This work describes two efficient and useful methods for solving fractional pantograph delay equations (FPDEs) with initial and boundary conditions. These two methods depend mainly on orthogonal polynomials, which are the method of the operational matrix of fractional derivative that depends on Bernstein polynomials and the operational matrix of the fractional derivative with Shifted Legendre polynomials. The basic procedure of this method is to convert the pantograph delay equation to a system of linear equations and by using, the operational matrices we get rid of the integration and differentiation operations, which makes solving the problem easier. The concept of Caputo has been used to describe fractional derivatives. Finally, some numerical examples are identified to show the utility and capability of the two proposed approaches. Mathematica®12 program has been relied upon in the calculations.
This paper deals with founding an estimation of best approximation of unbounded functions which satisfied weighted Lipschitz condition with respect to the convex polynomials by means of weighted moduli of smoothness of fractional order , ( , ) p f t . In addition we prove some properties of weighted moduli of smoothness of fractional order.
Background: Acne is a common disorder experienced by adolescents and persists into adulthood in approximately 12%–14% of cases with psychological and social implications of high gravity. Fractional resurfacing employs a unique mechanism of action that repairs a fraction of skin at a time. The untreated healthy skin remains intact and actually aids the repair process, promoting rapid healing with only a day or two of downtime. Aims: This study, was designed to evaluate the safety and effectiveness of fractional photothermolysis (fractionated Er: YAG laser 2940nm) in treating atrophic acne scars. Methods: 7 females and 3 males with moderate to severe atrophic acne scarring were enrolled in this study that attained private clinic for Derm
... Show MoreStress urinary incontinence (SUI) is involuntary urine leakage during activities that increase abdominal pressure such as coughing, sneezing and lifting of heavy weights. This is a very common disorder among women with history of multiple vaginal deliveries with an obstructed labor. SUI is considered one of the most distressing problems, especially for younger women, with severe quality of life implications, it caused by the loss of urethral support, usually as a consequence of the supporting structural muscles in the pelvis.
Objective: To prove and demonstrate the effect of a fractional CO2 micro-ablative laser (10600nm) in intra vaginal therapy for treating SUI and achieve a clinical improvement of t
... Show MoreBackground: Urinary incontinence (UI) is a common disorder that affects women of various ages and impacts all aspects of life. This condition negatively influences quality of life. Fractional CO2 laser (10600nm) is the recent method for treatment of stress urinary incontinence in women. Objectives: The purpose of the study was to evaluate the efficacy and safety of fractional CO2 laser (10600nm) in the treatment of female stress urinary incontinence. Materials & Methods: This study was done from July 2020 to February 2021conducted at the laser institute for postgraduate studies university of Baghdad, patients collected from a private clinic and the Department of
... Show MoreThis paper deals with the Magnetohydrodynyamic (Mill)) flow for a viscoclastic fluid of the generalized Oldroyd-B model. The fractional calculus approach is used to establish the constitutive relationship of the non-Newtonian fluid model. Exact analytic solutions for the velocity and shear stress fields in terms of the Fox H-function are obtained by using discrete Laplace transform. The effect of different parameter that controlled the motion and shear stress equations are studied through plotting using the MATHEMATICA-8 software.
This paper investigates the effect of magnetohydrodynamic (MHD) of an incompressible generalized burgers’ fluid including a gradient constant pressure and an exponentially accelerate plate where no slip hypothesis between the burgers’ fluid and an exponential plate is no longer valid. The constitutive relationship can establish of the fluid model process by fractional calculus, by using Laplace and Finite Fourier sine transforms. We obtain a solution for shear stress and velocity distribution. Furthermore, 3D figures are drawn to exhibit the effect of magneto hydrodynamic and different parameters for the velocity distribution.
This study was design to investigate the dimensional stability of heat-activated acrylic resin with different methods of flask cooling (15 minutes rapid cooling, one hour bench cooling, four hours delayed deflasking, and 24 hours delayed deflasking) at different time intervals (immediately, two days, seven days, 30 days) after deflasking. Heat-activated acrylic resin was used to prepare acrylic samples. Then, measurement of the distances where achieved between the centers of selected marks in the acrylic samples. They were measured at different time intervals for different methods of flask cooling. The results showed that the group samples of the four hours and 24 hours of delayed deflasking was insignificantly different from the control an
... Show MoreDue to its importance in physics and applied mathematics, the non-linear Sturm-Liouville problems
witnessed massive attention since 1960. A powerful Mathematical technique called the Newton-Kantorovich
method is applied in this work to one of the non-linear Sturm-Liouville problems. To the best of the authors’
knowledge, this technique of Newton-Kantorovich has never been applied before to solve the non-linear
Sturm-Liouville problems under consideration. Accordingly, the purpose of this work is to show that this
important specific kind of non-linear Sturm-Liouville differential equations problems can be solved by
applying the well-known Newton-Kantorovich method. Also, to show the efficiency of appl