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The aim of this paper is to present a new stable method for smoothing and differentiating noisy data defined on a bounded domain Ω ⊂ R N \Omega \subset {{\mathbb{R}}}^{N} with N ≥ 1 N\ge 1 . The proposed method stems from the smoothing properties of the classical diffusion equation; the smoothed data are obtained by solving a diffusion equation with the noisy data imposed as the initial condition. We analyze the stability and convergence of the proposed method and we give optimal convergence rates. One of the main advantages of this method lies in multivariable problems, where some of the other approaches are not easily generalized. Moreover, this approach does not require strong smoothness assumptions on the underlying data, which makes it appealing for detecting data corners or edges. Numerical examples demonstrate the feasibility and robustness of the method even with the presence of a large amounts of noise.

Multi-proxy signature schemes (MPSS) allow n proxy signers to generate a valid signature on behalf of the original signer. A new efficient self-certified MPSS with message recovery based on the Ell...

The profound characteristics of factoring problems and chaotic maps are utilized in this article to produce a new identity-based cryptographic scheme. These characteristics include random b...

We propose a new regularization scheme for stable numerical differentiation of noisy data defined on a bounded domain with . The method generates a sequence of smoothed (regularized) data obtained ...

The (extended) Nikiforov–Uvarov method is employed to find exact solutions of the Schrodinger operator for three $$\mathcal {PT}$$-invariant potentials (periodic exponential, cotangent and $$\mathcal {PT}$$-symmetric harmonic plus centrifugal). It is shown that their corresponding Schrodinger operators can exhibit real energy eigenvalues. The results are compared with similar works but with different methods. The comparisons led to Rodrigues formulas of some functions of interest. The eigenfunctions of these examples are expressed in terms of Hankel functions, Romanovski polynomials and Heun functions. The method is proved to be felicitous and leads to closed energy formulas for the potentials under study.

We consider the inverse problem of reconstructing the initial condition of a one-dimensional time-fractional diffusion equation from measurements collected at a single interior location over a finite time-interval. The method relies on the eigenfunction expansion of the forward solution in conjunction with a Tikhonov regularization scheme to control the instability inherent in the problem. We show that the inverse problem has a unique solution provided exact data is given, and prove stability results regarding the regularized solution. Numerical realization of the method and illustrations using a finite-element discretization are given at the end of this paper.

Classically, damping force is described as a function of velocity in the linear theory of mechanical models. In this work, a memory-dependent derivative model with respect to displacement is proposed to describe damping in various oscillatory systems of complex dissipation mechanisms where memory effects could not be ignored. A memory-dependent derivative is characterized by its time-delay τ and kernel function K( x, t) which can be chosen freely. Thus, it is superior to the fractional derivative in that it provides more access into memory effects and thus better physical meaning. To elucidate this, an equation of motion is proposed based on the prototype mass-spring model. The analytical solution is then attempted by the Laplace transform method. Due to the complexity of finding the inverse Laplace transform, a numerical inversion treatment is carried out using the fixed Talbot method and also compared with the finite difference discretization to validate the method. The calculations show that the response function is sensitive to different choices of τ and K( x, t). It is found that this proposed model supports the existence of memory-dependence in the structure of the material. The interesting case of resonance where the response function is classically increased rapidly is found to be weakened by an appropriate choice of τ and K( x, t).

In this paper, we are concerned with the backward problem of reconstructing the initial condition of a time-fractional diffusion equation from interior measurements. We establish uniqueness results and provide stability analysis. Our method is based on the eigenfunction expansion of the forward solution and the Tikhonov regularization to tackle the ill-posedness issue of the underlying inverse problem. Some numerical examples are included to illustrate the effectiveness of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.

In this study, we utilise the homotopy analysis method and the Sumudu transform to find the solution of fractional order partial differential equations. We focus primarily on the employment of the method for solving the fractional phi-4 equation in one-dimensional spatial domain. The method can be extended easily for more general nonlinear equations. Convergence and error analysis are given. Numerical experiments show that the method is both effective and accurate, and suits such type of applications.

Complex vibration phenomena appear so frequently in many engineering and physical experiments, and they are well modeled using nonlinear differential equations. However, contrary to the linear models, nonlinear models are difficult to analyze analytically or numerically and particularly for long-time spans. In this paper, we propose a novel method to provide approximate analytic solutions of an important class of nonlinear differential equations that describe the underdamped, overdamped, and oscillatory motions of massspring systems subjected to external excitations. The method is based on a novel modification of the Legendre operator matrix of differentiation technique which results in solutions that are accurate not only for short-time spans but also for long-time spans as well. We provide error analysis and present several examples to demonstrate the efficiency of the proposed method.

This paper deals with the anti-synchronisation of fractional order hyperchaotic systems. In particular, we employ the active control method to achieve complete anti-synchronisation between fractional order Lorenz and Chen hyperchaotic systems. Based on stability theorems in the fractional calculus, analysis of stability is performed for the proposed method. Numerical simulations demonstrate the feasibility of the proposed algorithm.

To estimate a parameter in an elliptic boundary value problem, the method of equation error chooses the value that minimizes the error in the PDE and boundary condition (the solution of the BVP having been replaced by a measurement). The estimated parameter converges to the exact value as the measured data converge to the exact value, provided Tikhonov regularization is used to control the instability inherent in the problem. The error in the estimated solution can be bounded in an appropriate quotient norm; estimates can be derived for both the underlying (infinite-dimensional) problem and a finite-element discretization that can be implemented in a practical algorithm. Numerical experiments demonstrate the efficacy and limitations of the method.