Linear Inhomogeneous Differential Equation Research Articles

Investigation of the dynamics of transverse oscillations of a vertical rod under gravity, friction, and thermal expansion

The paper considers the mathematical formulation of the problem of transverse oscillations of a vertical rod under gravity, friction and external pulse effect, leading to thermal expansion of the rod. The dynamics of the system under consideration corresponds to the behavior of a fuel element (FE) in a pulsed reactor and is related to the dynamic stability of the processes occurring in it.The FE dynamics is described by inhomogeneous linear differential equation of the fourth order with non-constant coefficients. Initial boundary conditions are not smooth since they correspond to the instant heating of the part of the FE surface exposed to neutron pulse radiation. The general solution of the homogeneous linear equation can be found using the concept of generalized functions expressed as a series expansion in terms of coordinate eigenfunctions dependent on p parameter. The p parameter is related with the FE mass and at some certain values results in bifurcation points of the boundary value problem for eigenfunctions. The partial solution can be found in several ways: using the Fourier transform, the method of Green's function, and in terms of series expansion by eigenfunctions.Eigenfunction expansion coefficients in an explicit form have been obtained and the numerical solution accuracy has been estimated for some important particular cases. The results of the obtained exact solutions appear to be very close to the results of the ANSYS numerical estimations. In the future the obtained exact solutions will be used as input data to simulate self-consistent system dynamics of more than 400 FE. Therefore, the advantage of the exact solution in practical use is that its calculation is much faster as compared with the finite element method done with ANSYS.

Analytical method for solving linear abstract differential equations of order [formula omitted]

In this paper, the concept of point-wise independent set of closed operators is introduced. The new concept together with the use of Hahn–Banach Theorem enable us to reduce an inhomogeneous linear abstract differential equation of order n namely, Anu(n)(t)+An−1u(n−1)(t)+⋯+A1u′(t)+A∘u(t)=f(t), into an inhomogeneous linear ordinary differential equation with constant coefficients of order n. The involved operators in the main abstract equation are assumed to be closed and linear on a given Banach space while the unknown function is assumed to be n-times differentiable.

Deflection of light by a Reissner–Nordström black hole and Painlevé VI equation

We consider the bending angle of the trajectory of a photon incident from and deflected to infinity around a Reissner–Nordström black hole. We treat the bending angle as a function of the squared reciprocal of the impact parameter and the squared electric charge of the background normalized by the mass of the black hole. It is shown that the bending angle satisfies a system of two inhomogeneous linear partial differential equations with polynomial coefficients. This system can be understood as an isomonodromic deformation of the inhomogeneous Picard–Fuchs equation satisfied by the bending angle in the Schwarzschild spacetime, where the deformation parameter is identified as the background electric charge. Furthermore, the integrability condition for these equations is found to be a specific type of the Painlevé VI equation that allows an algebraic solution. We solve the differential equations both at the weak and strong deflection limits. In the weak deflection limit, the bending angle is expressed as a power series expansion in terms of the squared reciprocal of the impact parameter and we obtain the explicit full-order expression for the coefficients. In the strong deflection limit, we obtain the asymptotic form of the bending angle that consists of the divergent logarithmic term and the finite O(1) term supplemented by linear recurrence relations which enable us to straightforwardly derive higher order coefficients. In deriving these results, the isomonodromic property of the differential equations plays an important role. Lastly, we briefly discuss the applicability of our method to other types of spacetimes such as a spinning black hole.

THERMAL ANALYSIS OF BI-DIRECTIONAL FUNCTIONALLY GRADED PLATES SUBJECTED TO HEAT GENERATION

The two-dimensional steady-state heat transfer problem is analyzed under the most general boundary conditions in a heterogeneous environment. It is assumed that the internal heat generation varies arbitrarily in two directions, while the thermal conductivity coefficient of the material is bi-directly (thickness-length) graded with the Mori-Tanaka model. The linear inhomogeneous partial differential equations produced under these conditions may not be solved by conventional methods. The governing equation of the system is numerically solved with high accuracy using the 2D pseudospectral Chebyshev method. Benchmark problems are used to check the accuracy of the method. It has been observed that the effect of volumetric heat generation on the temperature distribution is more than the heat transmission coefficient. Besides, this method is simple and effective and can be easily applied to such problems.

Параболічні крайові задачі математичної фізики в кусково-однорідному клиновидному циліндрично-круговому півпросторі з порожниною

The unique exact analytical solutions of parabolic boundary value problems of mathematical physics in piecewise homogeneous by the radial variable r wedge-shaped by the angular variable φ cylindrical-circular half-space with a cavity were constructed at first time by the method of classical integral and hybrid integral transforms in combination with the method of main solutions (matrices of influence and Green matrices) in the proposed article. The cases of assigning on the verge of the wedge the boundary conditions of the 1st kind (Dirichlet) and the 2nd kind (Neumann) and their possible combinations (Dirichlet – Neumann, Neumann – Dirichlet) are considered. Finite integral Fourier transform by an angular variable, a finite integral Fourier transform on the Cartesian semiaxis (0; +∞) by an applicative variable z and a Weber hybrid integral transform type on the polar axis (R0; +∞) with n points of conjugation by a radial variable were used to construct solutions of investigated boundary value problems. The consistent application of integral transforms by geometric variables allows us to reduce the three-dimensional initial boundary-value problems of conjugation to the Cauchy problem for a regular linear inhomogeneous 1st order differential equation whose unique solution is written in a closed form. The consistent application of inverse integral transforms to the obtained solution in the space of images restores the solutions of the considered parabolic boundary value problems through their integral image in an explicit form in the space of the originals. At the same time, the main solutions to the problems were obtained in an explicit form.

Cauchy Problem for a System of Linear Inhomogeneous Differential Equations of the First Order with Degenerate Matrix at the Derivatives and Pulsed Actions at Fixed Times

Cauchy Problem for a System of Linear Inhomogeneous Differential Equations of the First Order with Degenerate Matrix at the Derivatives and Pulsed Actions at Fixed Times

Phase Function Methods for Second Order Inhomogeneous Linear Ordinary Differential Equations

Phase Function Methods for Second Order Inhomogeneous Linear Ordinary Differential Equations

On periodic solutions of linear inhomogeneous differential equations with a small perturbation at the derivative

In a Banach space, using branching theory methods, a periodic solution of a linear inhomogeneous differential equation with a small perturbation at the derivative (perturbed equation) is constructed. Under the condition of presence of a complete generalized Jordan set, the uniqueness of this periodic solution is proven. It is shown that when a small parameter is equal to zero and certain conditions are met, the periodic solution of the perturbed equation transforms into the family of periodic solutions of the unperturbed equation. The result is obtained by representing the perturbed equation as an operator equation in Banach space and applying the theory of generalized Jordan sets and modified Lyapunov-Schmidt method. As is known, the latter method reduces the original problem to study of the Lyapunov-Schmidt resolving system in the root subspace. In this case, the resolving system splits into two inhomogeneous systems of linear algebraic equations, that have unique solutions at ε≠0 , and 2n -parameter families of real solutions at ε=0 , respectively.

On an Analytical Representation of an Integral Related to the Fock Integral That Appears in Calculations of the Electromagnetic Fields of Dipole Sources at the Interface between Two Half-Spaces

Abstract—The Fock integral is called after Fock who introduced it for the theoretical analysis of the electromagnetic field of magnetic dipoles at the boundary of a uniform conducting (nonmagnetic) half-space and obtained its analytical expression in terms of cylindrical functions. Detailed analytical representations of integrals, where all components of the fields of the vertical and horizontal magnetic dipoles are expressed, are reported in [A.V. Veshev et al., 1971]. To obtain analytical expressions for similar integrals representing the components of the fields of electric dipoles in a similar model, it is necessary to consider not only the Fock integral but also another related integral conditionally called the Fock integral 1 whose analytical expression is still unknown. The aims of this work are to derive an inhomogeneous linear first-order differential equation for this integral with the corresponding boundary conditions and to obtain the analytical representation of the Fock integral 1 by solving this equation. The result of this work will allow one to simplify the simulation of fields in a uniform half-space and to improve the interpretation of electromagnetic data due to more accurate and reliable estimates of the normal field in such models of a host medium.

Distributed “End-Edge-Cloud” structural car-following control system for intelligent connected vehicle using sliding mode strategy

This paper proposes a new car-following model to describe the characteristics of the urban Internet of vehicle (IoV) of the “End-Edge-Cloud” architecture. The stability of the proposed CF model is analyzed using the perturbation method. Furtherly, in order to overcome the disturbances in the operation of the newly proposed system in this paper, a sliding mode controller is designed based on non-homogeneous terminal sliding mode variables, and after setting up the controller, the dynamic performance of the system is analyzed by solving the linear inhomogeneous differential equation and the Lyapunov method, and it is proved that the system can converge in finite time. Finally, the rationality of the model proposed in this paper and the significant improvement in the dynamic performance of the system under the action of the controller are effectively demonstrated through numerical experiments conducted in MATLAB and simulation experiment carried out in the ROS-Gazebo environment.

Solution of Inhomogeneous Linear Fractional Differential Equations Involving Jumarie Fractional Derivative

This paper presents a method for solving inhomogeneous linear sequential fractional differential equations with constant coefficients (ILSFDE) involving Jumarie fractional derivatives in terms of Mittag-Leffler functions. For this purpose, the fundamental properties of the Jumarie derivative and Mittag-Leffler functions are given. After this, the successive jumarie fractional derivatives of Mittag-Leffler functions, fractional cosine, and sine functions are obtained. Further, we determined the particular integrals of these functions and then found the complete solutions of ILSFDE. in terms of Mittag-Leffler functions, fractional cosine, and sine functions. We have demonstrated this developed method with a few examples of ILSFDE. This method is similar to the method for finding the complete solutions of classical differential equations with constant coefficients.

Sufficient Condition for the Existence of an Additional Zone in Singularly Perturbated Second-Order Boundary Problem

Studies the Dirichlet, Neman and Robin boundary value problems for a singularly perturbed linear inhomogeneous second order ordinary differential equation. The considered boundary value problems have three features: the singular presence of a small parameter; the solution of the corresponding unperturbed equation has a k order pole and an additional boundary layer. The singular presence of a small parameter generates the classical boundary layer, and the singular point of the corresponding unperturbed equation generates the second boundary layer. As a result, we get a double boundary layer. A sufficient condition for the existence of an additional boundary layer is found.

Improved quantum algorithms for linear and nonlinear differential equations

We present substantially generalized and improved quantum algorithms over prior work for inhomogeneous linear and nonlinear ordinary differential equations (ODE). Specifically, we show how the norm of the matrix exponential characterizes the run time of quantum algorithms for linear ODEs opening the door to an application to a wider class of linear and nonlinear ODEs. In \cite{BCOW17}, a quantum algorithm for a certain class of linear ODEs is given, where the matrix involved needs to be diagonalizable. The quantum algorithm for linear ODEs presented here extends to many classes of non-diagonalizable matrices including singular matrices. The algorithm here is also exponentially faster than the bounds derived in \cite{BCOW17} for certain classes of diagonalizable matrices. Our linear ODE algorithm is then applied to nonlinear differential equations using Carleman linearization (an approach taken recently by us in \cite{Liue2026805118}). The improvement over that result is two-fold. First, we obtain an exponentially better dependence on error. This kind of logarithmic dependence on error has also been achieved by \cite{Xue_2021}, but only for homogeneous nonlinear equations. Second, the present algorithm can handle any sparse matrix (that models dissipation) if it has a negative log-norm (including non-diagonalizable matrices), whereas \cite{Liue2026805118} and \cite{Xue_2021} additionally require normality.

Mathematical modeling of vibration drying process taking into account the longitudinal movement of grain material

Purpose. Construction of a mathematical model of non-stationary thermal mass transfer in the drying process in the vibration equipment, taking into account the longitudinal movement of the grain material. Methods. Mathematical modeling methods based on a formalized description in the form of two nonlinear inhomogeneous differential equations of the non-stationary process of drying grain material in a vibrating dryer taking into account longitudinal movement. Results. A mathematical description of the process of drying grain material in a vibrating grain dryer has been formed, taking into account the longitudinal movement of grain in the form of a diffusion mathematical model for determining the distribution of moisture content and material temperature along the length of the vibration transporting support surface. Two-step identification of model parameters based on experimental data is applied. Conclusions. A mathematical description of the nonstationary drying process of grain material in a vibrating dryer taking into account longitudinal displacement is formulated, in the form of two linear inhomogeneous differential equations that can be used to optimize the regime parameters of existing grain dryers. Keywords: drying, grain material, mathematical model, diffusion displacement, moisture content, temperature, speed.

Digital Film Music Creation Model Based on Inhomogeneous First Order Constant Coefficient Linear Differential Equation

Abstract Computer music production is an essential branch of computer technology in artistic creation. This is an interdisciplinary subject with information science and art. This paper proposes a creation model for digital music with first-order inhomogeneous first-order constant coefficient linear differential equations. This mode includes acoustic modes for waveform envelopes and bands. It has a straight-line mapping of phonetic patterns and rhythmic patterns. Then this paper uses the signal reconstruction principle and iterative extrapolation method to restore the signal in the frequency domain to obtain the simulated music. The experimental results show that this digital music creation model of inhomogeneous first-order constant coefficient linear differentiation can save a lot of labor costs. And this way can improve the audio quality and output effect. Compared with the undetermined factor method and the Laplace transform method in conventional advanced mathematics, the inhomogeneous first-order constant coefficient linear differentiation is more convenient in music composition.

SOLUTIONS OF ONE PROBLEM OF A LINEAR INHOMOGENEOUS DIFFERENTIAL EQUATION OF THE SECOND ORDER

SOLUTIONS OF ONE PROBLEM OF A LINEAR INHOMOGENEOUS DIFFERENTIAL EQUATION OF THE SECOND ORDER

Analytic solution of the exact Daum–Huang flow equation for particle filters

State estimation for nonlinear systems, especially in high dimensions, is a generally intractable problem, despite the ever-increasing computing power. Efficient algorithms usually apply a finite-dimensional model for approximating the probability density of the state vector or treat the estimation problem numerically. In 2007 Daum and Huang introduced a novel particle filter approach that uses a homotopy-induced particle flow for the Bayesian update step. Multiple types of particle flows were derived since with different properties. The exact flow considered in this work is a first-order linear ordinary time-varying inhomogeneous differential equation for the particle motion. An analytic solution in the interval [0,1] is derived for the scalar measurement case, which enables significantly faster computation of the Bayesian update step for particle filters.

On Asymptotic Behavior of Solutions of Linear Inhomogeneous Stochastic Differential Equations with Correlated Inputs

On Asymptotic Behavior of Solutions of Linear Inhomogeneous Stochastic Differential Equations with Correlated Inputs

Numerical study of multi-order fractional differential equations with constant and variable coefficients

In this manuscript, a numerical method based on the conjunction of Paraskevopoulos's algorithm and operational matrices is developed to solve numerically the multi-order linear and nonlinear fractional differential equations. By means of this conjunction, the multi-order problem is decomposed into a system of differential equations of fractional order which are then solved by employing operational matrices approach. The accuracy and efficiency of the method is examined by taking some examples. In addition, the numerical results presented in Pak et al. [Analytical solutions of linear inhomogeneous fractional differential equation with continuous variable coefficients. Adv Differ Equ. 2019;2019(256):1–22] are improved in our work.

Hamiltanian minimum principle for optimum control of the distance learning process

A mathematical learning model based on control theory in the form of an inhomogeneous linear differential equation is proposed. Analytical formulas and graphs for optimal program control and optimal trajectory are obtained from the principle of the minimum of the Hamiltonian for autonomous systems.