Quasilinear Ordinary 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

Сингулярности квазилинейных дифференциальных уравнений

We study solutions of quasi-linear ordinary differential equations of the second order at their singular points, where the coefficient of the second-order derivative vanishes. Either solutions entering a singular point with definite tangential direction (proper solutions) or those without definite tangential direction (oscillating solutions) are considered. It is shown that oscillating solutions generically do not exist, and proper solutions enter a singular point in strictly definite tangential directions. A local representation for proper solutions in a form similar to Newton-Puiseux series is obtained.

Investigation of the problem of multifunctional work of a bulldozer-loader by reducing the mathematical model to pairwise nested numerical methods

The article attempts to substantiate the urgency of the developed bulldozer-loader with the facts of the consequences of emergencies associated with the objects of work, describes the technical, economic and overview characteristics of existing bulldozer-loaders. For a previously compiled mathematical model with a system in conveniently simplified quasilinear ordinary differential equations, the motions of the centers of mass of the working bodies with generalized coordinates and generalized velocities are described. The force fields of motion drives and equilibrium force fields are recommended and calculated. A numerical method of shooting was used, a general algorithm and a program for solving the problem were compiled, which were implemented. The results of the calculation were obtained, showing the tendencies of the correctness of the algorithms.

Kneser Solutions of Higher-Order Quasilinear Ordinary Differential Equations

Kneser Solutions of Higher-Order Quasilinear Ordinary Differential Equations

Алгоритмизация решения динамических краевых задач теории гибких пластин с учетом сдвига и инерции вращения

Most of the problems on flexible plates are solved in the Föppl-von Karman formulation, which is Love's special case. The constructed algorithms are not economical in terms of implementation on a computer. Therefore, construction of algorithms for the complete calculation of flexible plates with a given degree of accuracy with allowance for the shear and inertia of rotation is becoming a topical issue. The problem of creating an automated inference system and solving the equations of the theory of elasticity and plasticity were first posed in the monograph by V.K. Kabulov. In this work, for the first time, the main problems of algorithmization are formulated and ways of their machine solution are outlined. The problem of algorithmization is solved as follows: depending on geometric characteristics of the object and physical properties of the material, a design scheme of this model is selected; derivation of the initial differential equations and the corresponding boundary and initial conditions; selection of a computational algorithm and numerical solution of the obtained equations; analysis of the obtained numerical results describing the stress-strain state of the structure under consideration. This work consists of an introduction, three sections and a conclusion. In the first paragraph, the equations of motion of rectangular plates are given. Substituting the expression for the force of moments and shearing forces and introducing a dimensionless value, a system of equations in displacements is obtained. In the second section, using the central difference formulas, a system of quasilinear ordinary differential equations is obtained. Taking into account the boundary and initial conditions, the system of equations is reduced to matrix form, which can be solved by the Runge-Kutta method. In the third paragraph, an analysis of the results obtained is presented.

A thorough description of one-dimensional steady open channel flows using the notion of viscosity solution

Determining water surface profiles of steady open channel flows in a one-dimensional bounded domain is one of the well-trodden topics in conventional hydraulic engineering. However, it involves Dirichlet problems of scalar first-order quasilinear ordinary differential equations, which are of mathematical interest. We show that the notion of viscosity solution is useful in thoroughly describing the characteristics of possibly non-smooth and discontinuous solutions to such problems, achieving the conservation of momentum and the entropy condition. Those viscosity solutions are the generalized solutions in the space of bounded measurable functions. Generalized solutions to some Dirichlet problems are not always unique, and a necessary condition for the non-uniqueness is derived. A concrete example illustrates the non-uniqueness of discontinuous viscosity solutions in a channel of a particular cross-sectional shape.

Singular initial value problems for scalar quasi-linear ordinary differential equations

We discuss existence, non-uniqueness and regularity of one- and two-sided solutions of initial value problems for scalar quasi-linear ordinary differential equations where the initial condition corresponds to an impasse point of the equation. With a differential geometric approach, we reduce the problem to questions in dynamical systems theory. As an application, we discuss in detail second-order equations of the form g(x)u″=f(x,u,u′) with an initial condition imposed at a simple zero of g. This generalises results by Liang and also makes them more transparent via our geometric approach.

Singular Kneser solutions of higher-order quasilinear ordinary differential equations

Singular Kneser solutions of higher-order quasilinear ordinary differential equations

APPLICATION OF THE PRINCIPLE OF CONTRACTION MAPPING TO STUDY THE SOLUTION OF NONLINEAR FUNCTIONAL EQUATIONS IN BANACH SPACES

Processes of applied problems are often modeled by differential, integral and integro-differential equations, inequalities and inclusions with corresponding initial and boundary conditions determined by the environment.Functional models, where the unknown function and / or its derivatives have nonlinear dependences, often adequately describe the processes of the objects under study.The variety of mathematical models of nonlinear dependence requires a special scientific approach to their study and methods for finding solutions to problems. In applications, for an adequate course of the process, it becomes necessary to take into account the real circumstances in the essence of the varieties of nonlinear dependence.The questions investigated are related to the quasilinear differential equations solution.The theorem on the sufficient condition for the applicability of the principle of contraction mappings in space , , with reduction to a nonlinear integral equation, using the Green's function, to find the solution of the initial-boundary conditions of quasi-linear ordinary differential equations. The way is given in the proof of the theorem using the initial condition is determined, then with the same procedure of finding a sequence of functions , enables the approach to solving the problem with the desired accuracy. The above theorem and other as a byproduct, the results can be applied to research and find practical solutions of problems.This makes it possible to obtain a fairly wide application of the theorem, for example, when increasing the accuracy of the operation of automatic control and measuring devices.

Initial–Boundary Value Problem for a Nonlinear Beam Vibration Equation

We consider an initial–boundary value problem for the beam vibration equation, which is a fourth-order nonlinear equation with two independent variables. It is shown that under certain conditions on the initial data this problem can be reduced to the Cauchy problem for a countable system of quasilinear ordinary differential equations. Using the method of energy inequalities, we prove that this Cauchy problem has a solution. Based on this, we establish the existence of a local solution of the original initial–boundary value problem and construct it in closed form. A theorem on the uniqueness of a global solution is proved.

Strongly increasing solutions of higher-order quasilinear ordinary differential equations

Abstract In this paper we discuss the existence and asymptotic behavior of strongly increasing solutions of quasilinear ordinary differential equations of the form D ( α n , α n − 1 , … , α 1 ) x = p ( t ) | x | β sgn x , t ≥ a . $$\begin{array}{} \displaystyle D(\alpha_n, \alpha_{n-1}, \dots, \alpha_1)x = p(t)|x|^{\beta}\text{ sgn } x, \quad t \geq a. \end{array} $$ (1.1) It will be shown that there is an explicit difference between the cases α 1 α 2 ⋯ αn ≥ β and α 1 α 2 ⋯ αn < β for the structure of the totality of strongly increasing solutions of (1.1).

Regularization of the Movement of a Material Point Along a Flat Trajectory: Application to Robotics Problems

A control problem of the robot’s end-effector movement along a predefined trajectory is considered. The aim is to reduce the work against resistance forces and improve the comfortability of the motion. The integral of kinetic energy and weighted inertia forces for the whole period of motion is introduced as a cost functional. By applying variational methods, the problem is reduced to a system of quasilinear ordinary differential equations of the fourth order. Numerical examples of solving the problem for movement along straight, circular and elliptical trajectories are presented. For the sake of clarity, the problem is studied for a specific kind of a 3D printer in the 2DoF approximation. However, in the case of negligible masses of moving elements compared the mass of an end-effector, the solution is universal, i.e., it remains the same for given trajectories.

An identity for the Heaviside function and its application in representation of nonlinear Green’s function

An identity showing that any positive real power of the Heaviside function is almost everywhere equal to the Heaviside function of power 1 is proved in this paper. A similar identity is shown to hold for corresponding distributions. The identity is then used to attach a specific meaning to the nonlinear Green’s function of quasi-linear ordinary differential equations. As specific examples, nonlinear Green’s function is constructed for the forced cubic Duffing (explicitly) and fractionally nonlinear damped Lienard (implicitly) equations. Efficiency of the nonlinear Green’s function solution for various source functions is demonstrated numerically in comparison with the solution obtained by the well-known method of lines. Solutions obtained are unique in literature.

Generalized RK integrators for solving class of sixth-order ordinary differential equations

In this paper, the classical RK integrators for solving quasi-linear(QL) ordinary differential equations (ODEs) up to 5th-order have been generalized to solve class of 6th-order ODEs. The novel contribution of this work is the derivation of two efficient direct integrators for solving class of ODEs. Non-linear equations of order conditions (OCs) for the proposed RKM methods have derived up to the 7th-order using Taylor expansion. Two RKM integrators of three-stages, 5th-order and four-stages, 6th-order have been derived depend on these OCs. The proposed integrates have been tested using some implementations in order to compare them with classical methods which show that the proposed integrators are less than classical RK methods in term of time complexity and function evaluations.

Non-autonomous equations with unpredictable solutions

To make research of chaos more amenable to investigating differential and discrete equations, we introduce the concepts of an unpredictable function and sequence. The topology of uniform convergence on compact sets is applied to define unpredictable functions [1,2]. The unpredictable sequence is defined as a specific unpredictable function on the set of integers. The definitions are convenient to be verified as solutions of differential and discrete equations. The topology is metrizable and easy for applications with integral operators. To demonstrate the effectiveness of the approach, the existence and uniqueness of the unpredictable solution for a delay differential equation are proved as well as for quasilinear discrete systems. As a corollary of the theorem, a similar assertion for a quasilinear ordinary differential equation is formulated. The results are demonstrated numerically, and an application to Hopfield neural networks is provided. In particular, Poincaré chaos near periodic orbits is observed. The completed research contributes to the theory of chaos as well as to the theory of differential and discrete equations, considering unpredictable solutions.

Internal Layers for a Singularly Perturbed Second-Order Quasilinear Differential Equation with Discontinuous Right-Hand Side

A singularly perturbed boundary value problem for a second-order quasilinear ordinary differential equation is studied. We consider a new class of problems in which the nonlinearities experience discontinuities, which leads to the appearance of sharp transition layers in a neighborhood of the points of discontinuity. The existence of solutions is proved, and their asymptotic expansion with an internal transition layer is constructed.

A Combined Analytic and Numerical Procedure for Stagnation-Point Flow

The two dimensional vector form of the Navier-Stokes equation is reduced to a fourth–order equation for the streamfunction. Boundary conditions arise from considerations of the no-slip constraint at the surface as well the interaction of viscous flow with potential-flow at the edge of the boundary layer. By employing a separation of variables technique and introducing certain dimensionless variables, the stream function equation is converted into its dimensionless analog with appropriate boundary conditions. The resulting quasi-linear third-order ordinary differential equation facilitates the numerical computation of the velocity and the pressure terms. This is achieved by solving the nonlinear two-point boundary-value problem with a time-marching method involving a Crank-Nicolson and Newton-linearization schemes until steady-state solution is obtained. The velocity, stream-function and pressure profiles are discussed with reference to various computation parameters and are found to be in good agreement with the physics of the problem. It was also found that there is no penalty in accuracy for a broad range of CFL numbers. However as the CFL number exceeds a certain threshold, the approach to convergence becomes erratic as indicated by the spurious results produced by the solution residuals.

Asymptotic behavior of increasing positive solutions of second order quasilinear ordinary differential equations in the framework of regular variation

The existence and asymptotic behavior at infinity of increasing positive solutions of second order quasilinear ordinary differential equations $(p(t)\varphi (x'(t)) )^{\prime}+q(t)\psi(x(t))=0 $ are studied in the framework of regular variation, where p and q are continuous functions regularly varying at infinity and φ, ψ are both continuous functions regularly varying at zero and regularly varying at infinity, respectively.

Asymptotic forms of slowly decaying positive solutions of second-order quasilinear ordinary differential equations related to degenerate elliptic equations

We determine asymptotic forms of slowly decaying positive solutions of quasilinear ordinary differential equations related to so-called p-Laplace equation.

Extension in the Hilbert Space of the Differential Realization of a Countable Set of Nonlinear “Input–Output” Processes1

We analyze the algebraic extensions of a countable family of nonlinear dynamic control processes having differential realization in the class of quasi-linear ordinary differential equations (with software-positional control and without it) in a separable Hilbert space.