Composite Fractional Trapezoidal Rule with Romberg Integration

The aim of this research is to demonstrate a novel scheme for approximating the Riemann-Liouville fractional integral operator.This would be achieved by first establishing a fractional-order version of the 2-point Trapezoidal rule and then by proposing another fractional-order version of the(n+1)-composite Trapezoidal rule.In particular,the so-called divided-difference formula is typically employed to derive the 2-point Trapezoidal rule,which has accordingly been used to derive a more accurate fractional-order formula called the(n+1)-composite Trapezoidal rule.Additionally,in order to increase the accuracy of the proposed approximations by reducing the true errors,we incorporate the so-called Romberg integration,which is an extrapolation formula of the Trapezoidal rule for integration,into our proposed approaches.Several numerical examples are provided and compared with a modern definition of the Riemann-Liouville fractional integral operator to illustrate the efficacy of our scheme.

A New Medical Image Enhancement Algorithm Based on Fractional Calculus

The enhancement of medical images is a challenging research task due to the unforeseeable variation in the quality of the captured images.The captured images may present with low contrast and low visibility,which might inuence the accuracy of the diagnosis process.To overcome this problem,this paper presents a new fractional integral entropy(FITE)that estimates the unforeseeable probabilities of image pixels,posing as the main contribution of the paper.The proposed model dynamically enhances the image based on the image contents.The main advantage of FITE lies in its capability to enhance the low contrast intensities through pixels’probability.Initially,the pixel probability of the fractional power is utilized to extract the illumination value from the pixels of the image.Next,the contrast of the image is then adjusted to enhance the regions with low visibility.Finally,the fractional integral entropy approach is implemented to enhance the low visibility contents from the input image.Tests were conducted on brain MRI,lungs CT,and kidney MRI scans datasets of different image qualities to show that the proposed model is robust and can withstand dramatic variations in quality.The obtained comparative results show that the proposed image enhancement model achieves the best BRISQUE and NIQE scores.Overall,this model improves the details of brain MRI,lungs CT,and kidney MRI scans,and could therefore potentially help the medical staff during the diagnosis process.

Attractive Multistep Reproducing Kernel Approach for Solving Stiffness Differential Systems of Ordinary Differential Equations and Some Error Analysis

In this paper,an efficient multi-step scheme is presented based on reproducing kernel Hilbert space(RKHS)theory for solving ordinary stiff differential systems.The solution methodology depends on reproducing kernel functions to obtain analytic solutions in a uniform formfor a rapidly convergent series in the posed Sobolev space.Using the Gram-Schmidt orthogonality process,complete orthogonal essential functions are obtained in a compact field to encompass Fourier series expansion with the help of kernel properties reproduction.Consequently,by applying the standard RKHS method to each subinterval,approximate solutions that converge uniformly to the exact solutions are obtained.For this purpose,several numerical examples are tested to show proposed algorithm’s superiority,simplicity,and efficiency.The gained results indicate that themulti-step RKHSmethod is suitable for solving linear and nonlinear stiffness systems over an extensive duration and giving highly accurate outcomes.

Thermogram Adaptive Efficient Model for Breast Cancer Detection Using Fractional Derivative Mask and Hybrid Feature Set in the IoT Environment

In this paper,a novel hybrid texture feature set and fractional derivative filter-based breast cancer detection model is introduced.This paper also introduces the application of a histogram of linear bipolar pattern features(HLBP)for breast thermogram classification.Initially,breast tissues are separated by masking operation and filtered by Gr¨umwald–Letnikov fractional derivative-based Sobel mask to enhance the texture and rectify the noise.A novel hybrid feature set usingHLBP and other statistical feature sets is derived and reduced by principal component analysis.Radial basis function kernel-based support vector machine is employed for detecting the abnormality in the thermogram.The performance parameters are calculated using five-fold cross-validation scheme using MATLAB 2015a simulation software.The proposedmodel achieves the classification accuracy,sensitivity,specificity,and area under the curve of 94.44%,95.55%,92.22%,96.11%,respectively.A comparative investigation of different texture features with respect to fractional orderαto classify the breast malignancy is also presented.The proposed model is also compared with a few existing state-of-art schemes which verifies the efficacy of the model.Fractional orderαoffers extra adaptability in overcoming the limitations of thermal imaging techniques and assists radiologists in prior breast cancer detection.The proposed model is more generalized which can be used with different thermal image acquisition protocols and IoT based applications.

An Efficient Approach for Solving One-Dimensional Fractional Heat Conduction Equation

Several researchers have dealt with the one-dimensional fractional heat conduction equation in the last decades,but as far as we know,no one has investigated such a problem from the perspective of developing suitable fractional-order methods.This has actually motivated us to address this problem by the way of establishing a proper fractional approach that involves employing a combination of a novel fractional difference formula to approximate the Caputo differentiator of orderαcoupled with the modified three-point fractional formula to approximate the Caputo differentiator of order 2α,where 0<α≤1.As a result,the fractional heat conduction equation is then reexpressed numerically using the aforementioned formulas,and by dividing the considered mesh into multiple nodes,a system is generated and algebraically solved with the aid of MATLAB.This would allow us to obtain the desired approximate solution for the problem at hand.

Lie symmetry analysis,explicit solutions,and conservation laws of the time-fractional Fisher equation in two-dimensional space

In these analyses,we consider the time-fractional Fisher equation in two-dimensional space.Through the use of the Riemann-Liouville derivative approach,the well-known Lie point symmetries of the utilized equation are derived.Herein,we overturn the fractional fisher model to a fractional differential equation of nonlinear type by considering its Lie point symmetries.The diminutive equation’s derivative is in the Erdélyi-Kober sense,whilst we use the technique of the power series to conclude explicit solutions for the diminutive equations for the first time.The conservation laws for the dominant equation are built using a novel conservation theorem.Several graphical countenances were utilized to award a visual performance of the obtained solutions.Finally,some concluding remarks and future recommendations are utilized.