobi’s elliptic function expansion scheme are used to extract explicit solitons.     

Multilevel Circulant Preconditioner for High-Dimensional Fractional Diffusion Equations

High-dimensional two-sided space fractional diffusion equations with variable diffusion coefficients are discussed. The problems can be solved by an implicit finite difference scheme that is proven to be uniquely solvable, unconditionally stable and first-order convergent in the infinity norm. A nonsingular multilevel circulant preconditoner is proposed to accelerate the convergence rate of the Krylov subspace linear system solver efficiently. The preconditoned matrix for fast convergence is a sum of the identity matrix, a matrix with small norm, and a matrix with low rank under certain conditions. Moreover, the preconditioner is practical, with an O(N logN) operation cost and O(N) memory requirement. Illustrative numerical examples are also presented.     

Oscillation of the mirror and fractional RHEED reflections during homoepitaxy on the (2×4)-reconstructed GaAs(001) surface

Oscillations of the intensity of mirror and fractional RHEED reflections during homoepitaxy on the GaAs(001)-(2×4) reconstructed surface were studied. A considerable difference was observed in the patterns of intensity variation for the mirror and the fractional (0 1/4) and (0 3/4) reflections corresponding to the α and β phases on the reconstructed surface. A kinetic scheme of elementary processes occurring on the Ga(001) surface upon the homoepitaxial growth initiation is proposed. The activation energy for the nucleation process was experimentally determined (5-eV). It is shown that the temperature dependence of the probability of critical nucleus formation is determined by the desorption of As₂ dimers.     

A nonlinear fractional viscoelastic material model for polymers

In this contribution a test scheme based on tensile tests at different velocities, relaxation experiments and deformation controlled loading and unloading processes with intermediate relaxations has been used to experimentally characterize the nonlinear, inelastic material behavior. Based on the experimental observations a small strain nonlinear fractional viscoelastic material model is derived. In order to use the model within a finite element analysis, the constitutive equations have been generalized for the multiaxial case. The experimental test scheme and the fractional viscoelastic material model are subsequently applied to characterize and compute the mechanical behavior of the thermoplastic Polypropylene. After the identification of the material parameters several uniaxial and multiaxial simulations have been carried out and compared with experimental results.     

Lattice model of fractional gradient and integral elasticity: Long-range interaction of Grünwald–Letnikov–Riesz type

Lattice model with long-range interaction of power-law type that is connected with difference of non-integer order is suggested. The continuous limit maps the equations of motion of lattice particles into continuum equations with fractional Grünwald–Letnikov–Riesz derivatives. The suggested continuum equations describe fractional generalizations of the gradient and integral elasticity. The proposed type of long-range interaction allows us to have united approach to describe of lattice models for the fractional gradient and fractional integral elasticity. Additional important advantages of this approach are the following: (1) It is possible to use this model of long-range interaction in numerical simulations since this type of interactions and the Grünwald–Letnikov derivatives are defined by generalized finite difference; (2) The suggested model of long-range interaction leads to an equation containing the sum of the Grünwald–Letnikov derivatives, which is equal the Riesz’s derivative. This fact allows us to get particular analytical solutions of fractional elasticity equations.     

An artificial compressibility based fractional step method for solving time dependent incompressible flow equations. Temporal accuracy and similarity with a monolithic method

In this note, an artificial compressibility based fractional step method is analysed against a monolithic scheme for solving incompressible flow equations. The artificial compressibility (AC) procedure presented in this paper is stabilized via a characteristic based split (CBS), and thus it is referred to as the AC-CBS method. The monolithic method used for comparison in the present study is the pressure stabilized Petrov–Galerkin (PSPG) method. It is shown that the AC-CBS and PSPG procedures are identical in structure, except for the stabilization parameters. For unsteady problems, a dual time stepping algorithm is employed in the AC-CBS scheme. Unlike classical fractional step methods, this dual time stepping mechanism circumvents the temporal pressure splitting error, and thus provides the anticipated temporal accuracy. The temporal accuracy of the AC-CBS method is demonstrated via a standard benchmark problem. Up to fourth order time accurate schemes are introduced for a thorough analysis of the AC-CBS scheme.     

微小区系统交换通话优化处理方案及性能

提出了一种新型的微小区系统交换通话优化处理的方案,它是基于部分保护信道方案设计的,但额外地考虑了新建通话的拥塞问题.对该方案的性能进行了计算机仿算,并给出了与不同方案模型的性能比较.仿算结果表明,这个优化的交换通话处理方案相对于普遍的交换通话处理方案,更能够有效地改善系统的性能.     

Image encryption based on interference that uses fractional Fourier domain asymmetric keys

We propose an image encryption technique based on the interference principle and phase-truncation approach in the fractional Fourier domain. The proposed scheme offers multiple levels of security with asymmetric keys and is free from the silhouette problem. Multiple input images bonded with random phase masks are independently fractional Fourier transformed. Amplitude truncation of obtained spectrum helps generate individual and universal keys while phase truncation generates two phase-only masks analytically. For decryption, these two phase-only masks optically interfere, and this results in the phase-truncated function in the output. After using the correct random phase mask, universal key, individual key, and fractional orders, the original image is retrieved successfully. Computer simulation results with four gray-scale images validate the proposed method. To measure the effectiveness of the proposed method, we calculated the mean square error between the original and the decrypted images. In this scheme, the encryption process and decryption keys formation are complicated and should be realized digitally. For decryption, an optoelectronic scheme has been suggested.     

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.