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In this paper, we present the general Bernstein functions for the first time. The properties of generalized Bernstein basis functions are given and demonstrated. The classical Bernstein polynomial bases are merely a subset of the general Bernstein functions. Based on the new Bernstein base functions and the collocation method, we present a numerical method for solving linear and nonlinear χ-fractional differential equations (χ-FDEs) with variable coefficients. The fractional derivative used in this work is the χ-Caputo fractional derivative sense (χ-CFD). Combining the Bernstein functions basis and the collocation methods yields the approximation solution of nonlinear differential equations. These base functions can be used to solve many problems in applied mathematics, including calculus of variations, differential equations, optimal control, and integral equations. Furthermore, the convergence of the method is rigorously justified and supported by numerical experiments. [ABSTRACT FROM AUTHOR]

This article presents the cotangent fractional Gronwall inequality, a novel understanding of the Gronwall inequality within the context of the cotangent fractional derivative. We furnish an explanation of the cotangent fractional derivative and emphasize a selection of its distinct characteristics before delving into the primary findings. We present the cotangent fractional Gronwall inequality (Lemma 3.1) and a Corollary 3.2 using the Mittag-Leffler function, we establish singularity and compute an upper limit employing the Mittag-Leffler function for solutions in a nonlinear delayed cotangent fractional system, illustrating its practical utility. To underscore the real-world relevance of the theory, a tangible instance is given. [ABSTRACT FROM AUTHOR]

In this paper, we define and discuss the Orlicz sequence spaces ΓM(∆n u, q, s) and ΓM(p, ∆n u, q, s) of entire sequences, where (X, q) is a seminormed space and s is any real number such that s ≥ 0. Furthermore, we study some topological properties of these spaces. We discuss some inclusion relations between them and related results. [ABSTRACT FROM AUTHOR]

This paper presents an autonomous navigation system of Differential-Drive Wheeled Robot (DDWR) that is divided in to tracking a trajectory and static and dynamic obstacles avoidance. Three fuzzy controllers are used for this navigation system; tracking a desired path autonomously, which is obtained based on changing the robot's heading-angle toward the next desired step in the desired path using a tracking fuzzy controller. The second fuzzy controller is responsible for static obstacles avoidance based on changing the robot's heading-angle and the third one is used to avoid dynamic obstacles based on changing the robot's heading-angle and velocity. A tuned Kinematic Controller (KC) using the Genetic Algorithm (GA) is used for enhancing the trajectory-tracking task. An EV3 robot is utilized and three types of sensors are used; an accelerometer sensor, gyroscope sensor, and ultrasonic sensor. The results obtained verify that the robot is able to track a trajectory and avoid both static and dynamic obstacles correctly. [ABSTRACT FROM AUTHOR]

The morphing wing has recently received a lot of interest due to the development of smart materials and sophisticated control structures. During different phases of flight, such as take-off, cruising, and landing, the aerodynamic shape of a morphing wing can be improved. Various morphing wing structure concepts have been explored in recent years to improve flight performance. The morphing wing with a flexible leading edge, which is smooth and continuous without step or slot, is easier to construct and it promises more reliability than traditional high-lift configurations such as leading-edge slats. Significant geometric changes in an aircraft's wing during flight may enable efficient performance in disparate mission roles or enable new multi-role missions that would not be possible with a fixed-geometry aircraft. Traditional actuators and mechanisms are used in current aircraft to vary wing sweep for flight in different speed regimes, as well as to change the wing area and camber, and extend the flaps during landing and takeoff. An integrated actuator with problem detection, feedback and position control, onboard controls, and communication capabilities is called a smart actuator. This eliminates the need for external controllers and makes it simpler to install, operate, configure, and monitor the actuator. There are three different types of morphing mechanisms: airfoil morphing, in-plane morphing, and out-ofplane morphing. Each one of these categories has a different effect on aerodynamic qualities, and this project will focus on camber change and controlling it by using Intelligent Artificial then using it in model tests and ANSYS simulations. Copyright © 2023 Praise Worthy Prize S.r.l. - All rights reserved. [ABSTRACT FROM AUTHOR]

In this paper, straightforward and efficient brute force method is developed to slew spacecraft in the presence of static and dynamic constrained regions. Fuzzy logic is utilized to develop a fuzzy based controller that is used to control the maneuvers and test the proposed method. The controller performance is first tested in an unconstrained attitude maneuvers and then it is utilized to test the proposed method. The spacecraft is assumed to be a 3U CubeSat and the results show through Monte Carlo simulations that the controller has successfully avoided the undesired static and dynamic celestial objects. The proposed method inherits its simplicity from the geometric approach and has the ability to monitor continuously the constraints and it shows that the proposed method works in the presence of multiple constraints. [ABSTRACT FROM AUTHOR]

We present a new numerical approach to solving the fractional differential Riccati equations numerically. The approach—called the Mittag-Leffler–Galerkin method—comprises the finite Mittag-Leffler function and the Galerkin method. The error analysis of the method was studied. As a result, we present two theorems by which the error can be bounded. In addition to error analysis, the residual correction method, which allows us to estimate the error and obtain new approximate solutions, is also presented. To show how the method is applied, and the efficiency of the proposed method, some test examples were considered. When the numerical results obtained were examined, it was found that while the method achieves better results than some of the known methods in the literature, it also achieves results that are similar to those of others of the known methods. [ABSTRACT FROM AUTHOR]

Simultaneous Localization And Mapping (SLAM) is the ability of an autonomous robot to localize itself within the surroundings, while also constructing a map. It has several applications in a variety of industries, especially in indoor navigation, Autonomous driving, and robotics. This research uses an omnidirectional mobile robot to present the SLAM algorithm based on Extended Kalman Filter (EKF) in an indoor environment. Usually, SLAM combines data from odometry and other sensors, in order to overcome the accumulating error due to the wheel sliding that usually appears when using the odometry alone. Thus, in this study a Kinect sensor is used along with an odometer sensor in an unknown environment to reach a predefined point with a free collusion path using three main fuzzy controllers as part of its navigation system. The first fuzzy controller is responsible for the goal-seeking problem, where the robot moves towards the goal, the second and third controllers are responsible for static and dynamic obstacles avoidance. The robot changes its direction to avoid the static obstacles and its direction and speed to avoid the dynamic obstacles. An EV3 Lego robot is used to verify the effectiveness of the proposed study. Results shows the effectiveness of the used algorithm; the decentralized algorithm simplifies the control process, which decreases the error with the help of EKF. [ABSTRACT FROM AUTHOR]

The aim of this paper is to derive oscillation criteria of the following fourth-order differential equation with delay term r x z ′ ′ ′ x γ ′ + ∑ i = 1 n q i x f z η i x = 0 , under the assumption ∫ x 0 ∞ r − 1 / γ s d s = ∞. The results are based on comparison with the oscillatory behaviour of second-order delay equations and the generalised Riccati transformation. Not only do the provided theorems provide an entirely new technique but also they vastly improve on a number of previously published conclusions. We give three examples to illustrate our findings. [ABSTRACT FROM AUTHOR]

In this paper, the Bernstein collocation method (BCM) is used for the first time to solve the nonlinear magnetohydrodynamics (MHD) Jeffery–Hamel arterial blood flow issue. The flow model described by nonlinear partial differential equations is first transformed to a third-order one-dimensional equation. By using the Bernstein collocation method, the problem is transformed into a nonlinear system of algebraic equations. The residual correction procedure is used to estimate the error; it is simple to use and can be used even when the exact solution is unknown. In addition, the corrected Bernstein solution can be found. As a consequence, the solution is estimated using a numerical approach based on Bernstein polynomials, and the findings are verified by the 4th-order Runge–Kutta results. Comparison with the homotopy perturbation method shows that the present method gives much higher accuracy. The accuracy and efficiency of the proposed method were supported by the analysis of variance (ANOVA) and 95% of confidence on interval error. Finally, the results revealed that the MHD Jeffery–Hamel flow is directly proportional to the product of the angle between the plates α and Reynolds number Re. [ABSTRACT FROM AUTHOR]

This work presents a study in the theoretical and experimental performance of PV cells and tracking surfaces using the isotropic and anisotropic (HDKR) models' simulations for single and dual axis systems. The solar spatial and temporal coordinates are obtained for the experiment site during Julian days throughout the year. From there, further simulations are performed using Typical Meteorological Year (TMY) data to evaluate various incident irradiance parameters on tracking surfaces. Experiments with different systems including fixed, single-axis, and dual-axis systems are conducted and compared with simulation results. The simulation outcomes show a gain in PV power generation of around 6% in the anisotropic system compared to the isotropic model prototype. However, experiments show no difference between both models in dual-axis trackers. By using TMY data for optimization of tracking angles, the dual-axis trackers performance gain is 8.5% in PV power generation compared to single-axis system, and a 37% gain is observed compared to fixed-tilted PV cell. Finally, the tracking systems' optimized angles are not a function of the model used (isotropic/anisotropic) during summer months. [ABSTRACT FROM AUTHOR]

Heart disease is one of the leading causes of death. Early diagnosis improves the treatment quality and reduces the mortality rates. Using Electrocardiogram (ECG) signals in health diagnosis has attracted researchers to develop a qualified intelligent medical decision support system. In this research, a smart early detection system for three common heart diseases, namely Apnea, Atrial Fibrillation (AF), and Heart Failure (HF) is developed, by analyzing the ECG signal for the patients using artificial intelligence (AI) techniques. The designed system is implemented using three different artificial intelligence techniques, namely Artificial Neural Networks (ANN), Support Vector Machine (SVM), and K-Nearest Neighbor (KNN). Electrocardiogram (ECG) signals from PhysioNet and data collected by the researchers are analyzed, and four features are extracted; then, they are used as an input for the classification algorithms. The results show the many advantages of using the proposed artificial intelligence techniques, which can result in saving the lives of patients with heart diseases. The accuracy of classification using K-nearest Neighbor is 92.4% at k=1. Using the Artificial Neural Networks provides an accuracy of 95.7% at 33 epochs, while the highest classification accuracy is obtained using Support Vector Machine with a classification accuracy of 97.8%. [ABSTRACT FROM AUTHOR]

Wheelchairs have been an active-research problem in the maneuverability of mobile robots over the past decade. This study aims to develop the transportability of disabled people by controlling the motion of their smart wheelchairs. It presents a design of omnidirectional mobile robot where its wheels can be equipped to replace a conventional electric wheelchair in order to achieve flexible and reliable maneuverability. The decentralized algorithm is used for motion control of the omnidirectional robot, which deals with the three independent components of robot's motion in the body coordinate frame: rotational moving, horizontal moving and vertical moving, which can be controlled separately with separated and different sub-controllers. The simulation analysis of the proposed study has been tested in authors' previous work and two approaches of artificial intelligent-based controllers (PD-Fuzzy-P and GA-PID controllers) have been built in order to control optimally the maneuverability of such system. The results have showed that the PD-fuzzy-P controller has converged faster than the GA-PID controller and the robot has been able to track successfully the sharp curves' maneuverability, such as a 90° corner in the squared shape and a U-turn in the rose shape with an error approaching zero. In this paper, the experimental tests of the PD-fuzzy-P controller are carried out on a real robot developed using EV3 Lego kit parts and Vicon motion capturing system is used to capture the robot's motion in various trajectory scenarios in order to test the effectiveness of the proposed control design. [ABSTRACT FROM AUTHOR]

Braking system plays a major safety feature in any vehicle; it is considered as one of the key points in safe motoring. The main objective of a good braking system is to provide safe stopping, in addition to the fact that it provides repeatable stopping. Anti-lock Braking System (ABS) is one of the recently developed safety feature used in a vehicle in order to prevent wheels from locking. This article proposes an intelligent adaptive fuzzy logic (FL) genetic algorithm (GA) controller for the Anti-Lock Braking System (ABS) to control the braking of a vehicle by controlling the slip ratio based on quarter car model. The main objective of the proposed controller is to prevent wheels from locking, which is a major safety issue in any braking system for different road conditions. The proposed controller consists of a Fuzzy Logic (FL) Genetic Algorithm (GA) based controller. The Genetic Algorithm is utilized in tuning the membership functions of the Fuzzy logic controller, which is developed to calculate the amount of reduction of the applied torque due to applying brake based on the error of the measured slip ratio signal compared with the desired one. Then a Genetic Algorithm (GA) is utilized in tuning the Fuzzy Logic parameters, which are the center of the Gaussian membership function, and the spread (б). The simulation results of the proposed fuzzy logic genetic algorithm controller are compared with both the results of the Fuzzy logic controller and the PD controller developed in previous work of the authors. The simulation results present many advantages of using Fuzzy Logic Genetic Algorithm based controller over both the Fuzzy Logic and the PD controllers, and the simulation is carried out at different road conditions, namely dry, wet and icy road surface. [ABSTRACT FROM AUTHOR]

Suspension system plays a major role in both comfort and stability of a vehicle. This paper presents modeling and controlling for a 3 Degree of Freedom (DOF) active suspension system. Four controllers are designed to control the response of the active suspension system, namely PID, LQR, Fuzzy Logic Controller (FLC) and Artificial Neural Network (ANN). The response for both the active suspension system and the passive suspension system is compared. For passive suspension system, it has been found out that it is hard to improve both passenger comfort and road handling at the same time, because of the fixed parameters that cannot be changed during the work. On the other hand, in active suspension system, both ride comfort and road handling can be improved. This work has showed that ANN, FLC, LQR, and PID controllers can be used with an active suspension system in order to improve the performance, the stability, and the ride comfortability compared to the passive suspension system. All these controllers are simulated using MATLAB and Simulink. Different road profiles are used to test the active suspension system response, such as a step input of 0.1 m, and a sinewave of amplitude of 0.3m and a frequency of 0.318Hz. All the controllers show better response compared to passive suspension system. A compromise can be done to choose the controller depending on the desired states. [ABSTRACT FROM AUTHOR]

Mobile robots and their motion control are two of the challenging and active research areas nowadays. Wheeled robots can be very fast and stable when moving on smooth terrain. On the other hand, legged robots are far superior when it comes to rough terrains and environments that contain many obstacles, which is the case of many real-life situations. In this article, and the design and the implementation of a hexapod robot with 18 DOF is presented. This robot contains six legs attached to a main body known as the trunk. Each leg will have three links known as the coxa, the femur and the tibia, resulting in 3 joints and consequently 3 DOF for each leg powered by 3 DC servo motors. The amount of rotation for each servo motor is controlled by an Arduino microcontroller boards. Inverse and forward kinematics are studied to produce a mathematical model for the motion of the legs based on a predefined trajectory for the end effector of each leg. The robot is designed using CAD software and a simulation is carried out in order to validate the inverse and forward kinematics results. In addition, an experimental prototype setup is built and tested. Experimental results for the robot motion show smooth motion for the robot. It is tested on different types of terrains, namely, smooth and rough terrains, and the results are promising in each case. Video for the motion will be provided. [ABSTRACT FROM AUTHOR]

The study was conducted to examine the effect of storage temperature and duration on the viability and in vitro germination of pollen grains of seven pistachio cultivars. Pollens were stored at room temperature (24±2 oC), refrigerator (4 oC) and freezer (-5 oC) for 0, 1, 2, 3 and 4 weeks. Pollen viability was estimated by using staining methods including tetrazolium test (TTC), iodine and potassium iodine (IKI) and safranin solutions, and using an in vitro pollen germination. The results showed that at all storage methods and durations, pollen viability and in vitro pollen germination were significantly the highest for Batouri and Ashouri cultivars and the lowest for Marawhi and Elemi cultivars. The highest pollen viability as estimated by safranin staining was attained when pollens were stored under freezer condition. However, pollen viability by TTC was the lowest at room temperature storage. In addition, in vitro pollen germination and viability significantly decreased as storage duration increased. This study revealed no differences between in vitro germination percentages in refrigerated and freezer stored pollen up to 2 weeks. Meanwhile, in vitro germination of room-stored pollen gradually decreased when storage duration increased. At the end of storage period, pollen viability was reduced slightly under freezer conditions, whereas, the reduction in viability was the highest for refrigerated and room-stored pollen with no differences between them. This study showed a significant interaction effect of cultivar x storage temperature x storage duration on pollen viability but not for in vitro pollen germination. [ABSTRACT FROM AUTHOR]

The interaction between the human body motion and the robotic motion has gained much interest nowadays. Enhancing the accuracy of such a system requires the use of more complex systems in order to maintain the safety of the user, especially in critical systems such as disabled assistant systems. In this work, an investigation of motion control using Surface electromyography (sEMG) signals, which are obtained from the user's arm, is studied. This system can help disabled drivers who can use their arms and cannot use their feet for speed or braking pedals. The sEMG signal can be obtained by fixing a gesture with multi sEMG sensors on the user's arm between the elbow and the wrist. The controller receives the sEMG signal and the proper signal is sent to the motors responsible for accelerating or decelerating the vehicle through speed or braking pedal, respectively. The sEMG signal is usually time-varying noisy signal; hence, a proper filter is needed to attenuate the unwanted noisy signal. In addition, an intelligent controller is developed to control vehicles speeding and braking safely. In this paper a Fuzzy logic controller is chosen due to the high nonlinearity of such a system. The system can be improved to cover different activities and actions based on the user's needs. [ABSTRACT FROM AUTHOR]

Last decade witnessed a significant research effort directed towards heat sink design to improve its performance. Conventional heat sinks consist of parallel plates or pin fins. Ladder heat sink design is one of the effective recent designs formed by inserting a link between two parallel plates. In this work, the performance of ladder heat sink design is studied and compared with two designs, namely; elliptical and parallel plate heat sink designs. A computational simulation of the heat sink designs carried away using COMSOL Multiphysics software. Adaptive Neuro Fuzzy Inference System (ANFIS) is used to predict the pressure drop value with changing the dimensions of the heat sink fins, namely; length, width and height, and comparing the results with COMSOL values. Results show that ladder heat sink design has better performance in comparison with the other heat sink designs according to many parameters used to characterize the performance of the heat sink design, namely; pressure drop, temperature, cooling power and fluid velocity. Also, ANFIS shows accurate results in predicting the pressure drop value compared with the accurate value obtained from COMSOL Multiphysics. [ABSTRACT FROM AUTHOR]

In this paper, we propose an approximate solution method, called multistage Bernstein collocation method (MBCM), to solve strongly nonlinear damped systems. The method is given with an error analysis. The systems investigated include step function external excitation and periodical function external excitation. We found that the MBCM gives more accurate approximate solutions than the standard collocation method. Since MBCM splits the domain of the problem [a,b] into n parts, one can overcome oscillations arising from the standard collocation method by increasing n. The results obtained from MBCM compare favorably with that of the fourth-order Runge–Kutta method (RK4). [ABSTRACT FROM AUTHOR]

This study explored the effects of using online chat and word processors on students' writing skills that include; organizing a text, spelling, punctuation, grammar, phrasal verbs, idioms, idiomatic expressions, pragmatics, creativity, vocabulary growth, content, relational words, conjunctions, authenticity, figures of speech, imagination, coherence, style, socio-cultural aspects, language use, and the production of authentic text. The study group consisted of students in the Department of Languages and Translation at Taibah University who registered for the Writing Two course in the first semester of the 2012 - 2013 academic year. Forty subjects were divided into two sections: section one was assigned as an experimental group (supported by Facebook and Skype) and section two was assigned as a control group and was asked to write their essays with paper and pencil. Facebook and Skype accounts were created for every student in the experimental group. Data was analyzed from pre-test and post-test results to evaluate the question posed by the study: Does the use of online text chat assisted with word processors help undergraduate students develop their writing skills more than traditional methods of teaching? The results revealed that students who worked with Facebook and Skype showed a significant improvement in their writing skills when compared to the control group. In light of these findings, it is recommended that online discussions via Facebook, Skype, and other social media sites should be utilized when teaching writing and the other language skills. [ABSTRACT FROM AUTHOR]

This study explored the effects of using online chat and word processors on students' writing skills that include; organizing a text, spelling, punctuation, grammar, phrasal verbs, idioms, idiomatic expressions, pragmatics, creativity, vocabulary growth, content, relational words, conjunctions, authenticity, figures of speech, imagination, coherence, style, socio-cultural aspects, language use, and the production of authentic text. The study group consisted of students in the Department of Languages and Translation at Taibah University who registered for the Writing Two course in the first semester of the 2012 - 2013 academic year. Fourty subjects were divided into two sections: section one was assigned as an experimental group (supported by Face-book and Skype) and section two was assigned as a control group and was asked to write their essays with paper and pencil. Facebook and Skype accounts were created for every student in the experimental group. Data was analyzed from pre-test and post-test results to evaluate the question posed by the study: Does the use of online text chat assisted with word processors help undergraduate students develop their writing skills more than traditional methods of teaching? The results revealed that students who worked with Face-book and Skype showed a significant improvement in their writing skills when compared to the control group. In light of these findings, it is recommended that online discussions via Facebook, Skype, and other social media sites should be utilized when teaching writing and the other language skills. [ABSTRACT FROM AUTHOR]

This experimental study investigates the effect of using web-cam chat on the FL Learners' socio-cultural competence by answering the following question: What is the effect of using web-Cam chat on the socio-cultural on competence of tenth grade students? The sample of study consisted of 43 tenth grade students at AL-Sabila Private School for Boys, in the second semester of the scholastic year 2010/2011. Section (A) consisted of (20) students. It was chosen as an experimental group, while section (B) consisted of (23) students. It was decided as a control group. Data were collected within sixteen weeks. The pre-test and the post-test were of two parts: A written part and oral dialogues that test the FL students' ability to interact culturally appropriately in communicative settings. Scores were tabulated and prepared for statistical analysis. The results of the study have showed there is a statistically significant difference in the arithmetic mean in the achievement test for the experimental group. It means that the experimental group performed better than the control group. Then, the result of the study revealed that using web-Cam chat alongside the traditional method has great effect on the socio-cultural competence of the FL learners. The researcher has recommended making many studies on the effect of CMC another competencies of language. [ABSTRACT FROM AUTHOR]

We present a study of the dynamic behavior of a microelectromechanical systems (MEMS) device consisting of an imperfect clamped-clamped microbeam subjected to electrostatic and electrodynamic actuation. Our objective is to develop a theoretical analysis, which is able to describe and predict all the main relevant aspects of the experimental response. Extensive experimental investigation is conducted, where the main imperfections coming from microfabrication are detected, the first four experimental natural frequencies are identified and the nonlinear dynamics are explored at increasing values of electrodynamic excitation, in a neighborhood of the first symmetric resonance. Several backward and forward frequency sweeps are acquired. The nonlinear behavior is highlighted, which includes ranges of multistability, where the nonresonant and the resonant branch coexist, and intervals where superharmonic resonances are clearly visible. Numerical simulations are performed. Initially, two single mode reduced-order models are considered. One is generated via the Galerkin technique, and the other one via the combined use of the Ritz method and the Padé approximation. Both of them are able to provide a satisfactory agreement with the experimental data. This occurs not only at low values of electrodynamic excitation, but also at higher ones. Their computational efficiency is discussed in detail, since this is an essential aspect for systematic local and global simulations. Finally, the theoretical analysis is further improved and a two-degree-of-freedom reduced-order model is developed, which is also capable of capturing the measured second symmetric superharmonic resonance. Despite the apparent simplicity, it is shown that all the proposed reduced-order models are able to describe the experimental complex nonlinear dynamics of the device accurately and properly, which validates the proposed theoretical approach. [ABSTRACT FROM AUTHOR]

The idea of difference sequence spaces was introduced by Kizmaz [6], and this concept was generalized by Bektas and Colak [1]. In this paper, we define the sequence spaces c0(F, Δmu x) and l∞(F, Δmu x), where F = (fk) is a sequence of modulus functions, and examine some inclusion relations and properties of these spaces. [ABSTRACT FROM AUTHOR]

The propose of this paper is to define and study the Nθ(A, F, u, Δ) and Nθ0 (A, F, u, Δ) of lacunary strong (A, u, Δ)--convergent sequences with respect to a sequence of modulus functions, where A = (aik) is an infinite matrix of complex numbers, u = (ui) is an arbitrary sequence such that ui ≠ 0, i = 1, 2, 3, ..., and for any sequence x = (xi), the difference sequence Δx is given by Δx = (Δx)i=1∞ = (xi - xi-1)i=1∞. We also study some inclusion relations between lacunary strong (A, u, Δ)--convergence and lacunary (A, u, Δ)--statistical convergence. These new spaces will give as special case those spaces defined and studied by T. Bilgin in 2004. [ABSTRACT FROM AUTHOR]

In this paper, we define the sequence spaces W∞ (M, q; p, u, Δ), W0 (M, q; p, u, Δ), W(M, q; p, u, Δ), where for any sequence x = (xn), the difference sequence Δx is given by Δx = (Δxn)n=1∞ = (xn - xn-1)n=1∞, also we study some properties and theorems of these spaces. These are generalizations of those defined and studied by Z.U. Ahmad, Abroad H.A. Bataineh and some others previously. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two new methods to solve systems of ordinary differential equations. The first method is constituted of the generalized Bernstein functions, which are obtained by Bernstein polynomials, and operational matrix of differentiation with collocation method. The second method depends on tau method, the generalized Bernstein functions and operational matrix of differentiation. These methods produce a series which is obtained by non-polynomial functions set. We give the standard Bernstein polynomials to explain the generalizations for both methods. By applying the residual correction procedure to the methods, one can estimate the absolute errors for both methods and may obtain more accurate results. We apply the methods to some test examples including linear system, non-homogeneous linear system, nonlinear stiff systems, non-homogeneous nonlinear system and chaotic Genesio system. The numerical shows that the methods are efficient and work well. Increasing m yields a decrease on the errors for all methods. One can estimate the errors by using the residual correction procedure. [ABSTRACT FROM AUTHOR]

In this study, we introduce an efficient computational method to obtain an approximate solution of the time-dependent Emden-Fowler type equations. The method is based on the 2D-Bernstein polynomials (2D-BPs) and their operational matrices. In the cases of time-dependent Lane–Emden type problems and wave-type equations which are the special cases of the problem, the method converts the problem to a linear system of algebraic equations. If the problem has a nonlinear part, the final system is nonlinear. We analyzed the error and give a theorem for the convergence. To estimate the error for the numerical solutions and then obtain more accurate approximate solutions, we give the residual correction procedure for the method. To show the effectiveness of the method, we apply the method to some test examples. The method gives more accurate results whenever increasing n , m for linear problems. For the nonlinear problems, the method also works well. For linear and nonlinear cases, the residual correction procedure estimates the error and yields the corrected approximations that give good approximation results. We compare the results with the results of the methods, the homotopy analysis method, homotopy perturbation method, Adomian decomposition method, and variational iteration method, on the nodes. Numerical results reveal that the method using 2D-BPs is more effective and simple for obtaining approximate solutions of the time-dependent Emden-Fowler type equations and the method presents a good accuracy. [ABSTRACT FROM AUTHOR]

The aim of this research is to study the oscillatory properties of higher -order delay half linear differential equations with non-canonical operators. Two techniques for establishing new oscillation conditions for all solutions of higher-order differential equations will be presented. The first method to employ the Riccati transformations, which differ from those described in some published works. The second method employs comparison principles with first-order delay differential equations, from which it is straightforward to deduce oscillation for all studied equation solutions. In addition to improving, extending, and significantly simplifying the previously established criteria, the newly proposed criteria have the potential to serve as a benchmark for the theory of delay differential equations of higher order, which is still in its infancy of development. We were able to determine three fundamental theorems regarding the oscillation of this equation. Some examples will be provided to illustrate the findings. [ABSTRACT FROM AUTHOR]

We extend the spaces E(u; Δ²) defined and studied by Mursaleen [14] to E(p; u, Δ²) and propose to determine their α- and β-duals and to study some matrix maps. Our results generalize the results of Mursaleen [14] and yield them as a special case for pn = a, constant sequence. [ABSTRACT FROM AUTHOR]