Special Differential Equations Research Articles

Development and Analysis of Embedded Explicit Runge-Kutta Methods for Directly Solving Special Fifth-Order Quasi-Linear Ordinary Differential Equations

Development and Analysis of Embedded Explicit Runge-Kutta Methods for Directly Solving Special Fifth-Order Quasi-Linear Ordinary Differential Equations

A class of polynomial approximation methods to second-order delay differential equations

This paper studies special high-order methods to numerically solve special second-order ordinary differential equations and second-order differential equations with constant delay, which are based on vector field expansion along the shifted and scaled Legendre polynomial orthonormal basis. In the framework of method design, we obtain the second-order perturbation results by transforming the second-order differential equations into first-order ones, and the error accuracy estimation of numerical solutions is achieved by using perturbation results for the problem under consideration. In addition, some numerical tests on specific second-order delay Hamiltonian systems are shown, the purpose of which is to verify the effectiveness of the theoretical findings.

Characteristic Mode Solution of Complex-Coefficient Complex-Solution Differential Equations

Computation of complex-coefficient complex-solution differential equations is a problem that arises in various domains of science and engineering. This paper aims at applying the Theory of Characteristic Modes (TCM) approach along with the Method of Moments (MoM) in solving these problems with emphasis on procedures for higher differential equations. Several available methods, known in literatures, are available for solving the problem. The complexity of the available methods differs based on the accuracy of the solution. In this paper, the general method is first presented and then a simplified version of it is proposed to solve high order differential equations. Two examples are illustrated, a third and a fourth order complex-coefficients complex-solution differential equations, to show the simplicity of the proposed method. The proposed approach can be also introduced along with other methods to solve these special occurrences differential equations and other boundary value problems.

Image Compression and Protection Systems Based on Atomic Functions

Digital images are a particular type of data. They have numerous applications. Taking into account current challenges and trends, image compression and protection have to be ensured. Data format, which provides fast analysis of the image compressed, is needed. In order to satisfy a combination of these requirements, an appropriate information system should be developed. In this paper, we design such a system based on atomic functions (AF) that are solutions of special functional differential equations and, in terms of function theory, are as good constructive tools as trigonometric polynomials. AF-based image processing system (AFIPS), which satisfies the requirements considered, is developed. A core of this system is discrete atomic transform (DAT). Data protection feature of AFIPS is provided by the possibility to vary a structure of the procedure DAT. Constructive approximation properties of AF ensure high lossy and lossless image compression, as well as good image representation by DAT-coefficients. Software implementation of AFIPS is investigated. The results of test data processing are given.

Controllability of a generalized multi-pantograph system of non-integer order with state delay

<abstract><p>This paper presents the dynamical aspects of a nonlinear multi-term pantograph-type system of fractional order. Pantograph equations are special differential equations with proportional delays that are employed in many scientific disciplines. The pantograph mechanism, for instance, has been applied in numerous scientific disciplines like electrodynamics, engineering, and control theory. Because of its key rule in diverse fields, the current study establishes some necessary criteria for its controllability. The main idea of the proof is based on converting the system into a fixed point problem and introducing a suitable controllability Gramian matrix $ \mathcal{G}_{c} $. The Gramian matrix $ \mathcal{G}_{c} $ is used to demonstrate the linear system's controllability. Controllability criteria for the associated nonlinear system have been established in the sections that follow using the Schaefer fixed-point theorem and the Arzela-Ascoli theorem, as well as the controllability of the linear system and a few key assumptions. Finally, a computational example is listed.</p></abstract>

On Continued Fractions and Their Applications

Continued fractions constitute a very important subject in mathematics. Their importance lies in the fact that they have very interesting and beautiful applications in many fields in pure and applied sciences. This review article will reveal some of these applications and will reflect the beauty behind their uses in calculating roots of real numbers, getting solutions of algebraic Equations of the second degree, and their uses in solving special ordinary differential Equations such as Legendre, Hermite, and Laguerre Equations; moreover and most important, their use in physics in solving Schrodinger Equation for a certain potential. A comparison will also be given between the results obtained via continued fractions and those obtained through the use of well-known numerical methods. Advances in the subject will be discussed at the end of this review article.

Some Characterization Results of Nonlocal Special Random Impulsive Evolution Differential Equations

In this paper, we present existence and uniqueness of special random impulsive differential evolution equations with nonlocal condition in Hilbert spaces. Moreover we study the stability results for the same evolution equations. Existence and uniqueness results are proved using Banach fixed point theorem where as stability results using fixed point approach and semi group theory. Finally we give some applications of the nonlocal impulsive differential equations as well as evolution equations, which shows the importance of our theoretical results.

Underwater Robot Manipulator Control

This paper deals with the problem of bringing the end effector (grip center) of an underwater vehicle anthropomorphic manipulator to a predetermined position in a given time using the terminal state method. A dynamic model with the account of joint drives dynamics is formulated on the basis of obtained kinematic model constructed by using the Denavit-Hartenberg method (DH model). The DH model is used in a terminal nonlinear criterion that displays estimate of the proximity of the effector's orientation and position to the specified values. The dynamic model is adapted for effective application of the author's terminal state method (TSM) so that it forms a system of differential equations for the rotation angles of manipulator links around the longitudinal and transverse axes, having only desired TSM-controls in the right parts. The converted model provides simplifications of controls calculation by eliminating the numerical solution of special differential equations, that is needed in the case of using in TSM nonlinear dynamic models in general form. The found TSM-controls are further used in expressions for control actions on joints electric drives obtained on the basis of electric drives dynamic models. Unknown drives parameters as functions of links rotation angles or other unknown factors, are proposed to be determined experimentally. Such two-step procedure allowed to get drive control in the form of algebraic and transcendental expressions. Finally, by applying the developed software, simulation results of the manipulator end effector moving to the specified positions on the edge of the working area are presented. The resulting error (without accounting measurement error) does not exceed 2 centimeters at the 1.2 meters distance by arm reaching maximum of length ability. The work was performed under the Federal program of developing a robotic device for underwater research in shallow depths (up to 10 meters).

On the Approximated Solution of a Special Type of Nonlinear Third-Order Matrix Ordinary Differential Problem

Matrix differential equations are at the heart of many science and engineering problems. In this paper, a procedure based on higher-order matrix splines is proposed to provide the approximated numerical solution of special nonlinear third-order matrix differential equations, having the form Y(3)(x)=f(x,Y(x)). Some numerical test problems are also included, whose solutions are computed by our method.

Hamiltonian structure for a differential system from a modified Laguerre weight via the geometry of the modified third Painlevé equation

Recurrence coefficients of semi-classical orthogonal polynomials are often related to the solutions of special nonlinear second-order differential equations known as the Painlevé equations. Each Painlevé equation can be written in a standard form as a non-autonomous Hamiltonian system, so it is natural to ask whether differential systems satisfied by the recurrence coefficients also possess Hamiltonian structures. We consider recurrence coefficients for a modified Laguerre weight which satisfy a differential system known to be related to the modified third Painlevé equation and identify a Hamiltonian structure for it by constructing its space of initial conditions. We also discuss a transformation from this system to the modified third Painlevé equation which simultaneously identifies a discrete system for the recurrence coefficients with a discrete Painlevé equation.

The solution of nonlinear parabolic equation using variational iteration method

Variational iteration method is a semi analytic solution used to solve the parabolic differential equation both of homogen or nonhomogen. In the process of determining an approximation solution, this method did not use a linearization and a small pertubation. In this paper, the variational iteration method is implemented in the parabolic differential equation in the form of ut = uxx + f(u) + g(x, t) with appropriate intial condition. Furthermore, some examples of special parabolic differential equations are given to test the reliability and convergence of the method. Based on the result of study shows that the variational iteration method is able to solve the parabolic differential equation with a good accuration.

Embedded Exponentially-Fitted Explicit Runge-Kutta-Nyström Methods for Solving Periodic Problems

In this work, a pair of embedded explicit exponentially-fitted Runge–Kutta–Nyström methods is formulated for solving special second-order ordinary differential equations (ODEs) with periodic solutions. A variable step-size technique is used for the derivation of the 5(3) embedded pair, which provides a cheap local error estimation. The numerical results obtained signify that the new adapted method is more efficient and accurate compared with the existing methods.

Study of implicit delay fractional differential equations under anti-periodic boundary conditions

This research work is related to studying a class of special type delay implicit fractional order differential equations under anti-periodic boundary conditions. With the help of classical fixed point theory due to Schauder and Banach, we derive some results about the existence of at least one solution. Further, we also study some results including Hyers–Ulam, generalized Hyers–Ulam, Hyers–Ulam Rassias, and generalized Hyers–Ulam–Rassias stability. We provide some test problems for illustrating our analysis.

Common Hirota form Bäcklund transformation for the unified Soliton system

We study to unify soliton systems, KdV/mKdV/sinh-Gordon, through SO(2,1) ≅ GL(2, ) ≅ Möbius group point of view, which might be a keystone to exactly solve some special non-linear differential equations. If we construct the N-soliton solutions through the KdV type Bäcklund transformation, we can transform different KdV/mKdV/sinh-Gordon equations and the Bäcklund transformations of the standard form into the same common Hirota form and the same common Bäcklund transformation except the equation which has the time-derivative term. The difference is only the time-dependence and the main structure of the N-soliton solutions has the same common form for KdV/mKdV/sinh-Gordon systems. Then the N-soliton solutions for the sinh-Gordon equation is obtained just by the replacement from KdV/mKdV N-soliton solutions. We also give general addition formulae coming from the KdV type Bäcklund transformation which plays not only an important role to construct the trigonometric/hyperbolic N-soliton solutions but also an essential role to construct the elliptic N-soliton solutions. In contrast to the KdV type Bäcklund transformation, the well-known mKdV/sinh-Gordon type Bäcklund transformation gives the non-cyclic symmetric N-soliton solutions. We give an explicit non-cyclic symmetric 3-soliton solution for KdV/mKdV/sinh-Gordon equations.

Large spectral gap induced by small delay and its application to reduction

We consider general linear neutral differential equations with small delays in the view of pseudo exponential dichotomy. For the autonomous case, we first count the eigenvalues in a certain half plane, which generalized the previous works on serval special retarded differential equations. We next establish the existence of a pseudo exponential dichotomy for the nonautonomous case, and prove that the corresponding spectral gap approaches infinity as the delay tends to zero. The proof for this large spectral gap induced by small delay is owing to exact bounds and exponents for pseudo exponential dichotomy. Then based on above results, we give an invariant manifold reduction theorem for nonlinear neutral differential equations with small delays. Finally, our results are applied to a concrete example.

Multidimensional Bernstein polynomials and Bezier curves: Analysis of machine learning algorithm for facial expression recognition based on curvature

In this paper, by using partial derivative formulas of generating functions for the multidimensional unification of the Bernstein basis functions and their functional equations, we derive derivative formulas and identities for these basis functions and their generating functions. We also give a conjecture and some open questions related to not only subdivision property of these basis functions, but also solutions of a higher-order special differential equations. Moreover, we provide an implementation for a real world problem of human facial expression recognition with the help of curvature of Bezier curves whose machine learning supported by statistical evaluations on feature vectors using in the aforementioned machine learning algorithm.

Solving Oscillatory Delay Differential Equations Using Block Hybrid Methods

A set of order condition for block explicit hybrid method up to order five is presented and, based on the order conditions, two-point block explicit hybrid method of order five for the approximation of special second order delay differential equations is derived. The method is then trigonometrically fitted and used to integrate second-order delay differential equations with oscillatory solutions. The efficiency curves based on the log of maximum errors versus the CPU time taken to do the integration are plotted, which clearly demonstrated the superiority of the trigonometrically fitted block hybrid method.

An example of interplay between Physics and Mathematics: Exact resolution of a new class of Riccati Equations

A novel recipe for exactly solving in finite terms a class of special differential Riccati equations is reported. Our procedure is entirely based on a successful resolution strategy quite recently applied to quantum dynamical time-dependent SU(2) problems. The general integral of exemplary differential Riccati equations, not previously considered in the specialized literature, is explicitly determined to illustrate both mathematical usefulness and easiness of applicability of our proposed treatment. The possibility of exploiting the general integral of a given differential Riccati equation to solve an SU(2) quantum dynamical problem, is succinctly pointed out.

Laguerre Polynomial Solutions of a Class of Initial and Boundary Value Problems Arising in Science and Engineering Fields

In this study, we consider high-order nonlinear ordinary differential equations with the initial and boundary conditions. These kinds of differential equations are essential tools for modelling problems in physics, biology, neurology, engineering, ecology, economy, astrophysics, physiology and so forth. Each of the mentioned problems are described by one of the following equations with the specific physical conditions: Riccati, Duffing, EmdenFowler, Lane Emden type equations. We seek the approximate solution of these special differential equations by means of a operational matrix technique, called the Laguerre collocation method. The proposed method is based on the Laguerre series expansion and the collocation points. By using the method, the mentioned special differential equations together with conditions are transformed into a matrix form which corresponds to a system of nonlinear algebraic equations with unknown Laguerre coefficients, and thereby the problem is approximately solved in terms of Laguerre polynomials. In addition, some numerical examples are presented to demonstrate the efficiency of the proposed method and the obtained results are compared with the existing results in literature.

Efficient Geometric Numerical Methods for Solving Differential Systems

In this paper, we introduce numerical methods to solve differential equations systems for which good qualitative behaviour is essential. To do this, we equip the numerical methods introduced in [1] and [2] to the case that preserve the same qualitative behaviour. The proposed algorithms are efficient to solve special differential equations systems, like Lotka-Volterra problem.