Linear Differential Equations Research Articles

Symmetric solutions to symmetric partial difference equations

This paper studies discrete systems defined by linear difference equations on the lattice Z n that are invariant under a finite group of symmetries, and shows that there always exist solutions to such systems that are also invariant under this group of symmetries.

Uncertainty analysis of MHD oscillatory flow of ternary nanofluids through a diverging channel: a comparative study of nanofluid composites

Purpose This study aims to investigate the heat and mass transfer characteristics of a temperature-sensitive ternary nanofluid in a porous medium with magnetic field and the Soret–Dufour effect through a tapered asymmetric channel. The ternary nanofluid consists of Boron Nitride Nanotubes (BNNT), silver (Ag) and copper (Cu) nanoparticles, with a focus on understanding the thermal behaviour and performance across mono, hybrid and tri-hybrid nanofluids. This paper also examines the thermal behaviour of MHD oscillatory nanofluid flow and carries out an uncertainty analysis of the model using the Taguchi method. Design/methodology/approach The governing equations for this system are transformed into coupled linear partial differential equations using non-similarity transformations and solved numerically with the Crank–Nicolson scheme. The impact of temperature sensitivity at three distinct temperatures (5°C, 20°C and 60°C) is incorporated to analyse variations in viscosity and Prandtl number. The study also examines the combined effects of Soret–Dufour numbers and thermal radiation on heat and mass transfer within the nanofluid. Findings The results demonstrate that the inclusion of BNNT, Ag and Cu nanoparticles significantly enhances heat and mass transfer rate, with copper nanoparticles showing superior performance in terms of skin friction and heat transfer rates. The Soret and Dufour effects play critical roles in modulating heat and mass diffusion within tri-hybrid nanofluids. The study reveals that temperature sensitivity alters heat and mass transfer characteristics depending on the temperature range, with pronounced variations at elevated temperatures. The influence of thermal radiation and the Peclet number is found to significantly impact temperature distribution and overall heat transfer performance within the asymmetric channel. Originality/value To the best of the authors’ knowledge, this study is the first to analyse the heat and mass diffusion in a ternary nanofluid composed of BNNT, Ag and Cu nanoparticles, considering porous media, oscillatory flow and thermal radiation within a tapered asymmetric channel. The research extends to a novel examination of temperature sensitivity in mono, hybrid and tri-hybrid nanofluids at varying temperature gradients. Furthermore, a comparative analysis of skin friction and heat transfer rates between copper, alumina and ferro composites is presented for optimising the nanofluid performance.

The solution of a class of linear differential equations

The solution of a class of linear differential equations

A new tool for estimation of the in vivo distribution of intake radionuclides.

This study developed a more universal tool for estimating the in vivo distribution of intake radionuclides. The biokinetic models and transfer rates for intake radionuclides were cited from the publications of the International Commission on Radiological Protection (ICRP), and the models were transformed into a series of linear differential equations and numerically solved using the Scipy algorithm in Python. The user interfaces for model selection, data input and in vivo distribution calculations were designed using PyQt5. The calculation results of the new tool are completely consistent with those of the data viewer released by the ICRP and are also very similar to other reported values. The new tool allows users to select and change the biokinetic models and transfer rates and display the in vivo distribution of radionuclides in an image format. It is much more user friendly.

One loop effective actions in Kerr-(A)dS black holes

We compute new exact analytic expressions for one-loop scalar effective actions in Kerr (A)dS black hole (BH) backgrounds in four and five dimensions. These are computed by the connection coefficients of the Heun equation via a generalization of the Gelfand-Yaglom formalism to second-order linear ordinary differential equations with regular singularities. The expressions we find are in terms of Nekrasov-Shatashvili special functions, making explicit the analytic properties of the one-loop effective actions with respect to the gravitational parameters and the precise contributions of the quasinormal modes. The latter arise via an associated integrable system. In particular, we prove asymptotic formulas for large angular momenta in terms of hypergeometric functions and give a precise mathematical meaning to Rindler-like region contributions. Moreover, we identify the leading terms in the large distance expansion as the point particle approximation of the BH and their finite size corrections as encoding the BH tidal response. We also discuss the exact properties of the thermal version of the BH effective actions by providing a proof of the Denef-Hartnoll-Sachdev formula and explicitly computing it for new relevant cases. Although we focus on the real scalar field in dS-Kerr and (A)dS-Schwarzschild in four and five dimensions, similar formulas can be given for higher spin matter and radiation fields in more general gravitational backgrounds. Published by the American Physical Society 2024

Improvements on the Leighton‐type oscillation criteria for impulsive differential equations and extension to non‐canonical case

We investigate the oscillatory behavior of solutions of second‐order linear differential equations with impulses. These equations can be categorized into two types: canonical and non‐canonical. This study examines the canonical and non‐canonical impulsive differential equations in connection with the famous Leighton‐type oscillation theorem. The well‐known theorem states that every solution of where and are continuous with , is oscillatory if and This pioneering work of Leighton has received considerable attention since its inception, and hence, it has been extended to various types of equations, including delay differential equations, dynamic equations, and impulsive differential equations. There are also studies generalizing the Leighton oscillation theorem when the condition fails, that is, when the equation is of non‐canonical type. Our work focuses on refining the Leighton oscillation theorem to treat the canonical and non‐canonical cases. By doing so, we correct, supplement, and enhance the current literature on the oscillation theory of differential equations with impulses. Examples are also given to illustrate the significance of the obtained results.

Effects of rotation and chemical reactions on an electroconvection in Maxwell dielectric nanofluids within anisotropic porous media

ABSTRACT This study explores the complex interactions between rotation, chemical reactions, and electroconvection in Maxwell dielectric nanofluids within anisotropic porous media. The research aims to elucidate how these factors influence the heat transfer efficiency and fluid dynamics under the influence of external alternating current (AC) electric fields. The normal mode method is employed to analyse the stability of the system, allowing for the examination of oscillatory modes and their growth rates. By introducing perturbations in the physical parameters, the study derives a set of ordinary linear differential equations that characterize the system's response to external influences. The coupled differential equations were solved using the Galerkin approach for first approximation while meeting the necessary boundary conditions. Analytical solution of the representative equation for stress-free boundary states yields Rayleigh number formulas for nonoscillatory and oscillatory mode occur. Oscillatory modes are seen for both bottom and top-heavy nanoparticle distributions. The electric Rayleigh number, thermal Prandtl number, and stress relaxation parameter increases, whereas the Brinkman-Darcy number delays the onset of stationary and oscillatory convection. The study finds that positive concentration Rayleigh numbers stabilize the system, while negative ones lead to instability. Additionally, it demonstrates that external alternating current electric fields influence convection behavior and provides formulas for critical Rayleigh numbers, enhancing our understanding of fluid interactions in various engineering and technological applications.

Infinite order linear difference equation satisfied by a refinement of Goss zeta function

Infinite order linear difference equation satisfied by a refinement of Goss zeta function

Mohand Transform Approach to Ulam-Type Stability of Linear Differential Equations

Mohand Transform Approach to Ulam-Type Stability of Linear Differential Equations

A novel temporal finite element method to solve static viscoelastic problems

A novel temporal finite element method is presented to solve static viscoelastic problems, which is more flexible and appropriate to describe temporal variation of displacement than conventional finite difference or numerical integral methods. By using virtual work principle, a spatial finite element based governing equation is derived in terms of displacement and its derivatives. Then two temporal finite element models are developed using Gurtin variational principle and weighted residual technique. A kind of hybrid shape functions with polynomial and trigonometric basis is stressed to give more flexible descriptions of time varying variables. A criterion of stability analysis is derived, which can numerically be conducted when the constitution of shape functions and step size are prescribed. A recursive algorithm toward end is developed by which the temporal FE analysis can be conducted via a matrix power product with the initial condition, instead of step marching. The proposed approach is available for the viscoelastic model described by linear differential equations with order h ≤ 2, and can conveniently be combined with well-developed numerical algorithms to tackle with boundary value problems, such as FEM, SBFEM etc. Various numerical examples, including those with static/harmonic loads, creep, stress singularity and heterogeneous structures, etc. are provided to illustrate the efficiency of the proposed approaches, and impacts of the temporal FE model, constitution of shape functions, and step size, etc. are taken into account. Satisfactory results are achieved in comparison with analytical or ABAQUS-based solutions.

On Oscillations of Nonautonomous Linear Impulsive Differential Equations with General Piecewise Constant Deviating Arguments

On Oscillations of Nonautonomous Linear Impulsive Differential Equations with General Piecewise Constant Deviating Arguments

Modelling motion energy in psychotherapy: A dynamical systems approach.

In this study we introduce an innovative mathematical and statistical framework for the analysis of motion energy dynamics in psychotherapy sessions. Our method combines motion energy dynamics with coupled linear ordinary differential equations and a measurement error model, contributing new clinical parameters to enhance psychotherapy research. Our approach transforms raw motion energy data into an interpretable account of therapist-patient interactions, providing novel insights into the dynamics of these interactions. A key aspect of our framework is the development of a new measure of synchrony between the motion energies of therapists and patients, which holds significant clinical and theoretical value in psychotherapy. The practical applicability and effectiveness of our modelling and estimation framework are demonstrated through the analysis of real session data. This work advances the quantitative analysis of motion dynamics in psychotherapy, offering important implications for future research and therapeutic practice.

On Asymptotic Behavior of Solutions of Linear Multidimensional Stochastic Differential Equations with Multiplicative Noise

On Asymptotic Behavior of Solutions of Linear Multidimensional Stochastic Differential Equations with Multiplicative Noise

Generalized Linear Differential Equation using Hyers - Ulam Stability Approach

In this paper, We demonstrate the Hyers - Ulam stability of linear differential equation of fourth order. We interact with the differential equation\begin{align*}\gamma^{iv} (\omega) + \rho_1 \gamma{'''} (\omega)+ \rho_2 \gamma{''} (\omega) + \rho_3 \gamma' (\omega) + \rho_4 \gamma(\omega) = \chi(\omega),\end{align*}where $\gamma \in c^4 [\alpha,\beta], \chi \in [\alpha,\beta]$. Hyers-Ulam stability concerns the robustness of solutions of functional equations under small perturbations, ensuring that a solution approximately satisfying the equation is close to an exact solution. We extend this concept to fourth-order linear differential equations and continuous functions. Using fixed-point methods and various norms, we establish conditions under which such equations exhibit Hyers-Ulam stability. Several illustrative examples are provided to demonstrate the application of these results in specific cases, contributing to the growing understanding of stability in higher-order differential equations. Our findings have implications in both theoretical research and practical applications in physics and engineering.

The Non-Perturbative Approach in Examining the Motion of a Simple Pendulum Associated with a Rolling Wheel with a Time-Delay

The present study aims to examine the movement of a simple pendulum (SP) that is connected by a lightweight spring and connected with a rotating wheel. The motivation behind this topic is to gain a comprehensive understanding of intricate dynamic systems that involve interconnected mechanical components and feedback with temporal delays. This system is not only theoretically attractive but also practically applicable in domains such as robotics, engineering, and control systems. As well known, all classical perturbation methods exploit Taylor expansion to simplify the reality of restoring forces. In contrast, the non-perturbative approach (NPA), as a novel methodology, transforms any nonlinear ordinary differential equation (ODE) into a linear one. It scrutinizes the restoring forces, away from employing Taylor expansion; hence it eliminates the previous weakness. The concept of the NPA is based mainly on the He’s frequency formula (HFF). The confidence of the NPA comes from the numerical compatibility between the nonlinear and linear ODEs via the Mathematica Software (MS). Therefore, instead of handling the nonlinear ODE, we investigate the linear one. The achieved response is plotted over time to show the impact of the acted parameters during a specified time interval. Moreover, the phase plane curves that correspond to the plotted solution are presented and examined. The stability criteria of the analogous linear ODE are provided and drawn to explore the stability/instability zones. The performance is applicable in engineering and other fields due to its ease of adaptation to different nonlinear systems. Therefore, the NPA can be regarded as substantial, successful, and interesting and can extended to be applied in further categories within the field of couples dynamical systems.

Stability and convergence computational analysis of a new semi analytical-numerical method for fractional order linear inhomogeneous integro-partial-differential equations

Abstract The motive behind this research work, is to originate a semi-analytical-numerical approach for the solution of fractional order linear integro partial differential equations (FOLIPDEs), especially in-homogeneous FOLIPDEs of different types, like fractional version of Fredholm and Volterra type IEs etc. To attain the said purpose, we will study some fractional formulations of linear model integral equations from the existing literature. Then we will describe the proposed semi analytical-numerical procedure alongwith its stability analysis and convergence properties. We will then prove with the aid of some particular numerical examples that the said approach is not only lucid and efficient but also precise. The outcomes of the proposed technique will show that this scheme has notable prospectives for solving a large class of FOLIEs. At the end, the contribution of this work in the advancements of semi analytical-numerical approaches for
the solution of fractional order linear integro partial differential equations will be laid down and discussion for future research in this field.

Hyers-Ulam Stability of Fifth Order Linear Differential Equations

In this paper, we study the Hyers-Ulam stability for the fifth-order linear differential equation. In particular, we treat $\varsigma$ as an arrangement of differential equation and in the form%\begin{equation*}$$\varsigma^{v}(x)+\eta_1\varsigma^{iv}(x)+\eta_2\varsigma^{'''}(x)+\eta_3\varsigma^{''}(x)+\eta_4\varsigma^{'}(x)+\eta_5\varsigma(x)=\Omega(x)$$%\end{equation*}where $\varsigma \in c^{5} [k, l]$, $\Omega \in [k, l]$. We demonstrate that$\varsigma^{v}(x)+\eta_1\varsigma^{iv}(x)+\eta_2\varsigma^{'''}(x)+\eta_3\varsigma^{''}(x)+\eta_4\varsigma^{'}(x)+\eta_5\varsigma(x)=\Omega(x)$ has the Hyers-Ulam stability. Two illustrative examples are given to represent the effectiveness of the proposed method. Fifth-order linear differential equations find applications in a wide range of fields, from engineering and control theory to physics, biology, and beyond. These equations are powerful tools for modeling systems with complex dynamics that involve multiple interacting forces or rates of change. Understanding and analyzing their stability and behavior can lead to significant advancements in the design, control, and optimization of these systems.

Enhanced Adomian Decomposition Method for Accurate Numerical Solutions of PDE Systems

This research addresses critical challenges in numerical solutions, which are vital for various engineering and physical fields. The Modified Adomian Decomposition Method (MADM) is proposed as a novel approach for solving linear and nonlinear partial differential equations (PDEs). MADM builds upon the Adomian Decomposition Method (ADM) by incorporating a new integral operator that significantly improves convergence rates and accuracy. Numerical examples demonstrate the effectiveness of MADM in handling complex nonlinear PDEs. Compared to traditional ADM, MADM consistently achieves more accurate and rapidly converging solutions. This enhancement is attributed to the novel integral operator, which addresses the limitations of ADM for intricate nonlinear problems. The paper outlines the application of MADM, its solution procedure, and its effectiveness through numerical examples. Comparisons with standard ADM solutions and exact solutions validate MADM's accuracy and superiority. The results suggest that MADM is a promising tool for expanding the applicability of Adomian methods in the field of solving PDEs.

Динамічна нестійкість при взаємодії композитної оболонки, виготовленої за адитивними технологіями, з газовим потоком

The dynamic instability of a composite cylindrical-conical thin-walled structure interacting with a supersonic gas flow is analyzed. This structure consists of three layers. The middle layer is manufactured by FDM additive technologies from ULTEM material. The top and bottom layers are manufactured from carbon-filled plastic. Free linear vibrations of the thin-walled structure are studied by Rayleigh?Ritz semi-analytical method to obtain a model of dynamic instability. The free linear vibrations are analyzed numerically. The obtained eigenfrequencies and eigenmodes are in close agreement with the data obtained by the ANSYS commercial software. The calculated eigenmodes were used to construct a model of composite cylindrical-conical shell instability. This model of instability is a system of ordinary linear differential equations in the generalized coordinates of the thin-walled structure. The supersonic gas flow is described by a piston theory, which accounts for the angle of attack. The study of the dynamic instability of the composite cylindrical-conical shell reduces to analyzing the trivial equilibrium instability of the system of ordinary differential equations. Characteristic exponents are calculated to analyze the stability of the trivial solution. These characteristic exponents are calculated from an eigenvalue problem. If the angle of attack is 12° and the Mach number is small, the minimal value of the critical pressure is observed for three circular waves. If the Mach number is increased, the minimal critical pressure is observed for four and five circular waves. If the angle of attack is 6° and the Mach number is small, the minimal critical pressure is observed for two circular waves. If the Mach number is increased, the minimal critical pressure is observed for three and four circular waves. The dynamic stability is lost for eigenmodes with a small number of circular waves

On Autonomous Superposition Operators in Spaces of μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}–Almost Periodic Functions and Applications to Linear Differential Equations

In this article we focus mainly on the class of almost periodic functions in view of the Lebesgue measure (briefly: μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-a.p. functions) and on some if its subclasses. We are going to deal with autonomous superposition operators acting in the space of μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-a.p. functions. We will indicate necessary and sufficient conditions under which the autonomous superposition operator maps the space under consideration into itself as well as conditions under which it is continuous. As a corollary from these results we indicate when the autonomous superposition operator defined on that space is a bijection. Next, we will analyse in detail the situation when the composition of μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-a.p. function with a continuous function or with a homeomorphism gives a Stepanov almost periodic function. As an application of our results we indicate a subclass of μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-a.p. functions for which linear differential equations with a non-homogeneous term belonging to this subclass may not have μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-a.p. solutions.