Fractional Cosine Research Articles

Finite dimensional approximation to fractional stochastic integro-differential equations with non-instantaneous impulses

This manuscript proposes a class of fractional stochastic integro-differential equations (FSIDEs) with non-instantaneous impulses in an arbitrary separable Hilbert space. We use a projection scheme of increasing sequence of finite dimensional subspaces and projection operators to define approximations. In order to demonstrate the existence and convergence of approximate solutions, we utilize stochastic analysis theory, fractional calculus, theory of fractional cosine family of linear operators, and fixed point approach. Furthermore, we examine the convergence of the Faedo–Galerkin (F–G) approximate solutions to the mild solution of our given problem. Finally, a concrete example involving a partial differential equation is provided to validate the main abstract results.

NOVEL PERSPECTIVE TO THE FRACTIONAL SCHRÖDINGER EQUATION ARISING IN OPTICAL FIBERS

In this paper, the fractional Schrödinger equation is described with beta derivative, which is used to elucidate the dynamic interaction of ultra-short pulses with quantum properties in optical fibers. This work is to study the solitary wave and periodic solutions of the fractional Schrödinger equation by employing three powerful and simple mathematical approaches like fractional Kudryashov method, fractional cosine–sine method and fractional tanh function method. The acquired outcomes illustrate that the proposed three computational approaches are simple, efficient, concise and can be adopted to study more complex phenomena. Finally, the dynamical behavior of these acquired solitary wave solutions is illustrated by sketching some 3D figures with proper parameters.

Approximate controllability of higher order Riemann–Liouville fractional systems via cosine family

This work studies the controllability of a class of Riemann–Liouville fractional differential equations of order in a new Banach space using fractional cosine family. To derive the existence of solutions, we construct the definition of mild solutions of the system using Laplace transform theory. Then, by assuming the approximate controllability of corresponding linear equation, we prove that the nonlinear dynamical system is approximately controllable using an iterative technique. Lastly, we give an example to illustrate the established theory.

Solution of Inhomogeneous Linear Fractional Differential Equations Involving Jumarie Fractional Derivative

This paper presents a method for solving inhomogeneous linear sequential fractional differential equations with constant coefficients (ILSFDE) involving Jumarie fractional derivatives in terms of Mittag-Leffler functions. For this purpose, the fundamental properties of the Jumarie derivative and Mittag-Leffler functions are given. After this, the successive jumarie fractional derivatives of Mittag-Leffler functions, fractional cosine, and sine functions are obtained. Further, we determined the particular integrals of these functions and then found the complete solutions of ILSFDE. in terms of Mittag-Leffler functions, fractional cosine, and sine functions. We have demonstrated this developed method with a few examples of ILSFDE. This method is similar to the method for finding the complete solutions of classical differential equations with constant coefficients.

The discrete convolution for fractional cosine-sine series and its application in convolution equations

<abstract><p>The fractional sine series (FRSS) and the fractional cosine series (FRCS) were defined. Three types of discrete convolution operations for FRCS and FRSS were introduced, along with a detailed investigation into their corresponding convolution theorems. The interrelationship between these convolution operations was also discussed. Additionally, as an application of the presented results, two forms of discrete convolution equations based on the proposed convolution theorems were examined, resulting in explicit solutions for these equations. Furthermore, numerical simulations were provided to demonstrate that our proposed solution can be easily implemented with low computational complexity.</p> </abstract>

A study of nonlocal fractional delay differential equations with hemivariational inequality

<abstract><p>In this paper, we study an abstract system of fractional delay differential equations of order $ 1 < q < 2 $ with a hemivariational inequality in Banach spaces. To establish the existence of a solution to the abstract inequality, we employ the Rothe technique in conjunction with the surjectivity of multivalued pseudomonotone operators and features of the Clarke generalized gradient. Further, to show the existence of the fractional differential equation, we use the fractional cosine family and fixed point theorem. Finally, we include an example to elaborate the effectiveness of the findings.</p></abstract>

Fractional Fourier cosine and sine Laplace weighted convolution and its application

In this paper, four types of fractional Fourier cosine and sine Laplace weighted convolutions are defined, and the corresponding fractional Fourier cosine and sine Laplace convolution theorems associated with the fractional cosine transform, fractional sine transform and Laplace transform are derived in detail. Furthermore, the relationship between the fractional Fourier cosine (sine) Laplace weighted convolutions and existing convolutions are studied, and Young's type theorem is also investigated. In addition, as an application for fractional Fourier cosine (sine)-Laplace weighted convolution, the system of convolution-type integral equations is considered, the computational complexity is analysed and the explicit solutions for these equations are obtained.

Perturbation theory for fractional evolution equations in a Banach space

A strong inspiration for studying perturbation theory for fractional evolution equations comes from the fact that they have proven to be useful tools in modeling many physical processes. We study fractional evolution equations of order \(\alpha \in (1,2)\) associated with the infinitesimal generator of an operator fractional cosine (sine) function generated by bounded time-dependent perturbations in a Banach space. We show that the fractional abstract Cauchy problem associated with the infinitesimal generator A of a strongly continuous fractional cosine (sine) function remains uniformly well-posed under bounded time-dependent perturbation of A. We also provide some necessary special cases by using the Laplace transform of the generators of the given operator families.

Analytical Solution of the Fractional Linear Time-Delay Systems and their Ulam-Hyers Stability

We introduce the delayed Mittag-Leffler type matrix functions, delayed fractional cosine, and delayed fractional sine and use the Laplace transform to obtain an analytical solution to the IVP for a Hilfer type fractional linear time-delay systemD0,tμ,νzt+Azt+Ωzt−h=ftof order1<μ<2and type0≤ν≤1, with nonpermutable matricesAandΩ. Moreover, we study Ulam-Hyers stability of the Hilfer type fractional linear time-delay system. Obtained results extend those for Caputo and Riemann-Liouville type fractional linear time-delay systems with permutable matrices and new even for these fractional delay systems.

The explicit solutions for a class of fractional Fourier–Laplace convolution equations

In this paper, two types of fractional Fourier–Laplace convolutions are defined, and the corresponding fractional Fourier–Laplace convolution theorems associated with the fractional cosine transform (FRCT), fractional sine transform (FRST) and Laplace transform (LP) are investigated in detail. The relationship between the fractional Fourier–Laplace convolutions and the existing convolutions is given, and Young's type theorem as well as the weighted convolution inequality are also obtained. As an application for fractional Fourier–Laplace convolution, the filter design and the system of convolution-type integral equations are also considered, the computational complexity for the multiplicative filter is analysed, and explicit solutions for these equations are obtained.

Investigating existence results for fractional evolution inclusions with order r ∈ (1, 2) in Banach space

Abstract This manuscript investigates the issue of existence results for fractional differential evolution inclusions of order r ∈ (1, 2) in the Banach space. In the beginning, we analyze the existence results by referring to the fractional calculations, cosine families, multivalued function, and Martelli’s fixed point theorem. The result is also used to investigate the existence of nonlocal fractional evolution inclusions of order r ∈ (1, 2). Finally, a concrete application is given to illustrate our main results.

Robust Image Compression Algorithm using Discrete Fractional Cosine Transform

The discrete fractional Fourier transform become paradigm in signal processing. This transform process the signal in joint time-frequency domain. The attractive and very important feature of DFrCT is an availability of extra degree of one free parameter that is provided by fractional orders and due to which optimization is possible. Less execution time and easy implementation are main advantages of proposed algorithm. The merit of effectiveness of proposed technique over existing technique is superior due to application of discrete fractional cosine transform by which higher compression ratio and PSNR are obtained without any artifacts in compressed images. The novelty of the proposed algorithm is no artifacts in compressed image along with good CR and PSNR. Compression ratio (CR) and peak signal to noise ratio (PSNR) are quality parameters for image compression with optimum fractional order.

Discrete Convolution Associated with Fractional Cosine and Sine Series

Fractional sine series (FRSS) and fractional cosine series (FRCS) are the discrete form of the fractional cosine transform (FRCT) and fractional sine transform (FRST). The recent studies have shown that discrete convolution is widely used in optics, signal processing and applied mathematics. In this paper, firstly, the definitions of fractional sine series (FRSS) and fractional cosine series (FRCS) are presented. Secondly, the discrete convolution operations and convolution theorems for fractional sine and cosine series are given. The relationship of two convolution operations is presented. Lastly, the discrete Young’s type inequality is established. The proposed theory plays an important role in digital filtering and the solution of differential and integral equations.

Efficient hybrid passive method for the detection and localization of copy-move and spliced images

Digital passive image forgery methods are extensively used to verify the authenticity and integrity of images.Splicing and copy-move are the most common types of passive digital image forgeries. Several approaches have beenproposed to detect these forgeries separately, but very few approaches are available that can detect them simultaneously.However, a more e?icient method is still in demand to meet the day-to-day challenges to detect these forgeries at thesame time. So, a passive hybrid approach based on discrete fractional cosine transform (DFrCT) and local binarypattern (LBP) is proposed to detect copy-move and splicing forgeries simultaneously. The extra parameter i.e. fractionalparameter of DFrCT is utilized to enhance the accuracy and LBP is used to highlight the tampering artifacts effectively.Then, a support vector machine (SVM) is employed to categorize the images into authentic, copy-move, and splicedimages. Next, localization is performed on both the copy-move and spliced images to localize the duplicated areas inthe image. Experiments on six benchmark datasets, namely, CASIA v1.0, GRIP, CASIA v2.0, IMD, COVERAGE,and Columbia, attain accuracy rates of 99.67%, 99.23%, 99.76%, 98.81%, 95%, and 98.17%, respectively. To validatethe effectiveness of the proposed method, comparative analysis has been performed with existing methods in terms ofROC, precision, recall,F1score,F2score, and accuracy. Moreover, the robustness of the proposed work is tested underrotation attack and better results are attained in comparison to the existing techniques.

Mild solutions of a class of semilinear fractional integro-differential equations subjected to noncompact nonlocal initial conditions

In this paper, we prove the existence of mild solutions of a class of fractional semilinear integro-differential equations of order \(\beta\in(1,2]\) subjected to noncompact initial nonlocal conditions. We assume that the linear part generates an arbitrarily strongly continuous \(\beta\)-order fractional cosine family, while the nonlinear forcing term is of Caratheodory type and satisfies some fairly general growth conditions. Our approach combines the Monch fixed point theorem with some recent results regarding the measure of noncompactness of integral operators. Our conclusions improve and generalize many earlier related works. An example is provided to illustrate the main results.

MIMO-OFDM/OCDM low-complexity equalization under a doubly dispersive channel in wireless sensor networks

In this article, three novel systems for wireless sensor networks based on Alamouti decoding were investigated and then compared, which are Alamouti space–time block coding multiple-input single-output/multiple-input multiple-output multicarrier modulation (MCM) system, extended orthogonal space–time block coding multiple-input single-output MCM system, and multiple-input multiple-output system. Moreover, the proposed work is applied over multiple-input multiple-output systems rather than the conventional single-antenna orthogonal chirp division multiplexing systems, based on the discrete fractional cosine transform orthogonal chirp division multiplexing system to mitigate the effect of frequency-selective and time-varying channels, using low-complexity equalizers, specifically by ignoring the intercarrier interference coming from faraway subcarriers and using the LSMR iteration algorithm to decrease the equalization complexity, mainly with long orthogonal chirp division multiplexing symbols, such as the TV symbols. The block diagrams for the proposed systems are provided to simplify the theoretical analysis by making it easier to follow. Simulation results confirm that the proposed multiple-input multiple-output and multiple-input single-output orthogonal chirp division multiplexing systems outperform the conventional multiple-input multiple-output and multiple-input single-output orthogonal frequency division multiplexing systems. Finally, the results show that orthogonal chirp division multiplexing exhibited a better channel energy behavior than classical orthogonal frequency division multiplexing, thus improving the system performance and allowing the system to decrease the equalization complexity.

Fractional convolution associated with a class of integral equations

Three kinds of generalised convolution operations of fractional cosine transform and fractional sine transform are investigated, and the corresponding convolution theorems are derived, which can be seen as the generalisation of the classical results. The relationships of these generalised fractional convolution operations are also discussed. Additionally, the potential applications of the derived results in solving two kinds of generalised convolution integral equations are discussed, and the explicit solutions of these convolution-type integral equations are obtained.

Anti-forensic approach for JPEG compressed images with enhanced image quality and forensic undetectability

Most of the detectors employed in digital image forensics are based on JPEG compression. To determine the capability of these forensic detectors, proficient anti-forensic techniques that challenge and help in the upgradation of forensic techniques are required. This paper proposes an anti-forensic technique based on the shifted block Discrete Fractional Cosine Transform (DFrCT) approach. Afterwards, Total variation (TV) -based deblocking operation is used in order to remove the compression blocking artifacts. Due to the shifted block approach, the proposed method performs histogram smoothing without adding any dithering signal, which means that it is capable of applying dithering by itself. Further, to remove blocking artifacts which are left during the JPEG compression TV-based deblocking is used. The DFrCT approach provides an additional fractional parameter to improve the accuracy of the proposed approach. The proposed scheme is evaluated based on the UCID dataset images by considering the scalar based and machine learning-based forensic detectors. It is observed from the experimental results that the proposed approach provides improved performance in terms of PSNR, SSIM, and forensic undetectability when compared to existing techniques. The analysis performed in this paper challenges the security and robustness of JPEG compression forensic techniques.

Generalized DCT and DWT hybridization based robust feature extraction for face recognition

Pose, expression and light variations are the major complications in recognition of face images. Hence, a robust technique is introduced in this paper which deals with variations in light in the face images. Hybridization of fractional discrete Cosine transform (FrDCT) and DWT are utilized as the feature extraction technique which helps in normalizing the light variations in the face images. DWT helps in obtaining the global behaviour of the light compensated face image. Also, use of DWT with FrDCT helps in obtaining the redundancy which is difficult to obtain with FrDCT alone. Therefore, it is obtained with DWT and after that, FrDCT is exploited for the local correlation. The proposed method is experimented and investigated on YALE, Yale B and CMU PIE face databases and very good performance outcome has been achieved as compared to the other state of art methods.

Transform Based Speech Enhancement for Auditory Signals Based on Iterative Weiner Filtering

One of the important features of Speech processing is speech enhancement. In a noisy environment, speech enhancement plays a vital role. Many research works are being done in speech enhancement methods in recent years but still, it can't be attained. It mainly depends on Speech intelligibility which can improve the speech quality. In this research work, signal representation is considered and the various transforms are applied and compared. The analysis is done with the help of two parameters and the results are compared. Here the enhancement process is focused on using Advanced DCT (ADCT) and Discrete fractional Cosine transform. The ADCT has the advantage of energy compaction and flexible window switching. Iterative Wiener Filtering is used for filtering the coefficients. Pitch Synchronous Analysis (PSA) is combined for finding the exact pitch period.