Rabha W. Ibrahim

Fractional complex transforms for fractional differential equations

The fractional complex transform is employed to convert fractional differential equations analytically in the sense of the Srivastava-Owa fractional operator and its generalization in the unit disk. Examples are illustrated to elucidate the solution procedure including the space-time fractional differential equation in complex domain, singular problems and Cauchy problems. Here, we consider analytic solutions in the complex domain.MSC:30C45.

Classes of analytic functions with fractional powers defined by means of a certain linear operator

Motivated by the success of the familiar Dziok–Srivastava convolution operator, we introduce here a closely-related linear operator for analytic functions with fractional powers. By means of this linear operator, we then define and investigate a class of analytic functions. Finally, we determine certain conditions under which the partial sums of the linear operator of bounded turning are also of bounded turning. We also illustrate an application of a fractional integral operator.

On generalized Srivastava-Owa fractional operators in the unit disk

This article introduces a generalization for the Srivastava-Owa fractional operators in the unit disk. Conditions are given for the fractional integral operator to be bounded in Bergman space. Some properties for the above operator are also provided. Moreover, applications of these operators are posed in the geometric functions theory and fractional differential equations.

Denoising Algorithm Based on Generalized Fractional Integral Operator with Two Parameters

In this paper, a novel digital image denoising algorithm called generalized fractional integral filter is introduced based on the generalized Srivastava-Owa fractional integral operator. The structures of fractional masks of this algorithm are constructed. The denoising performance is measured by employing experiments according to visual perception and PSNR values. The results demonstrate that apart from enhancing the quality of filtered image, the proposed algorithm also reserves the textures and edges present in the image. Experiments also prove that the improvements achieved are competent with the Gaussian smoothing filter.

Texture Enhancement Based on the Savitzky-Golay Fractional Differential Operator

Texture enhancement for digital images is the most important technique in image processing. The purpose of this paper is to design a texture enhancement technique using fractional order Savitzky-Golay differentiator, which leads to generalizing the Savitzky-Golay filter in the sense of the Srivastava-Owa fractional operators. By employing this generalized fractional filter, texture enhancement is introduced. Consequently, it calculates the generalized fractional order derivative of the given image using the sliding weight window over the image. Experimental results show that the operator can extract more subtle information and make the edges more prominent. In general, the capability of the generalized fractional differential will be high because it is sensitive to the subtle fluctuations of values of pixels.

Fractional Alexander polynomials for image denoising

Image denoising is an important task in image processing. The interest in using a fractional mask window operator based on fractional calculus has grown for image denoising. This paper mainly introduces the concept of fractional calculus and proposes a new mathematical method in using fractional Alexander polynomials for image denoising. The structures of ni?n fractional mask windows on eight directions of this algorithm are constructed. Finally, we measure the denoising performance by employing experiments based on visual perception and by using peak signal-to-noise ratios. The experiments illustrate that the improvements achieved are compatible with other standard smoothing filters. An image denoising algorithm based on fractional Alexander polynomials is proposed.The denoising algorithm relies on the optimal values of fractional power parameters α and t.The denoising performance is measured based on the visual perception and the PSNR.Experiments demonstrate that the improvements achieved are compatible with the standard filters.

Ulam Stability for Fractional Differential Equation in Complex Domain

The present paper deles with a fractional differential equation , , where in sense of Srivastava-Owa fractional operators. The existence and uniqueness of holomorphic solutions are established. Ulam stability for the approximation and holomorphic solutions are suggested.

Fractional Differential Texture Descriptors Based on the Machado Entropy for Image Splicing Detection

Image splicing is a common operation in image forgery. Different techniques of image splicing detection have been utilized to regain people’s trust. This study introduces a texture enhancement technique involving the use of fractional differential masks based on the Machado entropy. The masks slide over the tampered image, and each pixel of the tampered image is convolved with the fractional mask weight window on eight directions. Consequently, the fractional differential texture descriptors are extracted using the gray-level co-occurrence matrix for image splicing detection. The support vector machine is used as a classifier that distinguishes between authentic and spliced images. Results prove that the achieved improvements of the proposed algorithm are compatible with other splicing detection methods.

Existence of Ulam Stability for Iterative Fractional Differential Equations Based on Fractional Entropy

In this study, we introduce conditions for the existence of solutions for an iterative functional differential equation of fractional order. We prove that the solutions of the above class of fractional differential equations are bounded by Tsallis entropy. The method depends on the concept of Hyers-Ulam stability. The arbitrary order is suggested in the sense of Riemann-Liouville calculus.

Some properties of certain multivalent analytic functions involving the Cho-Kwon-Srivastava operator

By making use of the principle of subordination between analytic functions and the Cho-Kwon-Srivastava operator, we introduce a certain subclass of multivalent analytic functions. Such results as subordination and superordination properties, convolution properties, inclusion relationships, distortion theorems, inequality properties and sufficient conditions for multivalent starlikeness are proved. The results presented here would provide extensions of those given in earlier works. Several other new results are also obtained.