Advanced Differential Equations Research Articles

An improved oscillation result for advanced differential equations on time scale

An improved oscillation result for advanced differential equations on time scale

Oscillation theorems for advanced differential equations

Oscillation theorems for advanced differential equations

On the Profitability of Selfish Blockchain Mining Under Consideration of Ruin

Mining blocks on a blockchain equipped with a proof of work consensus protocol is well known to be resource consuming. A miner bears the operational cost, mainly electricity consumption and IT gear, of mining and is compensated by a capital gain when a block is discovered. This paper aims at quantifying the profitability of mining when the possible event of ruin is also considered. This is done by formulating a tractable stochastic model and using tools from applied probability and analysis, including the explicit solution of a certain type of advanced functional differential equation. The expected profit at a future time point is determined for the situation when the miner follows the protocol as well as when the miner withholds blocks. The obtained explicit expressions allow us to analyze the sensitivity with respect to the different model components and to identify conditions under which selfish mining is a strategic advantage.

Analysis of Students Learning Difficulties in Advanced Differential Equation Courses Reviewed from the Use of Learning Media During the Covid-19 Pandemic

The inability of students to master the material concept of the advanced differential equation gives rise to internal and external factors that are formed in them so that it needs to require encouragement as effective use of learning media so that learning in the classroom becomes increasingly attractive to students. We used online learning media as zoom, offline or online learning videos, E-learning, and others. This research aimed to identify and describe students learning difficulties in advanced differential equation courses in terms of the use of learning media during the Covid-19 pandemic. The type of research used is qualitative research, with a descriptive approach. Results showed that students have great difficulty learning numerical methods to solve first-order differential equations from all three sub-chapters of matter. In this material, students are still less careful in answering questions (58,1%). Because students could not identify the type of methods used to solve problems, students need online learning media that are effective for math learning, such as offline learning videos (71%), as a solution to save internet quotas. So, this research is useful for educators/lecturers in observing the condition of student learning difficulties given the use of learning media used to motivate students to be active in learning advanced differential equation courses during the Covid-19 pandemic.

Unique Common Tripled Fixed Point for Three Mappings in <img src=image/13424616_01.gif> Spaces

It has been a century since the Banach fixed point theorem was established, and because of this, the result is the progenitor in some ways. This seems essential to revisit fixed point theorems in specific and in light of most of those. Those are numerous and prevalent in mathematics, as we will demonstrate. Fixed point theorems can be noticed in advanced mathematics, economics, micro-structures, geometry, dynamics, computational mathematics, and differential equations. <img src=image/13424616_03.gif> space is to broaden and extrapolate the paradigm of the concept of metric space. The characteristic of a <img src=image/13424616_03.gif> space, in essence, is to comprehend the topological features of three points rather than two points via the perimeter of a triangle, where the metric indicates the distance between two points. The domain of <img src=image/13424616_02.gif> space is significantly larger than that of the class of <img src=image/13424616_03.gif> space. Hence we utilised this generalized space in order to obtain common tripled fixed point for three mappings using rational type contractions in the setting of <img src=image/13424616_02.gif> spaces. Recently, Khomadram et al have developed coupled fixed point theorems in <img src=image/13424616_02.gif> spaces via rational type contractions. The main aim of our paper is to broaden and extrapolate the paradigm of Khomadram's results into tripled fixed point theorems. Therefore, examples are offered to support our findings.

Asymmetrical cell division with exponential growth

An advanced pantograph-type partial differential equation, supplemented with initial and boundary conditions, arises in a model of asymmetric cell division. Methods for solving such problems are limited owing to functional (nonlocal) terms. The separation of variables entails an eigenvalue problem that involves a nonlocal ordinary differential equation. We discuss plausible eigenvalues that may yield nontrivial solutions to the problem for certain choices of growth and division rates of cells. We also consider the asymmetric division of cells with linear growth rate which corresponds to "exponential growth” and exponential rate of cell division, and show that the solution to the problem is a certain Dirichlet series. The distribution of the first moment of the biomass is shown to be unimodal. doi:10.1017/S1446181121000109

On oscillation of higher-order advanced trinomial differential equations

We study the oscillatory property of the higher-order trinomial differential equation with advanced effects x(n)(t)+p(t)x′(t)+q(t)x(σ(t))=0,σ(t)≥t.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ x^{(n)}(t)+p(t)x'(t)+q(t)x \\bigl(\\sigma (t) \\bigr)=0,\\quad \\sigma (t) \\geq t. $$\\end{document} Suppose that all solutions of the corresponding (n-1)th-order two-term differential equation y(n−1)(t)+p(t)y(t)=0\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ y^{(n-1)}(t)+p(t)y(t)=0 $$\\end{document} are non-oscillatory. In order to supplement the research in the theory of oscillation proposed by (Džurina et al. in Electron. J. Differ. Equ. 2015:70, 2015), two types of clearly confirmable criteria for oscillatory behavior of the investigated equation are obtained. Some examples are offered to describe our main results.

A sharp oscillation criterion for second-order half-linear advanced differential equations

We study the half-linear advanced differential equation $$(r(t)| y'(t)|^{\alpha-1}y'(t))'+q(t)|y|^{\alpha-1}(\sigma(t))y(\sigma(t))= 0, \quad t \geq t_0 >0,$$ where $$\alpha>0, r(t)>0, q(t) >0, \sigma(t)\geq t$$ , and $$R(t):= \int_{t_0}^{t}r^{-1/\alpha}(s) \, {\rm d}s \to \infty$$ as $$t\to \infty$$ . We prove that such an equation is oscillatory if $$ \lambda_*:= \liminf_{t\to \infty}\frac{R(\sigma(t))}{R(t)}<\infty$$ and $$\liminf_{t\to \infty}r^{1/\alpha}(t)R^{\alpha+1}(t)q(t)> \max\{\alpha m^\alpha(1-m)\lambda_*^{-\alpha m}: 0<m<1 \}$$ or $$\lim_{t\to \infty}\frac{R(\sigma(t))}{R(t)}=\infty\quad \text{and}\quad \liminf_{t\to \infty}r^{1/\alpha}(t)R^{\alpha+1}(t)q(t)> 0.$$ The obtained criteria can be regarded as a natural extension of the well-known Kneser oscillation criterion for half-linear ordinary differential equations. Our oscillation constant is optimal for the correponding half-linear Euler-type delay differential equation.

Oscillation of a Class of Third-Order Neutral Differential Equations with Noncanonical Operators

The aim of this paper is to complement existing oscillation results for third-order neutral advanced differential equations under the condition of $$\gamma >0$$ ; in particular, the sufficient conditions are given in different way when $$\gamma =1$$ . Our main idea is by establishing sufficient conditions for nonexistence of so-called Kneser solutions. Then, combining with the results which guarantee the equation almost oscillation, we establish sufficient condition for oscillation of all solutions.

ASYMMETRICAL CELL DIVISION WITH EXPONENTIAL GROWTH

AbstractAn advanced pantograph-type partial differential equation, supplemented with initial and boundary conditions, arises in a model of asymmetric cell division. Methods for solving such problems are limited owing to functional (nonlocal) terms. The separation of variables entails an eigenvalue problem that involves a nonlocal ordinary differential equation. We discuss plausible eigenvalues that may yield nontrivial solutions to the problem for certain choices of growth and division rates of cells. We also consider the asymmetric division of cells with linear growth rate which corresponds to “exponential growth” and exponential rate of cell division, and show that the solution to the problem is a certain Dirichlet series. The distribution of the first moment of the biomass is shown to be unimodal.

Asymptotic Behavior of Solutions of Even-Order Advanced Differential Equations

In this paper, we establish the qualitative behavior of the even-order advanced differential equation a υ y κ − 1 υ β ′ + ∑ i = 1 j q i υ g y η i υ = 0 , υ ≥ υ 0 . The results obtained are based on the Riccati transformation and the theory of comparison with first- and second-order equations. This new theorem complements and improves a number of results reported in the literature. Two examples are presented to demonstrate the main results.

Oscillation of Second-Order Half-Linear Neutral Advanced Differential Equations

The purpose of this paper is to study the oscillation of second-order half-linear neutral differential equations with advanced argument of the form $$\begin{aligned} (r(t)((y(t)+p(t)y(\tau (t)))')^\alpha )'+q(t)y^\alpha (\sigma (t))=0,\ t\geqslant t_0, \end{aligned}$$ when $$\int _{}^\infty r^{-\frac{1}{\alpha }}(s){\text{d}}s<\infty $$ . We obtain sufficient conditions for the oscillation of the studied equations by the inequality principle and the Riccati transformation. An example is provided to illustrate the results.

A numerical technique for a general form of nonlinear fractional-order differential equations with the linear functional argument

Abstract In this paper, a numerical technique for a general form of nonlinear fractional-order differential equations with a linear functional argument using Chebyshev series is presented. The proposed equation with its linear functional argument represents a general form of delay and advanced nonlinear fractional-order differential equations. The spectral collocation method is extended to study this problem as a discretization scheme, where the fractional derivatives are defined in the Caputo sense. The collocation method transforms the given equation and conditions to algebraic nonlinear systems of equations with unknown Chebyshev coefficients. Additionally, we present a general form of the operational matrix for derivatives. A general form of the operational matrix to derivatives includes the fractional-order derivatives and the operational matrix of an ordinary derivative as a special case. To the best of our knowledge, there is no other work discussed this point. Numerical examples are given, and the obtained results show that the proposed method is very effective and convenient.

The limits of sustained self-excitation and stable periodic pulse trains in the Yamada model with delayed optical feedback.

We consider the Yamada model for an excitable or self-pulsating laser with saturable absorber and study the effects of delayed optical self-feedback in the excitable case. More specifically, we are concerned with the generation of stable periodic pulse trains via repeated self-excitation after passage through the delayed feedback loop and their bifurcations. We show that onset and termination of such pulse trains correspond to the simultaneous bifurcation of countably many fold periodic orbits with infinite period in this delay differential equation. We employ numerical continuation and the concept of reappearance of periodic solutions to show that these bifurcations coincide with codimension-two points along families of connecting orbits and fold periodic orbits in a related advanced differential equation. These points include heteroclinic connections between steady states and homoclinic bifurcations with non-hyperbolic equilibria. Tracking these codimension-two points in parameter space reveals the critical parameter values for the existence of periodic pulse trains. We use the recently developed theory of temporal dissipative solitons to infer necessary conditions for the stability of such pulse trains.

Oscillatory and asymptotic behavior of advanced differential equations

In this paper, a class of fourth-order differential equations with advanced type is studied. Applying the generalized Riccati transformation, integral averaging technique and the theory of comparison, a set of new criteria for oscillation or certain asymptotic behavior of solutions of this equations is given. Our results essentially improve and complement some earlier publications. Some examples are presented to demonstrate the main results.

Eigenfunction Families and Solution Bounds for Multiplicatively Advanced Differential Equations

A family of Schwartz functions W ( t ) are interpreted as eigensolutions of MADEs in the sense that W ( δ ) ( t ) = E W ( q γ t ) where the eigenvalue E ∈ R is independent of the advancing parameter q &gt; 1 . The parameters δ , γ ∈ N are characteristics of the MADE. Some issues, which are related to corresponding q-advanced PDEs, are also explored. In the limit that q → 1 + we show convergence of MADE eigenfunctions to solutions of ODEs, which involve only simple exponentials and trigonometric functions. The limit eigenfunctions ( q = 1 + ) are not Schwartz, thus convergence is only uniform in t ∈ R on compact sets. An asymptotic analysis is provided for MADEs which indicates how to extend solutions in a neighborhood of the origin t = 0 . Finally, an expanded table of Fourier transforms is provided that includes Schwartz solutions to MADEs.

An improved oscillation result for advanced differential equations

In this paper we study the advanced differential equations of the form x′(t)−q(t)xσ(t)=0,t≥t0,where q is a function of nonnegative real numbers and σ is a function of positive real numbers such that σ(t)>t, for all t≥t0. Based on a recursive sequence we have constructed, we improve the well known oscillation condition lim supt→∞∫tσ(t)q(s)ds>1 by lim supt→∞∫tσ(t)q(s)ds≥1.An example illustrating our results is also given.

Oscillation Theorems for Advanced Differential Equations with p-Laplacian Like Operators

The main objective of this paper is to establish new oscillation results of solutions to a class of even-order advanced differential equations with a p-Laplacian like operator. The key idea of our approach is to use the Riccati transformation and the theory of comparison with first and second-order delay equations. Some examples are provided to illustrate the main results.

New Oscillation Criteria for Advanced Differential Equations of Fourth Order

The main objective of this paper is to establish new oscillation results of solutions to a class of fourth-order advanced differential equations with delayed arguments. The key idea of our approach is to use the Riccati transformation and the theory of comparison with first and second-order delay equations. Four examples are provided to illustrate the main results.

On the Asymptotic Behavior of Advanced Differential Equations with a Non-Canonical Operator

In this paper, we aim to study the oscillatory behavior of a class of even-order advanced differential equations with a non-canonical operator. In addition, we present results on the asymptotic behavior of this type of equations and provide an example that illustrates our main results.