Linear Differential Equations Research Articles

Maxwell orthogonal polynomials

In the framework of the theory of semiclassical linear functionals in this contribution we deal with the sequence of orthogonal polynomials associated with the linear functional $ \langle{L, p}\rangle = \int_{0} ^{\infty} p(x) e^{- x^2}dx,$ where $p\in \mathbb{P},$ the linear space of polynomials with complex coefficients. The class of $L$ is one and we deduce a differential/difference equation (structure relation) for the sequence of orthogonal polynomials. The Laguerre-Freud equations that the coefficients of the three term recurrence relation satisfy are deduced. The connection with discrete Painlev\'e IV equations is emphasized. Finally, we analyze the lowering and raising operators (ladder operators) for such polynomials in order to find a second order linear differential equation they satisfy. As a consequence, an electrostatic interpretation of their zeros is formulated.

MATHEMATICAL MODELING OF THE DYNAMICS OF THE AEROELASTIC "PIPELINE – PRESSURE SENSOR" SYSTEM

This paper proposes mathematical models of mechanical systems "pipeline - pressure sensor" designed to control the pressure of the working medium in the combustion chambers of engines. In such systems, to mitigate the impact of vibration accelerations and high temperatures, the sensor is connected to the engine using a pipeline and is located at some distance from it. The movement of the working medium is described by linear models of fluid and gas mechanics. To describe the dynamics of an elastic sensitive element, linear models of the mechanics of a solid deformable body are used. Based on linear differential equations with partial derivatives, mathematical formulations of problems are proposed that correspond to three-dimensional models of pressure measurement systems in gas-liquid media for some pipeline cross-sectional shapes, namely, for a pipeline with a rectangular cross-section, with a section in the form of a sector and in the form of a ring. By introducing integral characteristics, the solution of problems is reduced to studying one-dimensional models. Equations have been obtained that make it possible to determine the pressure of the working medium in the combustion chamber at each moment of time by the value of deformation of the sensitive element of the sensor. Analytical and numerical-analytical methods for solving the corresponding initial-boundary value problems for systems of differential equations are proposed. In the analytical approach, the solution of the problem is reduced to solving a differential equation with a deviating argument. The numerical-analytical study of the problem is based on the application of the Galerkin method. Also, a numerical experiment was carried out and examples of calculating the deformation of the sensitive element of the sensor in the case of rigid fastening when specifying specific values of the mechanical parameters of the system are presented, also during setting the law of change in the excess pressure of the working medium in the engine.

Shadowing and hyperbolicity for linear delay difference equations

It is known that hyperbolic linear delay difference equations are shadowable on the half-line. In this article, we prove the converse and hence the equivalence between hyperbolicity and the positive shadowing property for the following two classes of linear delay difference equations: (a) for non-autonomous equations with finite delays and uniformly bounded compact coefficient operators in Banach spaces and (b) for Volterra difference equations with infinite delay in finite dimensional spaces.

An asymptotic result for linear nonhomogeneous mixed type differential equation

We consider a nonhomogeneous linear mixed type differential equation with variable coefficients and establish an asymptotic result for the solutions. Our result is obtained by the use of a solution of the so-called generalized characteristic equation of the corresponding homogeneous linear mixed type differential equation.

Dynamic analysis of the air–water interface instability in a wind-wave flume: Initial conditions for wave generation

This paper aims to investigate the dynamic mechanism for a previously calm water surface under a background wind field, which would lose stability due to minor perturbations. When spatial distribution and growth rate of the shear wind field satisfy a certain relationship, an Orr–Sommerfeld equation can be derived from linearized Navier–Stokes equations, for the amplitude of perturbation waves at the air–water interface. After transforming its coordinates to a finite interval, this fourth-order linear ordinary differential equation with variable coefficients can turn into a linear singular differential equation. It is solved under the background flow fields of air and water using the corrected Fourier method that we previously developed, to yield two sets of four analytical solutions with physical significance. The perturbation flow at the air–water interface must satisfy the kinematic and dynamic matching conditions (continuity of velocity and smooth transition of shear stress), which results in a fourth-order homogeneous linear algebraic equation for four undetermined constants. Setting various parameters in accordance with measured data with a wind-wave flume, the corresponding perturbation waves solution is identified. Our calculation densely distributes the most unstable and readily growing waves among perturbation waves with small phase speeds, consistent with the experimental results. Our perturbation solutions over a flat air–water interface are only applicable to the short period after instability onset, which provides an initial condition for wave generation. The consequent interaction between small-amplitude waves and wind field is beyond the scope of our linear theory, which has indeed been sufficiently addressed in the literature.

Optimal control governed impulsive neutral differential equations

We derive Pontryagin’s Maximum Principle for optimal control problems characterized by nonlinear impulsive neutral type differential equations. Our method utilizes the Dubovitskii-Milyutin theory, assuming that the linear variational impulsive differential equation along the optimal solution is exactly controllable. This principle offers necessary conditions for identifying optimal solutions.

Casting more light in the shadows: dual Somos-5 sequences

Abstract Motivated by the search for an appropriate notion of a cluster superalgebra, incorporating Grassmann variables, Ovsienko and Tabachnikov considered the extension of various recurrence relations with the Laurent phenomenon to the ring of dual numbers. 
Furthermore, by iterating recurrences with specific numerical values, some particular well-known integer sequences, such as the Fibonacci sequence, Markoff numbers, and Somos sequences, were shown to produce associated ``shadow'' sequences when they were extended to the dual numbers. Here we consider the most general version of the Somos-5 recurrence defined over the ring of dual numbers $\mathbb{D}$ with complex coefficients, that is the ring $\mathbb{C}[\varepsilon]$ modulo the relation $\varepsilon^2=0$. We present three different ways to present the general solution of the initial value problem for Somos-5 and its shadow part: in analytic form, using the Weierstrass sigma function with arguments in $\mathbb{D}$; in terms of the solution of a linear difference equation; and using Hankel determinants constructed from $\mathbb{D}$-valued moments, via a connection with a QRT map over the dual numbers.

A thresholding algorithm for Willmore-type flows via fourth-order linear parabolic equation

We propose a thresholding algorithm for Willmore-type flows in \mathbb{R}^{N} . This algorithm is constructed based on the asymptotic expansion of the solution to the initial value problem for a fourth-order linear parabolic partial differential equation whose initial data is the indicator function on the compact set \Omega_{0} . The main results of this paper demonstrate that the boundary \partial\Omega(t) of the new set \Omega(t) , generated by our algorithm, is included in O(t) -neighborhood of \partial\Omega_{0} for small t>0 and that the normal velocity from \partial\Omega_{0} to \partial\Omega(t) is nearly equal to the L^{2} -gradient of Willmore-type energy for small t>0 . Finally, numerical examples of planar curves governed by the Willmore flow are provided by using our thresholding algorithm.

Standard solutions of complex linear differential equations

A meromorphic solution of a complex linear differential equation (with meromorphic coefficients) for which the value zero is the only possible finite deficient/deviated value is called a standard solution. Conditions for the existence and the number of standard solutions are discussed for various types of deficient and deviated values.

Efficient relaxation scheme for the SIR and related compartmental models

In this paper, we introduce a novel numerical approach for approximating the Susceptible–Infectious–Recovered (SIR) model in epidemiology. Our method enhances the existing linearization procedure by incorporating a suitable relaxation term to tackle the transcendental equation of nonlinear type. Developed within the continuous framework, our relaxation method is explicit and easy to implement, relying on a sequence of linear differential equations. This approach yields accurate approximations in both discrete and analytical forms. Through rigorous analysis, we prove that, with an appropriate choice of the relaxation parameter, our numerical scheme is non-negativity-preserving; moreover, it is strongly convergent to the true solution. We also extend the applicability of our relaxation method to handle some variations of the traditional SIR model. Finally, we present numerical examples using simulated data to demonstrate the effectiveness of our proposed method.

Fourier Series for Nonperiodic Functions

We introduce a small change in the definition of the Fourier series so that we can guarantee the coincidence with the given function at the endpoints of the interval even if the function does not assume the same value at the endpoints. This definition of the Fourier series also eliminates the Gibbs phenomenom at the endpoints of the interval and proves useful in the resolution of antiperiodic and mixed boundary value problems with linear partial differential equations.

New general single, double and triple conformable integral transforms: Definitions, properties and applications

This study introduces an innovative general adaptive integral transform in single, double and triple types. This article outlines the definitions of these new transformations and establishes their main characteristics in each species. In addition, it examines the connections between newly introduced generic transformations and existing transformations. It is shown that previously developed adaptive transforms, including Laplace, Sumodo, Elzaki, G-transforms, Pourreza, and Aboodh, appear as special cases of this general adaptive transform. Furthermore, the effectiveness of the conformal generalized transform is demonstrated through its application in solving different types of linear and nonlinear fractional differential equations. Such as Black–Scholes and Berger’s equations, to demonstrating its proficiency in this domain. The proposed approach demonstrates versatility by encompassing nearly all conformable integral transforms of orders one, two, and three. As a result, it eliminates the need to derive new formulas for single, double, and triple conformable integral transforms, streamlining the process and enhancing the efficiency of solving related problems.

RESEARCH OF THE RELIABILITY OF THE GEODETIC NETWORK IN PROVIDING GEODETIC SUPPORT OF CONSTRUCTION

The safety and reliability of buildings and structures as a complex technical system depends not only on the correct consideration of data on the structural features of the building, but also on the behavior of the soil base and on the geodetic support of the construction. Geodetic works are an integral part of the technological process of construction production and belong to the main types of work. Ensuring reliable and safe, long-term operation of buildings and structures of residential, civil, industrial and agricultural purpose, located both in ordinary and in difficult engineering-geological, earthquake-hazardous, mining-geological conditions is an urgent scientific task in our time. Buildings located on artificial territories, on subsiding soils, landslide-prone slopes or in conditions of dense urban development need to determine the development of deformation processes based on the results of periodic geomonitoring at the stage of their operation. The purpose of the work is to create a Markov discrete-continuous stochastic model of the operation of the local geodetic network (GN) to provide geodetic support for construction and perform numerical calculations of the reliability and safety of the GN depending on the features of its restoration. Analyze regulatory and legal documents on the reliability and safety of construction objects. Perform an analysis of methods for assessing the reliability and safety of technical systems. Perform a numerical calculation of reliability, safety and efficiency indicators: availability ratio, limit probability states, mean time between failures, average duration of fault-free operation of the geodetic network (GN). Methodology. Analysis of regulatory documents on the reliability and safety of construction objects. Review of methods for assessing the reliability and safety of technical systems. Application of the state space method, construction of graphs of states and transitions of a renewable geodesic network. Construction of graphs of the readiness function, the probability of operation before the first failure and the frequency of getting into an emergency situation in the Mathcad software tool. Scientific novelty. The state space method allows simulating the behavior of the designed GN in order to identify the time spent in various states of the technical system depending on the intensity of failures of geodetic points and the intensity of their recovery. On the graph of states and transitions, in contrast to the “fault tree” method, it is possible to simultaneously see all possible situations separated by masks of emergency situations. Practical value. Due to the use of stable geodetic points selected based on the results of the performed reliability assessment, there is an opportunity to improve the quality of geodetic construction support. The most expedient scheme of GN restoration has been substantiated. The results of the analysis of the functioning models of the designed GN were used and the calculated values of: availability coefficient, frequency of GN failures and the probability of operation before the first failure. Results. A graphic model of the reliability behavior of GM in the form of a graph of states and transitions has been built. Geodetic network includes 8 points. A system of Kolmogorov-Chapman linear differential equations was compiled and solved. The distribution of probabilities of being in each state of GN has been obtained. The reasonable periodicity of GN restoration depending on the intensity of failures has been substantiated. It has been proven that periodic restoration of geodetic points allows maintaining a given level of GM reliability. This increases the accuracy of geodetic monitoring of construction and ensures functional safety.

Growth Estimates of Solutions of Linear Differential Equations with Dominant Coefficient of Lower (α, β, γ)-order

Growth Estimates of Solutions of Linear Differential Equations with Dominant Coefficient of Lower (α, β, γ)-order

Novel analytical perspectives on nonlinear instabilities of viscoelastic Bingham fluids in MHD flow fields

The nonlinear stability of a plane interface separating two Bingham fluids and fully saturated in porous media is inspected in the existing work. The two fluids are compressed by a normal magnetic field. The two fluids have diverse viscoelasticity, densities, magnetic, and porosity medium, with the existence of surface tension at the interface. The motivation of applied physics and engineering relations has encouraged the discussion of the current paper. Because the mathematical behavior is rather complex, the viscoelasticity involvement is reproduced only at the surface of separation, which is well-known as the viscous potential theory. Thereby, the equations of movement are scrutinized in a linear form, whereas a set of nonlinear boundary conditions are supposed. This procedure produces a nonlinear expressive nonlinear partial differential equation of the interface displacement. The non-perturbative approach which is based on the He’s frequency formula is employed to transform the nonlinear distinguishing ordinary differential equation with complex coefficients into a linear one. A novel process relying on the non-perturbative approach is utilized to examine the nonlinear stability and scrutinize the interface presentation. A non-dimensional analysis produces several dimensionless physical numerals. To validate the new approach, a comparison between the non-perturbative approach and its corresponding linear ordinary differential equation via the Mathematica Software is described and interpreted through a set of diagrams. Additionally, the Polar graphs have been elucidated. It is found that the mechanism of the stability does not change in the cases of real and complex coefficients.

Extension of the First-Order Recursive Filters Method to Non-Linear Second-Kind Volterra Integral Equations

A new numerical method for solving Volterra non-linear convolution integral equations (NLCVIEs) of the second kind is presented in this work. This new approach, named IIRFM-A, is based on the combined use of the Laplace transformation, a first-order decomposition, a bilinear transformation, and the Adomian decomposition. Unlike most numerical methods based on the Laplace transformation, the IIRFM-A method has the dual advantage of requiring neither the calculation of the Laplace transform of the source function nor that of intermediate inverse Laplace transforms. The application of this new method to the case of non-convolutive multiplicative kernels is also introduced in this work. Several numerical examples are presented to illustrate the great flexibility and efficiency of this new approach. A concrete thermal problem, described by a non-linear convolutive Volterra integral equation, is also solved numerically using the new IIRFM-A method. In addition, this new approach extends for the first time the field of use of first-order recursive filters, usually restricted to the case of linear ordinary differential equations (ODEs) with constant coefficients, to the case of non-linear ODEs with variable coefficients. This extension represents a major step forward in the field of recursive filters.

Boundary exponential stabilization of a time-delay ODE-KdV cascaded system

This paper deals with the well-posedness and exponential stabilization problems of a cascaded control system consisting of a linear ordinary differential equation (ODE) and the one-dimensional linear Korteweg–de Vries (KdV) partial differential equation posed on a bounded interval [0,l], where the states are subject to an arbitrary constant delay. The control input for the whole system acts at the left boundary of the KdV domain by Dirichlet condition, whereas the right boundary injects a Dirichlet term in the ODE subsystem. Based on the infinite dimensional backstepping method for the delay-free case, an explicit feedback control law is constructed. Under this feedback, we prove the well-posedness of the considered system in Hilbert space H=Rn×L2(0,l) by using semigroup theory and its exponential stability in the topology of ‖⋅‖H-norm by combining Lyapunov method with Halanay’s inequality and linear matrix inequalities (LMIs). A numerical example is provided to illustrate the result.

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.

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.