Advanced Differential Equations Research Articles

Asymptotic and Oscillatory Analysis of Second-Order Differential Equations with Distributed Deviating Arguments

This paper focuses on studying the oscillatory properties of a distinctive class of second-order advanced differential equations with distributed deviating arguments in a noncanonical case. Utilizing the Riccati method and the comparison method with first-order equations, in addition to other analytical methods, we have established criteria to test whether the solutions of the studied equation exhibit oscillatory behavior. To verify the validity of the results we obtained and determine their applicability, we present some examples to confirm the strength and accuracy of our proposed criteria.

Oscillatory and asymptotic behavior of even-order nonlinear differential equations with mixed neutral terms

Abstract This paper deals with the oscillation and asymptotic behaviour of even order nonlinear differential equations with mixed nonlinear neutral terms. The findings are obtained via utilising an integral criterion as well as a comparison theorem with the oscillatory properties of a first order advanced and/or delay differential equation. We provide novel oscillation criteria that improve, extend, and simplify previously published ones. The results are illustrated by two examples.

Oscillation criteria for the second-order linear advanced differential equation

New oscillatory criteria are established for the second-order linear advanced differential equation. Riccati’s technique and suitable estimates of non-oscillatory solutions are used for the proof of the results obtained. The presented criteria, in a certain sense, generalize the known ones.

Solutions of Inhomogeneous Multiplicatively Advanced ODEs and PDEs with a q‐Fredholm Theory and Applications to a q‐Advanced Schrödinger Equation

For q > 1, a new Green’s function provides solutions of inhomogeneous multiplicatively advanced ordinary differential equations (iMADEs) of form y(N)(t) − Ay(qt) = f(t) for t ∈ [0, ∞). Such solutions are extended to global solutions on ℝ. Applications to inhomogeneous separable multiplicatively advanced partial differential equations are presented. Solutions to a linear free forced q‐advanced Schrödinger equation are obtained, opening an avenue to applications in quantum mechanics. New q‐Mittag‐Leffler functions qEα,β and ΥN,p govern the allowable decay rate of the inhomogeneities f(t) in the above iMADE. This provides a refinement to standard distribution theory, as we show is necessary for this study of iMADEs. A q‐Fredholm theory is developed and related to the above approach. For f(t) whose antiderivatives provide eigenfuntions of the noncompact integral operator K below, we exhibit solutions of the iMADE. Examples are provided, including a certain class of Dirichlet series.

Oscillation of Third-Order Differential Equations with Advanced Arguments

The main objective of this work was to study some oscillatory and asymptotic properties of a new class of advanced neutral differential equations. Using new relations to link the solution and its corresponding function, we introduced new oscillatory criteria that aim to enhance, simplify, and complement some of current results. We provide some examples to demonstrate the significance of our results.

Existence and Stability of Ulam–Hyers for Neutral Stochastic Functional Differential Equations

The primary aim of this paper is to focus on the stability analysis of an advanced neural stochastic functional differential equation with finite delay driven by a fractional Brownian motion in a Hilbert space. We examine the existence and uniqueness of mild solution of dxa(s)+g(s,xa(s-ω(s)))=Ixa(s)+f(s,xa(s-ϱ(s)))ds+ς(s)dϖH(s),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\ extrm{d}}\\left[ {x}_{a}(s) + {\\mathfrak {g}}(s, {x}_{a}(s - \\omega (s)))\\right] =\\left[ {\\mathfrak {I}}{x}_a(s) + {\\mathfrak {f}}(s, {x}_a(s -\\varrho (s)))\\right] {\ extrm{d}}s + \\varsigma (s){\ extrm{d}}\\varpi ^{{\\mathbb {H}}}(s),$$\\end{document}0≤s≤T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0\\le s\\le {\\mathcal {T}}$$\\end{document}, xa(s)=ζ(s),-ρ≤s≤0.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${x}_a(s) = \\zeta (s),\\ -\\rho \\le s\\le 0. $$\\end{document} The main goal of this paper is to investigate the Ulam–Hyers stability of the considered equation. We have also provided numerical examples to illustrate the obtained results. This article also discusses the Euler–Maruyama numerical method through two examples.

Some remarks on the solution of the cell growth equation

A process of growth and division of cells is modelled by an initial boundary value problem that involves a first-order linear functional partial differential equation, the so-called sell growth equation. The analytical solution to this problem was given in the paper Zaidi et al. (Zaidi et al . 2015 Solutions to an advanced functional partial differential equation of the pantograph type ( Proc. R. Soc. A 471 , 20140947 ( doi:10.1098/rspa.2014.0947 )). In this note, we simplify the arguments given in the paper mentioned above by using the theory of operator semigroups. This theory enables us to prove the existence and uniqueness of the solution and to express this solution in terms of Dyson–Phillips series. The asymptotics of the solution is also discussed from the point of view of the theory of operator semigroups.

Advanced fractional calculus, differential equations and neural networks: analysis, modeling and numerical computations

Most physical systems in nature display inherently nonlinear and dynamical properties; hence, it would be difficult for nonlinear equations to be solved merely by analytical methods, which has given rise to the emerging of engrossing phenomena such as bifurcation and chaos. Conjointly, due to nonlinear systems’ exhibiting more exotic behavior than harmonic distortion, it becomes compelling to test, classify and interpret the results in an accurate way. For this reason, avoiding preconceived ideas of the way the system is likely to respond is of pivotal importance since this facet would have effect on the type of testing run and processing techniques used in nonlinear systems. Paradigms of nonlinear science may suggest that it is ‘the study of every single phenomenon’ due to its interdisciplinary nature, which is another challenge encountered and needs to be addressed by generating and designing a systematic mathematical framework where the complexity of natural phenomena hints the requirement of identifying their commonalties and classifying their various manifestations in different nonlinear systems. Studying such common properties, concepts or paradigms can enable one to gain insight into nonlinear problems, their essence and consequences in a broad range of disciplines all forthwith. Fractional differential equations associated with non-local phenomena in physics have arisen as a powerful mathematical tool within a multidisciplinary research framework. Fractional differential equations, as one extension of the fractional calculus theory, can yield the evolution of various systems properly, which reinforces its position in mathematics and science while setting stage for the description of dynamic, complicated and nonlinear events. Through the reflection of the systems’ actual properties, fractional calculus manifests unforeseeable and hidden variations, and thus, enables integration and differentiation, with the solutions to be approximated by numerical methods along with modeling and predicting the dynamics of multiphysics, multiscale and physical systems. Neural Networks (NNs), consisting of hidden layers with nonlinear functions that have vector inputs and outputs, are also considerably employed owing to their versatile and efficient characteristics in classification problems as well as their sophisticated neural network architectures, which make them capable of tackling complicated governing partial differential equation problems. Furthermore, partial differential equations are used to provide comprehensive and accurate models for many scientific phenomena owing to the advancements of data gathering and machine learning techniques which have raised opportunities for data-driven identification of governing equations derived from experimentally observed data. Given these considerations, while many problems are solvable and have been solved, efforts are still needed to be able to respond to the remaining open questions in the fields that have a broad range of spectrum ranging from mathematics, physics, biology, virology, epidemiology, chemistry, engineering, social sciences to applied sciences. With a view of different aspects of such questions, our special issue provides a collection of recent research focusing on the advances in the foundational theory, methodology and topical applications of fractals, fractional calculus, fractional differential equations, differential equations (PDEs, ODEs, to name some), delay differential equations (DDEs), chaos, bifurcation, stability, sensitivity, machine learning, quantum machine learning, and so forth in order to expound on advanced fractional calculus, differential equations and neural networks with detailed analyses, models, simulations, data-driven approaches as well as numerical computations.

Calibrating Distribution Models from PELVE

The Value at Risk (VaR) and the expected shortfall (ES) are the two most popular risk measures in banking and insurance regulation. To bridge between the two regulatory risk measures, the probability equivalent level of VaR-ES (PELVE) was recently proposed to convert a level of VaR to that of ES. It is straightforward to compute the value of PELVE for a given distribution model. In this article, we study the converse problem of PELVE calibration; that is, to find a distribution model that yields a given PELVE, which may be obtained either from data or from expert opinion. We discuss separately the cases when one-point, two-point, n-point, and curve constraints are given. In the most complicated case of a curve constraint, we convert the calibration problem to that of an advanced differential equation. We apply the model calibration techniques to estimation and simulation for datasets used in insurance. We further study some technical properties of PELVE by offering a few new results on monotonicity and convergence.

Oscillation and Asymptotic Behavior of a Special Delay Third Order Nonlinear Neutral Functional Differential Equation

Abstract The oscillation theory of differential equations is an important branch of performance of differential equations, which is widely used in engineering control, vibration mechanics, mechanics, and industry. Therefore, the vibration performance of different parts has attracted people’s attention, and a lot of research work has been done. For a special class of delay differential equations - advanced piecewise continuous differential equations, the oscillation of numerical solution is discussed. The θ − method is used to discretize the equation, and the numerical method is obtained to keep the oscillation of the analytical solution of the equation, progressive conditions. At the same time, four different states of the dynamic behavior are discussed in detail for the analytical solution and the numerical solution respectively. Some numerical examples further verify the corresponding conclusions.

Stability Analysis of a One Degree of Freedom Robot Model With Sampled Digital Acceleration Feedback Controller in Turning and Milling

Abstract This study is interested in the stability of robots in machining. The goal is to improve the dynamic performance of robots using an additional acceleration signal fed back through the conventional built-in proportional-derivative controller provided by the manufacturer. The structure of the robot is modelled with a simple one degree-of-freedom lumped model and the control signals are fed back via a linear spring and damping. The time delays of the feedback controllers are considered as zero-order holds, which results in sawtooth-like time-periodic time delays. The resulting equation of motion is an advanced delay differential equation. The semidiscretization method is shown for such systems having multiple sampled digital delays and continuous delays. First, we establish the stable regions in the plane of the sampling delay and the gain of the acceleration signal without machining. Then, we show the possibility to improve stability in turning and milling using the additional acceleration feedback controller compared to the cases without any controller or using only the built-in proportional-derivative controller.

A Comprehensive Proof of Bertrand's Theorem

A cornerstone result in Newtonian mechanics is Bertrand's Theorem concerning the behavior of the solutions of the classical two-body problem. It states that among all possible gravitational laws there are only two exhibiting the property that all bounded orbits are closed. One of these is Newtonian gravitation, the other being Hookean gravitation. We present a comprehensive proof of Bertrand's Theorem that is accessible to undergraduate students who are familiar with basic notions from advanced calculus and differential equations.

Oscillation Criteria for Advanced Half-Linear Differential Equations of Second Order

In this paper, we find new oscillation criteria for second-order advanced functional half-linear differential equations. Our results extend and improve recent criteria for the same equations established previously by several authors and cover the existing classical criteria for related ordinary differential equations. We give some examples to illustrate the significance of the obtained results.

Oscillatory Properties of Fourth-Order Advanced Differential Equations

This paper presents a study on the oscillatory behavior of solutions to fourth-order advanced differential equations involving p-Laplacian-like operator. We obtain oscillation criteria using techniques from first and second-order delay differential equations. The results of this work contribute to a deeper understanding of fourth-order differential equations and their connections to various branches of mathematics and practical sciences. The findings emphasize the importance of continued research in this area.

Asymptotic behavior of solutions of the third-order nonlinear advanced differential equations

<abstract><p>The aim of this work is to study some asymptotic properties of a class of third-order advanced differential equations. We present new oscillation criteria that complete, simplify and improve some previous results. We also provide many different examples to clarify the significance of our results.</p></abstract>

Computational solution of advanced stochastic time-delay differential equations using hybrid block extended Adams Moulton methods for customer’s satisfaction when using electronic payment systems in Nigerian banks

This study examined the implementation of Hybrid Block Extended Adams Moulton Methods (HBEAMM) for the computational solution of Advanced Stochastic Time-Delay Differential Equations (ASTDDEs) through the use of various electronic payment systems such as ATM, POS and internet banking offered by the Nigerian banks. This work attempts to proffer solution to customer challenges in using the ATM of most banks in Nigeria. The challenges are considered in this work as capable of resulting to advanced time-delay and volatility noise. It is therefore modeled as advanced stochastic time-delay differential equation (ASTDDE) and solved using Hybrid Extended Block Adams Moulton Methods (HEBAMMs). The discrete schemes of the proposed method (HBEAMM) were obtained by continuous formulations of multistep collocation method by matrix inversion approach. The necessary and sufficient conditions for convergence and stability of the method were analyzed and proved satisfactory. The discrete schemes obtained were applied in solving some advanced stochastic time-delay differential equations and the results obtained revealed the degree of customers’ satisfaction in the use of e-payment systems. The accuracy of the method is compared with other existing methods and presented graphically to prove its superiority.

Some New Oscillation Results for Higher-Order Nonlinear Differential Equations with a Nonlinear Neutral Term

The authors study the oscillatory behaviors of solutions of higher-order nonlinear differential equations with a nonlinear neutral term. The right hand side of their equation contains both an advanced and a delay term, and either (or both) of them can be sublinear or superlinear. The influence of these terms on the oscillatory and asymptotic behaviors of solutions is investigated by using a comparison to first-order advanced and delay differential equations. New oscillation criteria are presented that improve and extend many known oscillation criteria in the literature. An example is provided to illustrate the results.

Solutions of a Class of Multiplicatively Advanced Differential Equations II: Fourier Transforms

For a wide class of solutions to multiplicatively advanced differential equations (MADEs), a comprehensive set of relations is established between their Fourier transforms and Jacobi theta functions. In demonstrating this set of relations, the current study forges a systematic connection between the theory of MADEs and that of special functions. In a large subset of the general case, we introduce a new family of Schwartz wavelet MADE solutions W μ , λ t for μ and λ rational with λ > 0 . These W μ , λ t have all moments vanishing and have a Fourier transform related to theta functions. For low parameter values derived from λ , the connection of the W μ , λ t to the theory of wavelet frames is begun. For a second set of low parameter values derived from λ , the notion of a canonical extension is introduced. A number of examples are discussed. The study of convergence of the MADE solution to the solution of its analogous ODE is begun via an in depth analysis of a normalized example W − 4 / 3 , 1 / 3 t / W − 4 / 3 , 1 / 3 0 . A useful set of generalized q -Wallis formulas are developed that play a key role in this study of convergence.

Extended Approach to the Asymptotic Behavior and Symmetric Solutions of Advanced Differential Equations

We studied the asymptotic behavior of fourth-order advanced differential equations of the form aυw′′′υβ′+qυgwδυ=0. New results are presented for the oscillatory behavior of these equations in the form of Philos-type and Hille–Nehari oscillation criteria. Some illustrative examples are presented.

Oscillation of noncanonical second-order advanced differential equations via canonical transform

In this paper, we develop a new technique to deduce oscillation of a second-order noncanonical advanced differential equation by using established criteria for second-order canonical advanced differential equations. We illustrate our results by presenting two examples.