Numerical Study of Rosenau-KdV Equation Using Finite Element Method Based on Collocation Approach

In the present paper, a numerical method is proposed for the numerical solution of Rosenau-KdV equation with appropriate initial and boundary conditions by using collocation method with septic B-spline functions on the uniform mesh points. The method is shown to be unconditionally stable using von-Neumann technique. To check accuracy of the error norms L2 and L∞ are computed. Interaction of two and three solitary waves are used to discuss the effect of the behavior of the solitary waves during the interaction. Furthermore, evolution of solitons is illustrated by undular bore initial condition. These results show that the technique introduced here is suitable to investigate behaviors of shallow water waves

Impact of opioid-free analgesia on pain severity and patient satisfaction after discharge from surgery: multispecialty, prospective cohort study in 25 countries

Background: Balancing opioid stewardship and the need for adequate analgesia following discharge after surgery is challenging. This study aimed to compare the outcomes for patients discharged with opioid versus opioid-free analgesia after common surgical procedures.Methods: This international, multicentre, prospective cohort study collected data from patients undergoing common acute and elective general surgical, urological, gynaecological, and orthopaedic procedures. The primary outcomes were patient-reported time in severe pain measured on a numerical analogue scale from 0 to 100% and patient-reported satisfaction with pain relief during the first week following discharge. Data were collected by in-hospital chart review and patient telephone interview 1 week after discharge.Results: The study recruited 4273 patients from 144 centres in 25 countries; 1311 patients (30.7%) were prescribed opioid analgesia at discharge. Patients reported being in severe pain for 10 (i.q.r. 1-30)% of the first week after discharge and rated satisfaction with analgesia as 90 (i.q.r. 80-100) of 100. After adjustment for confounders, opioid analgesia on discharge was independently associated with increased pain severity (risk ratio 1.52, 95% c.i. 1.31 to 1.76; P < 0.001) and re-presentation to healthcare providers owing to side-effects of medication (OR 2.38, 95% c.i. 1.36 to 4.17; P = 0.004), but not with satisfaction with analgesia (beta coefficient 0.92, 95% c.i. -1.52 to 3.36; P = 0.468) compared with opioid-free analgesia. Although opioid prescribing varied greatly between high-income and low- and middle-income countries, patient-reported outcomes did not.Conclusion: Opioid analgesia prescription on surgical discharge is associated with a higher risk of re-presentation owing to side-effects of medication and increased patient-reported pain, but not with changes in patient-reported satisfaction. Opioid-free discharge analgesia should be adopted routinely

Derivation of septic B-spline function in n-dimensional to solve n-dimensional partial differential equations

In this study, a new structure for the septic B-spline collocation algorithm in n-dimensional is presented as a continuation of generating B-spline functions in n-dimensional to solve mathematical models in n-dimensional. The septic B-spline collocation algorithm is displayed in three forms: one dimensional, two dimensional, and three dimensional. In various domains, these constructs are essential for solving mathematical models. The effectiveness and correctness of the suggested method are demonstrated using a few two- and three-dimensional test problems. The proposed new structure provides better results than other methods because it deals with a larger number of points than the field. To create comparisons, we use different numerical approaches accessible in the literature

Solving Natural Convection Of Darcian Fluid In Porous Media Using Rational Chebyshev Collocation Method

In this paper, numerical technique is introduced for solving natural convection of Darcian fluid about a vertical full cone embedded in porous media with a prescribed wall temperature or surface heat flux boundary conditions. The power function of distance from the vertex of the inverted cone gives us nonlinear differential equation. Rational Chebyshev collocation method is used to solve the produced third order nonlinear differential equation with boundary conditions transformed to a system of nonlinear equations. The proposed base is specified by its ability of deal with boundary conditions with an independent variable that may tend to infinity with easy manner without divergent. Moreover, we compare the proposed method with other methods to investigate applicability and accuracy

Solving Fully Rough Interval Multi-level Multi-objective linear Fractional Programming Problems via FGP

This paper introduces an algorithm for solving fully rough intervalmulti-level multi-objective linear fractional programming problems where all of its coefficients in objective functionsand in constraints are rough intervals. At the first phases of the solution approach and to avoid the complexity of the problem, the shifting method proposed by Osman and El-sherbiny [20] will be used to split the rough problem into four crisp problems which will be solved simultaneously. At the second phase, for each problem, a membership function was constructed to develop a fuzzy goal programming model for obtaining the satisfactory solution of the multilevel multi-objective fractional programming problem. The linearization process introduced by Pal et. al [1] will be applied to linearize the membership functions. Finally, an illustrative numerical example is given to demonstrate the algorithm

An exponential Chebyshev second kind approximation for solving high-order ordinary differential equations in unbounded domains, with application to Dawson's integral

A new exponential Chebyshev operational matrix of derivatives based on Chebyshev polynomials of second kind (ESC) is investigated. The new operational matrix of derivatives of the ESC functions is derived and introduced for solving high-order linear ordinary differential equations with variable coefficients in unbounded domain using the collocation method. As an application the introduced method is used to evaluate Dawson's integral by solving its differential equation. The corresponding differential equation to Dawson's integral is a boundary value problem with conditions tends to infinity. The obtained numerical results are compared with the exact solution and showed good accuracy

Solving Systems of Linear Fractional Integro-Differential Equations with Delay using Least Squares Method and Chebyshev Polynomials

In this paper, a numerical scheme is introduced for solving a linear system of fractional delay integro-differential equations (FDIDEs). The fractional derivative is considered in the Caputo sense. The spectral least squares method with the aid of the first kind Chebyshev polynomials was introduced to treat the proposed systems. The suggested method reduces this type of system to the solution of a block system of linear algebraic equations. To demonstrate the accuracy and applicability of the presented method some test examples are provided. Numerical results show that this approach is easy to implement and accurate when applied to the proposed system of FDIDEs

On the Resonant Vibrations Control of the Nonlinear Rotor Active Magnetic Bearing Systems

Nonlinear vibration control of the twelve-poles electro-magnetic suspension system was tackled in this study, using a novel control strategy. The introduced control algorithm was a combination of three controllers: the proportional-derivative (PD) controller, the integral resonant controller (IRC), and the positive position feedback (PPF) controller. According to the presented control algorithm, the mathematical model of the controlled twelve-poles rotor was established as a nonlinear four-degree-of-freedom dynamical system coupled to two first-order filters. Then, the derived nonlinear dynamical system was analyzed using perturbation analysis to extract the averaging equations of motion. Based on the extracted averaging equations of motion, the efficiency of different control strategies (i.e., PD, PD+IRC, PD+PPF, and PD+IRC+PPF) for mitigating the rotor’s undesired vibrations and improving its catastrophic bifurcation was investigated. The acquired analytical results demonstrated that both the PD and PD+IRC controllers can force the rotor to respond as a linear system; however, the controlled system may exhibit the maximum oscillation amplitude at the perfect resonance condition. In addition, the obtained results demonstrated that the PD+PPF controller can eliminate the rotor nonlinear oscillation at the perfect resonance, but the system may suffer from high oscillation amplitudes when the resonance condition is lost. Moreover, we report that the combined control algorithm (PD+IRC+PPF) has all the advantages of the individual control algorithms (i.e., PD, PD+IRC, PD+PPF), while avoiding their drawbacks. Finally, the numerical simulations showed that the PD+IRC+PPF controller can eliminate the twelve-poles system vibrations regardless of both the excitation force magnitude and the resonant conditions at a short transient time

On the Performance of a Nonlinear Position-Velocity Controller to Stabilize Rotor-Active Magnetic-Bearings System

The performance of a nonlinear position-velocity controller in stabilising the lateral vibrations of a rotor-active magnetic-bearings system (RAMBS) is investigated. Cubic nonlinear position-velocity and linear position-velocity controllers are introduced to stabilise RAMBS lateral oscillations. According to the proposed control law, the nonlinear system model is established and then investigated with perturbation analysis. Nonlinear algebraic equations that govern the steady-state oscillation amplitudes and the corresponding phases are derived. Depending on the obtained algebraic equations, the different frequency response curves and bifurcation diagrams are plotted for the studied model. Sensitivity analysis for the linear and nonlinear controllers’ gains is explored. Obtained analytical results demonstrated that the studied model had symmetric bifurcation behaviours in both the horizontal and vertical directions. In addition, the integration of the cubic position controller made the control algorithm more flexible to reshape system dynamical behaviours from the hardening spring characteristic to the softening spring characteristic (or vice versa) to avoid resonance conditions. Moreover, the optimal design of the cubic position gain and/or cubic velocity gain could stabilise the unstable motion and eliminate the nonlinear effects of the system even at large disc eccentricities. Lastly, numerical validations for all acquired results are performed, where the presented simulations show accurate correspondence between numerical and analytical investigations

Control Performance, Stability Conditions, and Bifurcation Analysis of the Twelve-Pole Active Magnetic Bearings System

The active magnetic bearings system plays a vital role in high-speed rotors technology, where many research articles have discussed the nonlinear dynamics of different categories of this system such as the four-pole, six-pole, eight-pole, and sixteen-pole systems. Although the twelve-pole system has many advantages over the eight-pole one (such as a negligible cross-coupling effect, low power consumption, better suspension behaviors, and high dynamic stiffness), the twelve-pole system oscillatory behaviors have not been studied before. Therefore, this article is assigned to explore the effect of the magneto-electro-mechanical nonlinearities on the oscillatory motion of the twelve-pole system controlled via a proportional derivative controller for the first time. The normalized equations of motion that govern the system vibrations are established by means of classical mechanics. Then, the averaging equations are extracted utilizing the asymptotic analysis. The influence of all system parameters on the steady-state oscillation amplitudes is explored. Stability charts in a two-dimensional space are constructed. The stable margin of both the system and control parameters is determined. The obtained investigations reveal that proportional gain plays a dominant role in reshaping the dynamics and motion bifurcation of the twelve-pole systems. In addition, it is found that stability charts of the system can be controlled by simply utilizing both the proportional and derivative gains. Moreover, the numerical simulations showed that the twelve-poles system can exhibit both quasiperiodic and chaotic oscillations besides the periodic motion depending on the control parameters’ magnitude