Scalar Nonlinear Differential Equations Research Articles

Existence and Stability Results for Differential Equations with a Variable-Order Generalized Proportional Caputo Fractional Derivative

An initial value problem for a scalar nonlinear differential equation with a variable order for the generalized proportional Caputo fractional derivative is studied. We consider the case of a piecewise constant variable order of the fractional derivative. Since the order of the fractional integrals and derivatives depends on time, we will consider several different cases. The argument of the variable order could be equal to the current time or it could be equal to the variable of the integral determining the fractional derivative. We provide three different definitions of generalized proportional fractional integrals and Caputo-type derivatives, and the properties of the defined differentials/integrals are discussed and compared with what is known in the literature. Appropriate auxiliary systems with constant-order fractional derivatives are defined and used to construct solutions of the studied problem in the three cases of fractional derivatives. Existence and uniqueness are studied. Also, the Ulam-type stability is defined in the three cases, and sufficient conditions are obtained. The suggested approach is more broadly based, and the same methodology can be used in a number of additional issues.

Biological self-heating in industrial compost piles: A modelling case study

We consider a simple 'toy model' for the spontaneous combustion of industrial compost stockpiles. The model is a scalar non-linear differential equation which can be analysed using techniques taught in an introductory subject on non-linear ordinary differential equations. This model was used as a case study in a third year subject. We discuss how students approached some of the questions. Could they transfer their prior knowledge about differential equations to an industrial case study? How would they cope with a problem which required both pen-and-paper and numerical calculations? Students used a variety of approaches but the worked solutions only showed one. It would be beneficial for students to see that there is not one correct method to solve such problems.

Ulam-Type Stability for a Boundary-Value Problem for Multi-Term Delay Fractional Differential Equations of Caputo Type

A boundary-value problem for a couple of scalar nonlinear differential equations with a delay and several generalized proportional Caputo fractional derivatives is studied. Ulam-type stability of the given problem is investigated. Sufficient conditions for the existence of the boundary-value problem with an arbitrary parameter are obtained. In the study of Ulam-type stability, this parameter was chosen to depend on the solution of the corresponding fractional differential inequality. We provide sufficient conditions for Ulam–Hyers stability, Ulam–Hyers–Rassias stability and generalized Ulam–Hyers–Rassias stability for the given problem on a finite interval. As a partial case, sufficient conditions for Ulam-type stability for a couple of multi-term delay, Caputo fractional differential equations are obtained. An example is illustrating the results.

Boundary Value Problem for Multi-Term Nonlinear Delay Generalized Proportional Caputo Fractional Differential Equations

A nonlocal boundary value problem for a couple of two scalar nonlinear differential equations with several generalized proportional Caputo fractional derivatives and a delay is studied. The exact solution of the scalar nonlinear differential equation with several generalized proportional Caputo fractional derivatives with different orders is obtained. A mild solution of the boundary value problem for the multi-term nonlinear couple of the given fractional equations is defined. The connection between the mild solution and the solution of the studied problem is discussed. As a partial case, several results for the nonlocal boundary value problem for the linear and non-linear multi-term Caputo fractional differential equations are provided. The results generalize several known results in the literature.

ОЦІНКИ СТІЙКОСТІ В НЕЛІНІЙНИХ ДИФЕРЕНЦІАЛЬНИХ РІВНЯННЯХ СПЕЦІАЛЬНОГО ВИДУ

Quite a lot of works have been devoted to problems of stability theory and, in particular, to the use of the second Lyapunov method for this. The main ones are the following [1-7]. The main attention in these works is paid to obtaining stability conditions. At the same time, when solving practical problems, it is important to obtain quantitative characteristics of the convergence of solutions to an equilibrium position. In this paper, we consider nonlinear scalar differential equations with nonlinearity of a special form (weakly nonlinear equations). Differential equations of this type are encountered in the study of processes in neurodynamics [8,9]. In this paper, we obtain stability conditions for a stationary solution of scalar equations of this type. And also the characteristics of the convergence of the process are calculated. It is shown that the solution of stability problems is closely related to optimization problems [10-12].

Stability of some differential equations of the fourth-order and fifth-order

The article is devoted to the study of the problem of stability of nonlinear ordinary differential equations by the method of semi-definite Lyapunov’s functions. The types of fourth-order and fifth-order scalar nonlinear differential equations of general form are singled out, for which the sign-constant auxiliary functions are defined. Sufficient conditions for stability in the large are obtained for such equations. The results coincide with the necessary and sufficient conditions in the corresponding linear case. Studies emphasize the advantages in using the semi-positive functions in comparison with the classical method of applying Lyapunov’s definite positive functions.

Nonautonomous nonlinear ODEs: Nonresonance conditions and rotation numbers

We deal with the T -periodic problem associated with a nonlinear scalar differential equation like (1) x ″ + f ( t , x ) = 0 , where, for x → 0 and | x | → + ∞ , the nonlinearity f is assumed behave linearly, with a time-dependent coefficient. We prove that the Landesman–Lazer conditions at zero and at infinity possess a rotational effect on “small” and “large” (in the phase-plane) solutions of (1) . As a consequence, we are able to generalize previous multiplicity results in the resonant case - i.e., when the linearizations at zero and/or at infinity are resonant, through the use of the Poincaré–Birkhoff fixed point theorem.

$$L^p$$ L p -solutions of a nonlinear third order differential equation and the Poincaré–Perron problem

In this paper we prove the well-posedness and we study the asymptotic behavior of nonoscillatory $$L^p$$ -solutions for a third order nonlinear scalar differential equation. The equation consists of two parts: a linear third order with constant coefficients part and a nonlinear part represented by a polynomial of fourth order in three variables with variable coefficients. The results are obtained assuming three hypotheses: (1) the characteristic polynomial associated with the linear part has simple and real roots, (2) the coefficients of the polynomial satisfy asymptotic integral smallness conditions, and (3) the polynomial coefficients are in $$L^p([t_0,\infty [)$$ . These results are applied to study a fourth order linear differential equation of Poincare type and a fourth order linear differential equation with unbounded coefficients. Moreover, we give some examples where the classical theorems cannot be applied.

On the Differential Equation for the Square Modulus of the Extremal Angular Velocity Vector in the Least-Energy Reorientation Problem for a Circular Top

We investigate the optimal control problem for the reorientation of a perfectly rigid circular top. For the criterion of the maneuver’s efficiency, we take the integral-square functional characterizing the total energy consumption. The control is the principal momentum of the applied external forces. In this problem, we obtain (for the first time) a scalar nonlinear differential equation such that its solution characterizes the second power of the absolute value of the extremal angular velocity vector. For polynomials depending on time, the family of solutions of the equation is considered. Numerical examples are provided.

Modeling transcriptional co-regulation of mammalian circadian clock.

The circadian clock is a self-sustaining oscillator that has a period of about 24 hours at the molecular level. The oscillator is a transcription-translation feedback loop system composed of several genes. In this paper, a scalar nonlinear differential equation with two delays, modeling the transcriptional co-regulation in mammalian circadian clock, is proposed and analyzed. Sufficient conditions are established for the asymptotic stability of the unique nontrivial positive equilibrium point of the model by studying an exponential polynomial characteristic equation with delay-dependent coefficients. The existence of the Hopf bifurcations can be also obtained. Numerical simulations of the model with proper parameter values coincide with the theoretical result.

Necessary and sufficient conditions for the existence and uniqueness of a bounded solution of the equation

Necessary and sufficient conditions for a bounded solution of the nonlinear scalar differential equation , , to exist and be unique are presented in the case when is a continuous function and the functions and are bounded and continuous. The case when and are almost periodic functions is also investigated. Bibliography: 31 titles.

Conditions of Solvability for Nonlinear Differential Equations with Perturbations of the Solutions in the Space of Functions Bounded on the Axis

We present the conditions of existence and uniqueness of bounded solutions for a nonlinear scalar differential equation $$ \frac{dx(t)}{dt}=f\left(x(t)+h(t)\right) $$ , t ∈ ℝ, in the case where a function f is continuous on ℝ and a function h is bounded and continuous. In addition, we study the case of an almost periodic function h.

A dynamic human motion: coordination analysis.

This article is concerned with the generic structure of the motion coordination system resulting from the application of the method of virtual holonomic constraints (VHCs) to the problem of the generation and robust execution of a dynamic humanlike motion by a humanoid robot. The motion coordination developed using VHCs is based on a motion generator equation, which is a scalar nonlinear differential equation of second order. It can be considered equivalent in function to a central pattern generator in living organisms. The relative time evolution of the degrees of freedom of a humanoid robot during a typical motion are specified by a set of coordination functions that uniquely define the overall pattern of the motion. This is comparable to a hypothesis on the existence of motion patterns in biomechanics. A robust control is derived based on a transverse linearization along the configuration manifold defined by the coordination functions. It is shown that the derived coordination and control architecture possesses excellent robustness properties. The analysis is performed on an example of a real human motion recorded in test experiments.

On strongly monotone solutions of a class of cyclic systems of nonlinear differential equations

The n-dimensional cyclic systems of first order nonlinear differential equations(A)xi′+pi(t)xi+1αi=0,i=1,…,n(xn+1=x1),(B)xi′=pi(t)xi+1αi,i=1,…,n(xn+1=x1), are analyzed in the framework of regular variation. Under the assumption that α1⋯αn<1 and pi(t), i=1,…,n, are regularly varying functions, it is shown that the situation in which system (A) (resp. (B)) possesses decreasing (resp. increasing) regularly varying solutions of negative (resp. positive) indices can be completely characterized, and moreover that the asymptotic behavior of such solutions is governed by the unique formula describing their order of decay (resp. growth) precisely. Examples are presented to demonstrate that the main results for (A) and (B) can be applied effectively to some higher order scalar nonlinear differential equations to provide new accurate information about the existence and the asymptotic behavior of their positive strongly monotone solutions.

Asymptotic behavior of increasing solutions to a system of [formula omitted] nonlinear differential equations

We consider the system xi′=ai(t)|xi+1|αisgnxi+1, i=1,…,n, n≥2, where ai, i=1,…,n, are positive continuous functions on [a,∞), αi∈(0,∞), i=1,…,n, with α1⋯αn<1, and xn+1 means x1. We analyze the asymptotic behavior of the solutions to this system whose components are eventually positive increasing. In particular, we derive exact asymptotic formulas for the extreme case, where all the solution components tend to infinity (the so-called strongly increasing solutions). We offer two approaches: one is based on the asymptotic equivalence theorem, and the other utilizes the theory of regular variation. The above-mentioned system includes, as special cases, two-term nonlinear scalar differential equations of arbitrary order n and systems of n/2 second-order nonlinear equations (provided n is even), which are related to elliptic partial differential systems. Applications to these objects are presented and a comparison with existing results is made. It turns out that some of our results yield new information even in the simplest case, a second-order Emden–Fowler differential equation.

Quasilinearization for initial value problems involving differential equations with “maxima”

An approximate method for solving initial value problems for first order scalar nonlinear differential equations with “maxima” is given. This method uses lower and upper solutions of the given problem, and in a sense is a variant of the method of quasilinearization. The main features of proposed method are that it is constructive in nature, and the obtained successive approximations of the unknown solution converge quadratically. Each successive approximation is a solution of an appropriately chosen initial value problem. In fact, a new scheme for constructing the initial value problems is suggested. An example which illustrates the advantages of the suggested algorithm is also included.

Fourth-order accurate bicompact schemes for hyperbolic equations

The construction of highresolution differenceschemes for hyperbolic equations has been addressedin numerous works (see [1, 2] and the referencestherein). Several methods are available for improvingthe accuracy of schemes in space. They can be basedon multipoint stencils; differential consequences ofthe original equations; compact approximations ofderivatives [3]; and combinations of grid functionsobtained on different grids, for example, the Richardson method [4, 5]. There are also approaches thatcombine these methods. For example, in the wellknown work [6] concerning gasdynamic computations, a fourthorder compact approximation and differential consequences of the original equation wereused to construct a fourthorder accurate scheme on atwopoint stencil for a thirdorder scalar nonlineardifferential equation. An analysis of the literature shows that [6] is apparently the first to construct a fourthorder accuratescheme on a twopoint stencil in space. Importantly,in the computation of highgradient flows on the basisof hyperbolic conservation laws [1, 2], twopoint compact schemes seem more promising than threepointones [3] because the former have the following important properties. First, twopoint stencil schemes preserve the order of accuracy in the transition from auniform to a nonuniform grid. Second, interpolationacross a discontinuity in compact twopoint schemescan be avoided by placing a grid node at the discontinuity point. The major elements of the technique of [6] for constructing difference schemes include the introductionof derivatives as additional unknown functions inorder to reduce highorder equations to systems offirstorder equations, the use of the integrointerpolation method, and the Simpson and Maclaurin quadrature rules. Note that in this technique replacing theSimpson and Maclaurin formulas by the trapezoidalrule yields secondorder accurate twopoint schemes.In [6, 7], a secondorder scheme on a twopointminimum stencil for hyperbolic equations wasobtained via the introduction of the derivative of theunknown function as an additional independent variable. In [8] the same technique was used to construct afourthorder scheme on a twopoint stencil for theheat equation. In the recent paper [9], higher orderaccurate schemes on a twopoint stencil were called,for brevity, bicompact. Below, we will follow this (inour view apt) terminology.In [10] the bicompact scheme of [6] for a thirdorder scalar differential equation was modified andextended to systems of secondorder nonlinear differential equations. This modified scheme was used in[11, 12] to solve inner and outer problems in viscousgas dynamics.In contrast to [6–12], for deriving bicompactschemes, we use, as auxiliary functions, primitives ofthe original unknown functions rather than their spatial derivatives. Importantly, the auxiliary functionsthus introduced are once more differentiable than theoriginal unknown functions. For example, if a gasdynamic variable is a piecewise continuous function,then its primitive is a piecewise smooth function. In this work, we construct unconditionally stablefourthorder accurate implicit bicompact schemes forlinear and nonlinear equations and systems of equations of the hyperbolic type written in conservativeform. Based on these schemes, the solution to a mixedCauchy problem [13] is determined by the runningcalculation method. The properties of these schemesare examined. Test computations confirm the accuracy and high quality of the resulting solutions.Below, we construct and analyze bicompactschemes for linear and nonlinear hyperbolic equationsand systems of equations wri tten in conservative form.

New Hybrid FE-FV Method for Computing Current Distribution in 2-D Superconductors: Application to an HTS Cylinder in Transverse Magnetic Field

This paper presents a new numerical method based on finite elements - finite volumes (FE-FV) for solving 2-D diffusion problems in high temperature superconductors (HTS). The approach does not involve directly the resistivity term (rho), generally used to model the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">J</i> ) characteristic as a power law, i.e <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">J</i> ) = rho( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">J</i> ) <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">J</i> , with rho( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">J</i> ) prop <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">J</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n-1</sup> . Instead, we use a <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">J</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</i> ) constitutive law <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">J</i> prop <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(1/n)</sup> , with <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">rarr</sup> = <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Ee</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">rarr</sup> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</sub> (a single component), which leads to a scalar non-linear differential equation. After presenting in details the developments, the method is tested in the case of a superconducting cylinder submitted to a transverse magnetic field. The current density obtained is compared to another numerical technique (the semi-analytical method) in order to validate the results. Although not fully optimized yet, it appears that the proposed method is very stable, especially for large n-values (greater than 100).

Instability of solutions for certain nonlinear vector differential equations of fourth order

The main purpose of this paper is to give a result with explanatory example that deals directly with the instability of the trivial solution of a certain nonlinear vector differential equation of the fourth order. The result established here improves and includes a well-known instability result established for a scalar nonlinear differential equation of the fourth order.

Periodic Solutions for Nonlinear Differential Equation with Functional Delay

Abstract We use the modification of Krasnoselskii's fixed point theorem due to T. A. Burton ([Proc. Amer. Math. Soc. 124: 2383–2390, 1996]) to show that the scalar nonlinear differential equation with functional delay 𝑥′(𝑡) = –𝑎(𝑡)𝑥3(𝑡) + 𝐺(𝑡, 𝑥3(𝑡 – 𝑟(𝑡))) has a periodic solution. It is not required that 𝑟(𝑡) be differentiable, while 𝑎 and 𝐺 are continuous with respect to their arguments.