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  1. Fulltext Analytic Study of Complex Fractional Tsallis' Entropy with Applications in CNNs
    Ibrahim RW, Darus M
    Entropy (Basel), 2018 Sep 20;20(10).
    PMID: 33265810 DOI: 10.3390/e20100722
    In this paper, we study Tsallis' fractional entropy (TFE) in a complex domain by applying the definition of the complex probability functions. We study the upper and lower bounds of TFE based on some special functions. Moreover, applications in complex neural networks (CNNs) are illustrated to recognize the accuracy of CNNs.
  2. Fulltext Fractional dynamic system simulating the growth of microbe
    Hadid SB, Ibrahim RW
    Adv Differ Equ, 2021;2021(1):351.
    PMID: 34341660 DOI: 10.1186/s13662-021-03498-3
    There are different approaches that indicate the dynamic of the growth of microbe. In this research, we simulate the growth by utilizing the concept of fractional calculus. We investigate a fractional system of integro-differential equations, which covers the subtleties of the diffusion between infected and asymptomatic cases. The suggested system is applicable to distinguish the presentation of growth level of the infection and to approve if its mechanism is positively active. An optimal solution under simulation mapping assets is considered. The estimated numerical solution is indicated by employing the fractional Tutte polynomials. Our methodology is based on the Atangana-Baleanu calculus (ABC). We assess the recommended system by utilizing real data.
  3. Fulltext Periodicity computation of generalized mathematical biology problems involving delay differential equations
    Jasim Mohammed M, Ibrahim RW, Ahmad MZ
    Saudi J Biol Sci, 2017 Mar;24(3):737-740.
    PMID: 28386204 DOI: 10.1016/j.sjbs.2017.01.050
    In this paper, we consider a low initial population model. Our aim is to study the periodicity computation of this model by using neutral differential equations, which are recognized in various studies including biology. We generalize the neutral Rayleigh equation for the third-order by exploiting the model of fractional calculus, in particular the Riemann-Liouville differential operator. We establish the existence and uniqueness of a periodic computational outcome. The technique depends on the continuation theorem of the coincidence degree theory. Besides, an example is presented to demonstrate the finding.
  4. Hybrid time-space dynamical systems of growth bacteria with applications in segmentation
    Ibrahim RW, Nashine HK, Kamaruddin N
    Math Biosci, 2017 10;292:10-17.
    PMID: 28728968 DOI: 10.1016/j.mbs.2017.07.007
    A biological dynamic system carries engineering properties such as control systems and signal processing (or image processing) addicted to molecular biology at the level of bio-molecular communication networks. Dynamical system features and signal reply functions of cellular signaling pathways are some of the main topics in biological dynamic systems (for example the biological segmentation). In the present paper, we introduce new generalized hybrid time-space dynamical systems of growing bacteria. We impose the approximate analytic solution for the system. The generalization adapted the concepts of the Riemann-Liouville fractional operators for time and the Srivastava-Owa fractional operators for space. Moreover, we introduce a numerical perturbation method of two operators to obtain the approximate solutions. We establish the existence and uniqueness results and impose some applications in the sequel. Moreover, we study the Ulam stability and apply these stable solutions to improve the segmentation of a class of growing bacteria.
  5. Fulltext Some properties for integro-differential operator defined by a fractional formal
    Abdulnaby ZE, Ibrahim RW, Kılıçman A
    Springerplus, 2016;5(1):893.
    PMID: 27386341 DOI: 10.1186/s40064-016-2563-0
    Recently, the study of the fractional formal (operators, polynomials and classes of special functions) has been increased. This study not only in mathematics but extended to another topics. In this effort, we investigate a generalized integro-differential operator [Formula: see text] defined by a fractional formal (fractional differential operator) and study some its geometric properties by employing it in new subclasses of analytic univalent functions.
  6. A new deformable model based on fractional Wright energy function for tumor segmentation of volumetric brain MRI scans
    Ibrahim RW, Hasan AM, Jalab HA
    Comput Methods Programs Biomed, 2018 Sep;163:21-28.
    PMID: 30119853 DOI: 10.1016/j.cmpb.2018.05.031
    BACKGROUND AND OBJECTIVES: The MRI brain tumors segmentation is challenging due to variations in terms of size, shape, location and features' intensity of the tumor. Active contour has been applied in MRI scan image segmentation due to its ability to produce regions with boundaries. The main difficulty that encounters the active contour segmentation is the boundary tracking which is controlled by minimization of energy function for segmentation. Hence, this study proposes a novel fractional Wright function (FWF) as a minimization of energy technique to improve the performance of active contour without edge method.

    METHOD: In this study, we implement FWF as an energy minimization function to replace the standard gradient-descent method as minimization function in Chan-Vese segmentation technique. The proposed FWF is used to find the boundaries of an object by controlling the inside and outside values of the contour. In this study, the objective evaluation is used to distinguish the differences between the processed segmented images and ground truth using a set of statistical parameters; true positive, true negative, false positive, and false negative.

    RESULTS: The FWF as a minimization of energy was successfully implemented on BRATS 2013 image dataset. The achieved overall average sensitivity score of the brain tumors segmentation was 94.8 ± 4.7%.

    CONCLUSIONS: The results demonstrate that the proposed FWF method minimized the energy function more than the gradient-decent method that was used in the original three-dimensional active contour without edge (3DACWE) method.

  7. Fulltext Periodicity and positivity of a class of fractional differential equations
    Ibrahim RW, Ahmad MZ, Mohammed MJ
    Springerplus, 2016;5(1):824.
    PMID: 27390664 DOI: 10.1186/s40064-016-2386-z
    Fractional differential equations have been discussed in this study. We utilize the Riemann-Liouville fractional calculus to implement it within the generalization of the well known class of differential equations. The Rayleigh differential equation has been generalized of fractional second order. The existence of periodic and positive outcome is established in a new method. The solution is described in a fractional periodic Sobolev space. Positivity of outcomes is considered under certain requirements. We develop and extend some recent works. An example is constructed.
  8. Fulltext Differential inequalities imposed by the extended hypergeometric function
    Ibrahim RW, Ahmad MZ, Al-Janaby HF
    Springerplus, 2016;5:375.
    PMID: 27066382 DOI: 10.1186/s40064-016-1996-9
    Recently, the generalized hypergeometric function is extended by utilizing the Beta function. Based on this type of function, we introduce a new operator in the open unit disk. The present article investigates some subordination and superordination results for certain normalized analytic functions in the open unit disk, which are acted upon by the generalized Noor integral operator. Some of outcomes improve and generalize previously known outcomes.
  9. Fulltext Mathematical model for adaptive evolution of populations based on a complex domain
    Ibrahim RW, Ahmad MZ, Al-Janaby HF
    Saudi J Biol Sci, 2016 Jan;23(1):S45-9.
    PMID: 26858564 DOI: 10.1016/j.sjbs.2015.09.012
    A mutation is ultimately essential for adaptive evolution in all populations. It arises all the time, but is mostly fixed by enzymes. Further, most do consider that the evolution mechanism is by a natural assortment of variations in organisms in line for random variations in their DNA, and the suggestions for this are overwhelming. The altering of the construction of a gene, causing a different form that may be communicated to succeeding generations, produced by the modification of single base units in DNA, or the deletion, insertion, or rearrangement of larger units of chromosomes or genes. This altering is called a mutation. In this paper, a mathematical model is introduced to this reality. The model describes the time and space for the evolution. The tool is based on a complex domain for the space. We show that the evolution is distributed with the hypergeometric function. The Boundedness of the evolution is imposed by utilizing the Koebe function.
  10. Fulltext Invariant domain watermarking using heaviside function of order alpha and fractional Gaussian field
    Abbasi A, Woo CS, Ibrahim RW, Islam S
    PLoS One, 2015;10(4):e0123427.
    PMID: 25884854 DOI: 10.1371/journal.pone.0123427
    Digital image watermarking is an important technique for the authentication of multimedia content and copyright protection. Conventional digital image watermarking techniques are often vulnerable to geometric distortions such as Rotation, Scaling, and Translation (RST). These distortions desynchronize the watermark information embedded in an image and thus disable watermark detection. To solve this problem, we propose an RST invariant domain watermarking technique based on fractional calculus. We have constructed a domain using Heaviside function of order alpha (HFOA). The HFOA models the signal as a polynomial for watermark embedding. The watermark is embedded in all the coefficients of the image. We have also constructed a fractional variance formula using fractional Gaussian field. A cross correlation method based on the fractional Gaussian field is used for watermark detection. Furthermore the proposed method enables blind watermark detection where the original image is not required during the watermark detection thereby making it more practical than non-blind watermarking techniques. Experimental results confirmed that the proposed technique has a high level of robustness.
  11. Fulltext Mathematical model based on fractional trace operator for COVID-19 image enhancement
    Karim FK, Jalab HA, Ibrahim RW, Al-Shamasneh AR
    J King Saud Univ Sci, 2022 Oct;34(7):102254.
    PMID: 35957965 DOI: 10.1016/j.jksus.2022.102254
    The medical image enhancement is major class in the image processing which aims for improving the medical diagnosis results. The improving of the quality of the captured medical images is considered as a challenging task in medical image. In this study, a trace operator in fractional calculus linked with the derivative of fractional Rényi entropy is proposed to enhance the low contrast COVID-19 images. The pixel probability values of the input image are obtained first in the proposed image enhancement model. Then the covariance matrix between the input image and the probability of a pixel intensity of the input image to be calculated. Finally, the image enhancement is performed by using the convolution of covariance matrix result with the input image. The proposed enhanced image algorithm is tested against three medical image datasets with different qualities. The experimental results show that the proposed medical image enhancement algorithm achieves the good image quality assessments using both the BRISQUE, and PIQE quality measures. Moreover, the experimental results indicated that the final enhancement of medical images using the proposed algorithm has outperformed other methods. Overall, the proposed algorithm has significantly improved the image which can be useful for medical diagnosis process.
  12. Fulltext A new mathematical evaluation of smoking problem based of algebraic statistical method
    Mohammed MJ, Rakhimov IS, Shitan M, Ibrahim RW, Mohammed NF
    Saudi J Biol Sci, 2016 Jan;23(1):S11-5.
    PMID: 26858555 DOI: 10.1016/j.sjbs.2015.08.015
    Smoking problem is considered as one of the hot topics for many years. In spite of overpowering facts about the dangers, smoking is still a bad habit widely spread and socially accepted. Many people start smoking during their gymnasium period. The discovery of the dangers of smoking gave a warning sign of danger for individuals. There are different statistical methods used to analyze the dangers of smoking. In this study, we apply an algebraic statistical method to analyze and classify real data using Markov basis for the independent model on the contingency table. Results show that the Markov basis based classification is able to distinguish different date elements. Moreover, we check our proposed method via information theory by utilizing the Shannon formula to illustrate which one of these alternative tables is the best in term of independent.
  13. Fulltext A New Local Fractional Entropy-Based Model for Kidney MRI Image Enhancement
    Al-Shamasneh AR, Jalab HA, Palaiahnakote S, Obaidellah UH, Ibrahim RW, El-Melegy MT
    Entropy (Basel), 2018 May 05;20(5).
    PMID: 33265434 DOI: 10.3390/e20050344
    Kidney image enhancement is challenging due to the unpredictable quality of MRI images, as well as the nature of kidney diseases. The focus of this work is on kidney images enhancement by proposing a new Local Fractional Entropy (LFE)-based model. The proposed model estimates the probability of pixels that represent edges based on the entropy of the neighboring pixels, which results in local fractional entropy. When there is a small change in the intensity values (indicating the presence of edge in the image), the local fractional entropy gives fine image details. Similarly, when no change in intensity values is present (indicating smooth texture), the LFE does not provide fine details, based on the fact that there is no edge information. Tests were conducted on a large dataset of different, poor-quality kidney images to show that the proposed model is useful and effective. A comparative study with the classical methods, coupled with the latest enhancement methods, shows that the proposed model outperforms the existing methods.
  14. MRI Brain Classification Using the Quantum Entropy LBP and Deep-Learning-Based Features
    Hasan AM, Jalab HA, Ibrahim RW, Meziane F, Al-Shamasneh AR, Obaiys SJ
    Entropy (Basel), 2020 Sep 15;22(9).
    PMID: 33286802 DOI: 10.3390/e22091033
    Brain tumor detection at early stages can increase the chances of the patient's recovery after treatment. In the last decade, we have noticed a substantial development in the medical imaging technologies, and they are now becoming an integral part in the diagnosis and treatment processes. In this study, we generalize the concept of entropy difference defined in terms of Marsaglia formula (usually used to describe two different figures, statues, etc.) by using the quantum calculus. Then we employ the result to extend the local binary patterns (LBP) to get the quantum entropy LBP (QELBP). The proposed study consists of two approaches of features extractions of MRI brain scans, namely, the QELBP and the deep learning DL features. The classification of MRI brain scan is improved by exploiting the excellent performance of the QELBP-DL feature extraction of the brain in MRI brain scans. The combining all of the extracted features increase the classification accuracy of long short-term memory network when using it as the brain tumor classifier. The maximum accuracy achieved for classifying a dataset comprising 154 MRI brain scan is 98.80%. The experimental results demonstrate that combining the extracted features improves the performance of MRI brain tumor classification.
  15. A medical image enhancement based on generalized class of fractional partial differential equations
    Ibrahim RW, Jalab HA, Karim FK, Alabdulkreem E, Ayub MN
    Quant Imaging Med Surg, 2022 Jan;12(1):172-183.
    PMID: 34993069 DOI: 10.21037/qims-21-15
    Background: The interest in using fractional calculus operators has grown in the field of image processing. Image enhancement is one of image processing tools that aims to improve the details of an image. The enhancement of medical images is a challenging task due to the unforeseeable variation in the quality of the captured images.

    Methods: In this study, we present a mathematical model based on the class of fractional partial differential equations (FPDEs). The class is formulated by the proportional-Caputo hybrid operator (PCHO). Moreover, some properties of the geometric functions in the unit disk are applied to determine the upper bound solutions for this class of FPDEs. The upper bound solution is indicated in the relations of the general hypergeometric functions. The main advantage of FPDE lies in its capability to enhance the low contrast intensities through the proposed fractional enhanced operator.

    Results: The proposed image enhancement algorithm is tested against brain and lungs computed tomography (CT) scans datasets of different qualities to show that it is robust and can withstand dramatic variations in quality. The quantitative results of Brisque, Piqe, SSEQ, and SAMGVG were 40.93%, 41.13%, 66.09%, and 31.04%, respectively for brain magnetic resonance imaging (MRI) images and 39.07, 41.33, 30.97, and 159.24 respectively for the CT lungs images. The comparative results show that the proposed image enhancement model achieves the best image quality assessments.

    Conclusions: Overall, this model significantly improves the details of the given datasets, and could potentially help the medical staff during the diagnosis process.

  16. Fulltext Retraction Note: A brief comparative examination of tangent hyperbolic hybrid nanofluid through a extending surface: numerical Keller-Box scheme
    Jamshed W, Prakash M, Devi SSU, Ibrahim RW, Shahzad F, Nisar KS, et al.
    Sci Rep, 2023 Apr 13;13(1):6099.
    PMID: 37055498 DOI: 10.1038/s41598-023-33083-5
  17. Fulltext Irreversibility analysis of hydromagnetic nanofluid flow past a horizontal surface via Koo-Kleinstreuer-Li (KKL) model
    Hussain SM, Shahzad F, Katbar NM, Jamshed W, Eid MR, Kamel A, et al.
    Heliyon, 2023 Jul;9(7):e17668.
    PMID: 37483748 DOI: 10.1016/j.heliyon.2023.e17668
    The goal of this research is to investigate the effects of Ohmic heating, heat generation, and viscous dissipative flow on magneto (MHD) boundary-layer heat transmission flowing of Jeffrey nanofluid across a stretchable surface using the Koo-Kleinstreuer-Li (KKL) model. Engine oil serves as the primary fluid and is suspended with copper oxide nanomolecules. The governing equations that regulate the flowing and heat transmission fields are partial-differential equations (PDEs) that are then converted to a model of non-linear ordinary differential equations (ODEs) via similarity transformation. The resultant ODEs are numerically resolved using a Keller box technique via MATLAB software that is suggested. Diagrams and tables are used to express the effects of various normal liquids, nanomolecule sizes, magneto parameters, Prandtl, Deborah, and Eckert numbers on the velocity field and temperature field. The outcomes display that the copper oxide-engine oil nanofluid has a lower velocity, drag force, and Nusselt number than the plain liquid, although the introduction of nanoparticles raises the heat. The heat transference rate is reduced by Eckert number, size of nanomolecules, and magneto parameter rising. Whilst, Deborah number is shown to enhance both the drag-force factor and the heat transfer rate. Furthermore, the discoveries reported are advantageous to upgrading incandescent lighting bulbs, heating, and cooling equipment, filament-generating light, energy generation, multiple heating devices, and other similar devices.
  18. Fulltext A brief comparative examination of tangent hyperbolic hybrid nanofluid through a extending surface: numerical Keller-Box scheme
    Jamshed W, Prakash M, Devi SSU, Ibrahim RW, Shahzad F, Nisar KS, et al.
    Sci Rep, 2021 Dec 15;11(1):24032.
    PMID: 34912014 DOI: 10.1038/s41598-021-03392-8
    A novel hybrid nanofluid was explored in order to find an efficient heat-transmitting fluid to replace standard fluids and revolutionary nanofluids. By using tangent hyperbolic hybrid combination nanoliquid with non-Newtonian ethylene glycol (EG) as a basis fluid and a copper (Cu) and titanium dioxide (TiO2) mixture, this work aims to investigate the viscoelastic elements of the thermal transferring process. Flow and thermal facts, such as a slippery extended surface with magnetohydrodynamic (MHD), suction/injection, form factor, Joule heating, and thermal radiation effects, including changing thermal conductivity, were also integrated. The Keller-Box method was used to perform collective numerical computations of parametric analysis using governing equivalences. In the form of graphs and tables, the results of TiO2-Cu/EG hybrid nanofluid were compared to those of standard Cu/EG nanofluid in important critical physical circumstances. The entropy generation study was used to examine energy balance and usefulness for important physically impacting parameters. Detailed scrutiny on entropy development get assisted with Weissenberg number, magnetic parameter, fractional volumes, injection parameter, thermal radiation, variable thermal conductivity, Biot number, shape variation parameter, Reynolds and Brinkman number. Whereas the entropy gets resisted for slip and suction parameter. In this case, spotted entropy buildup with important parametric ranges could aid future optimization.
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