System Of Linear Differential Equations Research Articles

Symplectic solutions of the plane annular sectors in micropolar elasticity

The symplectic approach is utilized to derive the solutions of the plane annular sector in micropolar elasticity. According to the Hellinger-Reissner variational principle, the Hamiltonian canonical equations are obtained for the plane problem in an annular sectorial region. Through the method of separation of variables, we obtain an eigenvalue equation of the Hamiltonian operator matrix which is a system of linear differential equations with variable coefficients. By the method of series solutions, its general solutions can be expressed as the linear combination of Bessel functions and power functions. The corresponding eigensolutions for three types of homogeneous boundary conditions are derived. According to the adjoint symplectic orthogonality of the eigensolutions, the solution of the micropolar plane annular sector is expressed as a series expansion of eigensolutions. Numerical analyses are presented to demonstrate the validity of the proposed method and confirm its ability to accurately capture the size effect.

Optimal Stabilization in Systems of Linear Differential Equations

Optimal Stabilization in Systems of Linear Differential Equations

Analytical Solutions of Systems of Linear Delay Differential Equations by the Laplace Transform: Featuring Limit Cycles

In this paper we develop an approach for obtaining the solutions to systems of linear retarded and neutral delay differential equations. Our analytical approach is based on the Laplace transform, inverse Laplace transform and the Cauchy residue theorem. The obtained solutions have the form of infinite non-harmonic Fourier series. The main advantage of the proposed approach is the closed-form of the solutions, which are capable of accurately evaluating the solution at any time. Moreover, it allows one to study the asymptotic behavior of the solutions. A remarkable discovery, which to the best of our knowledge has never been presented in the literature, is that there are some particular linear systems of both retarded and neutral delay differential equations for which the solution asymptotically approaches a limit cycle. The well-known method of steps in many cases is unable to obtain the asymptotic behavior of the solution and would most likely fail to detect such cycles. Examples illustrating the Laplace transform method for linear systems of DDEs are presented and discussed. These examples are designed to facilitate a discussion on how the spectral properties of the matrices determine the manner in which one proceeds and how they impact the behavior of the solution. Comparisons with the exact solution provided by the method of steps are presented. Finally, we should mention that the solutions generated by the Laplace transform are, in most instances, extremely accurate even when the truncated series is limited to only a handful of terms and in many cases become more accurate as the independent variable increases.

A falling fluid droplet in an oscillating flow field

We examine the flow in and around a falling fluid droplet in a vertically oscillating flow. We assume axisymmetric Stokes flow, and for small deformations to the droplet, the governing equations can be linearized leading to an infinite system of linear ordinary differential equations. In this study, we have analytically solved the problem in the small-capillary limit. We note that the solution locally breaks down at the poles of the droplet. The drag and center of the mass were also obtained. In the case when only odd modes are present, the droplet shows three-dimensional axisymmetric heart-shaped solutions oscillating vertically in time. When only even modes are present, the droplet exhibits axisymmetric stretching and squeezing.

Sensor Fault Detection and Diagnosis of Linear Parabolic PDE Systems With Unknown Inputs

This paper investigates the sensor bias fault detection and diagnosis problem for linear parabolic partial differential equation (PDE) systems under the existence of unknown input signals. A variation of Wirtinger's inequality is used to design a Luenberger-type PDE observer and radial basis function (RBF) neural networks are applied to approximate the unknown inputs, guaranteeing the boundedness of the state estimation error. Taking the measurement error as the fault indication signal, a residual evaluation logic is developed to achieve fault detection. An adaptive diagnostic law is constructed to estimate the values of sensor bias faults once they are detected. <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\sigma$</tex-math></inline-formula> -modification RBF neural networks are designed to compensate for the unknown inputs in the presence of sensor faults, ensuring that both state and sensor bias faults estimation errors converge to some adjustable sets. The effectiveness and applicability of the developed fault diagnosis strategy are demonstrated by simulation results on a heat diffusion process.

On a Particular Particular Solution

Summary The usual process for solving a nonhomogeneous system of linear differential equations is to find the general complementary homogeneous solution first and then construct a particular solution from it. When can we flip the script on this process and compute a particular solution first? This paper argues this is possible when the nonhomogeneous forcing function is analytic. It presents two different series expansions for this particular particular solution using power series methods, and reveals a surprising connection to the more familiar variation of parameters formula.

Research of direct and inverse problems of sensitivity using methods of practical stability by direction

The general statements of the problems of construction of extreme ranges of tolerances on parameters for a linear system of differential equations in the presence of dynamic restrictions on the spread of the state vector are considered. It was found that such sensitivity problems are closely related to the problems of estimating the maximum area of stability in the corresponding function space. In order to obtain upper estimates of the sought values, the problem of estimating tolerances from the positions of stability in a single direction is formalized. The cases of linear and nonlinear restrictions on the spread of the phase coordinate vector with initial linear initial conditions relative to the vector of the system parameters are studied. Necessary and sufficient conditions are formulated in the form of criteria for estimating tolerances on parameters covering the case of fixed initial conditions. The tasks of estimating the maximum volume of the stability region with respect to the deviations of the state vectors and system parameters are considered. For the case of compatible admissible constraints, the paper gives the corresponding numerical calculations for the upper estimates. In particular, for fixed initial conditions, the maximum set of tolerances for system parameters with compatible dynamic restrictions on fluctuations of state vectors and parameters are determined according to these evaluation criteria. From the standpoint of practical stability in direction, direct sensitivity problems are investigated, numerical estimates of the spread of the phase coordinate vector of the system in the presence of linear and fixed initial conditions relative to the parameter vector are given. It has been demonstrated that in the given specification of the problem of maximum estimation of tolerances it is completely covered by problem statements of practical stability by direction. Thus, to calculate areas of guaranteed sensitivity, the concept of practical stability in a single direction is extended to the space of sensitivity functions at the initial moment of time. The above algorithms are used to estimate the range of tolerances for the adjustment parameters of the discrete model of the induction acceleration system with the given limitations on the spread of the quality criterion; the variation of the quality indicator within the tolerance field is presented in the form of a linear form relative to the spread of the parameter vector.

DECOUPLING A SYSTEM OF LINEAR DIFFERENTIAL EQUATIONS INTO BLOCKS

DECOUPLING A SYSTEM OF LINEAR DIFFERENTIAL EQUATIONS INTO BLOCKS

On Solution of First-Order Linear Systems of Partial Differential Equations

On Solution of First-Order Linear Systems of Partial Differential Equations

Numerical investigation of systems of fractional partial differential equations by new transform iterative technique

&lt;p&gt;This research introduced a new method, the Aboodh Tamimi Ansari transform method ($ (AT)^2 $ method), for solving systems of linear and nonlinear fractional partial differential equations. The method combined the Aboodh transform method and the Tamimi Ansari method, allowing for the simultaneous solution of linear and nonlinear terms without restrictions. The Caputo sense was considered for fractional derivatives. The effectiveness of the proposed method was demonstrated through numerical solutions, graphical representations, and tabular data, showing strong agreement with exact solutions. The approach was deemed precise, easy to apply, and could be extended to address further challenges in fractional-order problems. Computational tasks were carried out using Mathematica 13.&lt;/p&gt;

МОДЕЛЬ АДСОРБЦИОННОЙ ОЧИСТКИ СТОКОВ С УЧЕТОМ ХИМИЧЕСКОЙ РЕАКЦИИ

The article poses and solves the general problem of the dynamics of adsorption from a liquid in a fixed layer of adsorbent, taking into account a first-order chemical reaction and a linear adsorption isotherm. The adsorption model is based on the mass balance equations of the target component for an infinitesimal layer element and adsorption kinetics, written as a system of linear first-order partial differential equations with corresponding boundary conditions. The problem is solved by the Laplace transform method. This made it possible to reduce the joint solution of a system of equations in the originals to an independent solution in images of the Cauchy problem for an ordinary differential equation and an algebraic equation. As a result, expressions were obtained for the concentrations of the target component in the adsorbent layer and in the flow at any time and in any section of the layer. This made it possible to simulate a practical problem of the adsorption of chloroform by zeolites for the treatment of wastewater from a real swimming pool and obtain interesting graphic materials based on the solution and the developed algorithm. The results of the solution can be translated into numbers by programming in any available high-level algorithmic language. It is shown that the shape of the curves of changes in the concentrations of the target component in the adsorbent and adsorbent over the thickness of the adsorbent layer and over time does not change, although the numerical values of these parameters change in a fairly wide range. It is also noted that in the range of concentration changes considered in the article, the concentration of the adsorbent initially decreases sharply, and then smoothly along the entire length of the fixed adsorbent layer decreases slightly. It has been revealed that the use of methods, techniques and theorems of the Laplace transform method in relation to the problem of adsorption dynamics makes it possible to deeply penetrate the mechanism of the problem being solved, which is not possible with other solution methods. This allows, with a known set of initial data, to find the change in the concentrations of the target component in both phases and select the required thickness of the adsorbent layer.

STOCHASTIC GALERKIN METHOD AND PORT-HAMILTONIAN FORM FOR LINEAR FIRST-ORDER ORDINARY DIFFERENTIAL EQUATIONS

We consider linear first-order systems of ordinary differential equations (ODEs) in port-Hamiltonian (pH) form. Physical parameters are remodeled as random variables to conduct an uncertainty quantification. A stochastic Galerkin projection yields a larger deterministic system of ODEs, which does not exhibit a pH form in general. We apply transformations of the original systems such that the stochastic Galerkin projection becomes structure-preserving. Furthermore, we investigate meaning and properties of the Hamiltonian function belonging to the stochastic Galerkin system. A large number of random variables implies a high-dimensional stochastic Galerkin system, which suggests itself to apply model order reduction (MOR) generating a low-dimensional system of ODEs. We discuss structure preservation in projection-based MOR, where the smaller systems of ODEs feature pH form again. Results of numerical computations are presented using two test examples.

On the regularity of the flow of analytic vector fields

The flow of vector field X is the solution to the system of linear partial differential equations ∂_t u(t,z)=X(u(t,z)) with initial condition u(0,z)=z. It is known that for an analytic vector field, its flow is given by a convergent Lie series. Recently, Carillo gave a quick and elementary proof of this fact by introducing a special family of norms. This paper gives an even more concise proof by directly estimating the formal series expansion and using a family of majorant functions studied by Lax.

CANONICAL APPROXIMATIONS IN IMPULSE STABILIZATION FOR A SYSTEM WITH AFTEREFFECT

For optimal stabilization of an autonomous linear system of differential equations with aftereffect and impulse controls, the formulation of the problem in the functional state space is used. For a system with aftereffect, approximating systems of ordinary differential equations proposed by S.N. Shimanov and J. Hale are used. A method for constructing approximations for optimal stabilizing control of an autonomous linear system with aftereffect and impulse controls is proposed. Matrix Riccati equations are used to find approximating controls.

Exterior Boundary-Value Poincaré Problem for Elliptic Systems of the Second Order with Two Independent Variables

This paper offers a number of examples showing that in the case of two independent variables the uniform ellipticity of a linear system of differential equations with partial derivatives of the second order, which fulfills condition (3), do not always cause the normal solvability of formulated exterior elliptic problems in the sense of Noether.Nevertheless, from the system of differential equations with partial derivatives of elliptic type it is possible to choose, under certain additional conditions, classes which are normally solvable in the sense of Noether.This paper also shows that for the so-called decomposed system of differential equations, with partial derivatives of an elliptic type in the case of exterior regions, the Noether theorems are valid.

SAWI TRANSFORMATION FOR SOLVING A SYSTEM OF LINEAR ORDINARY DIFFERENTIAL EQUATIONS

There are many problems in nature whose solutions are obtained through mathematical concepts. One of the most common mathematical concepts is a mathematical concept that is classified under initial value problems, such as a system of linear ordinary differential equations equipped with initial values. One tool that can solve the initial value problem is the Sawi transformation. This article describes the study of the initial value problem as a system of linear ordinary differential equations and its solution using the Sawi transformation. In addition, as part of applying the resulting theory, 2 (two) case examples are given (a first-order chemical reaction system with three certain chemicals and a mass-spring system with forced motion) to be solved using the Sawi transformation. So that problem solving can be interpreted and easily understood, in the 2 (two) case studies discussed and focused on the concentration of the chemical reactants, simulations were carried out for several different initial values and reaction rate constants. Compared to other methods (Laplace transform), the results obtained from using the Sawi transformation for the cases discussed show that the analytical solutions for the selected initial values have similar solutions.

On auxiliary fields and Lagrangians for relativistic wave equations

We address the problem of the existence of a Lagrangian for a given system of linear partial differential equation with constant coefficients. As a subtask, this involves bringing the system into a pre-Lagrangian form, wherein the number of equations matches the number of unknowns. We introduce a class of overdetermined systems, called co-flat, and show that they always admit a pre-Lagrangian form, which can be explicitly constructed by means of auxiliary variables. Moreover, we argue that such systems enjoy pre-Lagrangian formulations without auxiliary variables at all. As an application of our method, we construct new pre-Lagrangian and Lagrangian formulations for free massive fields of arbitrary integer spin. In contrast to the well-known models of Singh and Hagen, our Lagrangians involve much fewer auxiliary fields.

Use of Reversion Method to Solve Mathematical Models on Series Electrical Circuits (Rl) with Nonlinear Inductors

Forming a mathematical model of a series electrical circuit with a nonlinear inductor obtained based on Kirchoff's laws one and two takes the form of a nonlinear differential equation. Approach methods are used to determine the electric current in the circuit, including the Reversion Method, namely by converting nonlinear differential equations into a system of linear differential equations.

Free vibration of cylindrical shells and panels with cutout via the differential transform method

Semi-numerical methods hold significant value due to their efficiency in time-consuming calculations. Accordingly, this article proposes an effective strategy to derive the vibration parameters of cylindrical shells (CSs) with cutout utilizing a semi-analytical approach entitled differential transform method (DTM). To achieve this, the governing equations of motion are first introduced based on Sander’s theory of shells. These equations give rise to a coupled system of linear ordinary differential equations, which pose a challenge in solving the eigenvalue problem. It should be emphasized that the classical version of the differential transform method could not be directly applied to solve the eigenvalue problem related to the above-mentioned coupled differential equation. An innovative technique is employed and discussed in the paper to remedy these difficulties, which could be widely used in similar problems. As a result, the natural frequencies and mode shapes of some CSs with cutouts, various internal angles, and diverse material characteristics are attained. The outcomes undergo validation through the hybrid finite element method, and the proposed method’s accuracy is rigorously proved. In conclusion, the proposed DTM provides an efficient solution to the eigenvalue problem of coupled differential equations and offers a valuable tool for obtaining accurate vibration parameters of CSs.

An oscillating reaction network with an exact closed form solution in the time domain

BackgroundOscillatory behavior is critical to many life sustaining processes such as cell cycles, circadian rhythms, and notch signaling. Important biological functions depend on the characteristics of these oscillations (hereafter, oscillation characteristics or OCs): frequency (e.g., event timings), amplitude (e.g., signal strength), and phase (e.g., event sequencing). Numerous oscillating reaction networks have been documented or proposed. Some investigators claim that oscillations in reaction networks require nonlinear dynamics in that at least one rate law is a nonlinear function of species concentrations. No one has shown that oscillations can be produced for a reaction network with linear dynamics. Further, no one has obtained closed form solutions for the frequency, amplitude and phase of any oscillating reaction network. Finally, no one has published an algorithm for constructing oscillating reaction networks with desired OCs.ResultsThis is a theoretical study that analyzes reaction networks in terms of their representation as systems of ordinary differential equations. Our contributions are: (a) construction of an oscillating, two species reaction network [two species harmonic oscillator (2SHO)] that has no nonlinearity; (b) obtaining closed form formulas that calculate frequency, amplitude, and phase in terms of the parameters of the 2SHO reaction network, something that has not been done for any published oscillating reaction network; and (c) development of an algorithm that parameterizes the 2SHO to achieve desired oscillation, a capability that has not been produced for any published oscillating reaction network.ConclusionsOur 2SHO demonstrates the feasibility of creating an oscillating reaction network whose dynamics are described by a system of linear differential equations. Because it is a linear system, we can derive closed form expressions for the frequency, amplitude, and phase of oscillations, something that has not been done for other published reaction networks. With these formulas, we can design 2SHO reaction networks to have desired oscillation characteristics. Finally, our sensitivity analysis suggests an approach to constructing a 2SHO for a biochemical system.