Linear Differential Equations Research Articles

Best Ulam constants for damped linear oscillators with variable coefficients

An associated Riccati equation is used to study the Ulam stability of non-autonomous linear differential equations that model the damped linear oscillator. In particular, the best (minimal) Ulam constants for these equations are derived. These robust results apply to equations with solutions that blow up in finite time, as well as to equations with solutions that exist globally on (−∞,∞). Illustrative, non-trivial examples are presented, highlighting the main results.

Cole–Hopf linearization of the thermocapillary Marangoni dynamics of a two-dimensional bubble with insoluble surfactant

The Marangoni stress-induced dynamics of a two-dimensional inviscid bubble loaded with insoluble surfactant moving in a linear temperature gradient is studied in the limit of small Reynolds, capillary and thermal Péclet numbers. The bubble moves due to the combined effects of thermocapillary Marangoni stresses and those induced by the advective-diffusive spreading of an initial concentration of surfactant on the bubble boundary. It is shown that this nonlinear multiphysics initial value problem is linearizable at any finite non-zero value of a surface Péclet number Pes\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Pe_s$$\\end{document} governing the size of surface diffusion of the insoluble surfactant relative to its advective spreading. This is done by showing that the dynamics can be encoded in the evolution of a function, analytic and single-valued outside the bubble, that satisfies a complex partial differential equation of Burgers type. It is shown that this equation can be linearized by a complex generalization of the classical Cole–Hopf transformation. A numerical method is formulated to solve this linear partial differential equation and determine the Marangoni dynamics of a bubble with some initial surfactant concentration and illustrative calculations are carried out. Results are shown to be consistent with exact equilibrium solutions available from the formulation as Pes→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Pe_s \\rightarrow 0$$\\end{document} and Pes→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Pe_s \\rightarrow \\infty $$\\end{document} and a perturbative formula for the bubble migration velocity at large but finite Pes\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Pe_s$$\\end{document}.

Differential equations for Carrollian amplitudes

Differential equations are powerful tools in the study of correlation functions in conformal field theories (CFTs). Carrollian amplitudes behave as correlation functions of Carrollian CFT that holographically describes asymptotically flat spacetime. We derive linear differential equations satisfied by Carrollian MHV gluon and graviton amplitudes. We obtain non-distributional solutions for both the gluon and graviton cases. We perform various consistency checks for these differential equations, including compatibility with conformal Carrollian symmetries.

Development and Analysis of novel Integrable Nonlinear Dynamical Systems on Quasi-One-Dimensional Lattices. Parametrically Driven Nonlinear System of Pseudo-Excitations on a Two-Leg Ladder Lattice

Following the main principles of developing the evolutionary nonlinear integrable systems on quasi-one-dimensional lattices, we suggest a novel nonlinear integrable system of parametrically driven pseudo-excitations on a regular two-leg ladder lattice. The initial (prototype) form of the system is derivable in the framework of semi-discrete zero-curvature equation with the spectral and evolution operators specified by the properly organized 3 × 3 square matrices. Although the lowest conserved local densities found via the direct recursive method do not prompt us the algebraic structure of system’s Hamiltonian function, but the heuristically substantiated search for the suitable two-stage transformation of prototype field functions to the physically motivated ones has allowed to disclose the physically meaningful nonlinear integrable system with time-dependent longitudinal and transverse inter-site coupling parameters. The time dependencies of inter-site coupling parameters in the transformed system are consistently defined in terms of the accompanying parametric driver formalized by the set of four homogeneous ordinary linear differential equations with the time-dependent coefficients. The physically meaningful parametrically driven nonlinear system permits its concise Hamiltonian formulation with the two pairs of field functions serving as the two pairs of canonically conjugated field amplitudes. The explicit example of oscillatory parametric drive is described in full mathematical details.

A feasible numerical computation based on matrix operations and collocation points to solve linear system of partial differential equations

In this work, a system of linear partial differential equations with constant and variable coefficients via Cauchy conditions is handled by applying the numerical algorithm based on operational matrices and equally-spaced collocation points. To demonstrate the applicability and efficiency of the method, four illustrative examples are tested along with absolute error, maximum absolute error, RMS error, and CPU times. The approximate solutions are compared with the analytical solutions and other numerical results in literature. The obtained numerical results are scrutinized by means of tables and graphics. These comparisons show accuracy and productivity of our method for the linear systems of partial differential equations. Besides, an algorithm is described that summarizes the formulation of the presented method. This algorithm can be adapted to well-known computer programs.

Solving linear differential equations with mixed arguments

Solving linear differential equations with mixed arguments

An Observation About Weak Solutions of Linear Differential Equations in Hilbert Spaces

An Observation About Weak Solutions of Linear Differential Equations in Hilbert Spaces

Higher order fractal differential equations

This paper provides a summary of the fractal calculus framework. It presents higher-order homogeneous and nonhomogeneous linear fractal differential equations with [Formula: see text]-order. Solutions for these equations with constant coefficients are obtained through the method of variation of parameters and the method of undetermined coefficients. The solution space for higher [Formula: see text]-order linear fractal differential equations is defined, showcasing its non-integer dimensionality. The solutions to [Formula: see text]-order linear fractal differential equations are graphically depicted to illustrate their non-differentiability. Additionally, equations of motion governing the behavior of two masses in fractal time are proposed and solved.

A novel least squares approach generating approximations orthogonal to the null space of the operator

We introduce a novel least squares functional specifically formulated to solve linear partial differential equations with operators that have a nonempty null space. Our method involves projecting the solution onto the orthogonal complement of the operator's null space to overcome challenges encountered by conventional numerical methods when nonzero null components are present. We describe the theoretical framework of the proposed method and validate it through numerical examples that show improved accuracy and usability in cases where traditional methods are less effective due to significant null space components. Overall, this approach provides a practical and reliable solution for partial differential equations with substantial null space components.

Euler Method for a Class of Linear Impulsive Neutral Differential Equations

This paper presents a new numerical scheme for a class of linear impulsive neutral differential equations with constant coefficients based on the Euler method. We rigorously establish the first-order convergence of the proposed numerical approach. Additionally, the asymptotical stability of the exact solutions and numerical solutions of impulsive neutral differential equations are studied. To substantiate our findings, two illustrative examples are provided, demonstrating the theoretical conclusions of this paper.

Exponential stability of a non-uniform Euler-Bernoulli beam with axial force

The aim of this paper is to investigate the boundary stabilization of a non-uniform Euler-Bernoulli beam with axial force generated by the equation ρ(x)utt+(σ(x)uxx)xx−(q(x)ux)x=0 defined in (0,1)×(0,∞), where the coefficients ρ(x)>0, σ(x)>0, and q(x)≥0. By adopting a new technique based on a qualitative theory of fourth-order linear differential equations, we prove that the spectrum of the system operator is confined to the open left-half complex plane and that all the associated eigenvalues are geometrically simple. Using this result and the Riesz basis approach, we obtain the exponential stability and the optimal decay rate of the system. This study extends the earlier paper by B.-Z. Guo [SIAM J. Control Optim., 40 (2002), pp. 1905–1923], where no axial force acting on the system (i.e., q≡0).

CoOMBE: A suite of open-source programs for the integration of the optical Bloch equations and Maxwell-Bloch equations

The programs described in this article and distributed with it aim (1) at integrating the optical Bloch equations governing the time evolution of the density matrix representing the quantum state of an atomic system driven by laser or microwave fields, and (2) at integrating the 1D Maxwell-Bloch equations for one or two laser fields co-propagating in an atomic vapour. The rotating wave approximation is assumed. These programs can also be used for more general quantum dynamical systems governed by the Lindblad master equation. They are written in Fortran 90; however, their use does not require any knowledge of Fortran programming. Methods for solving the optical Bloch equations in the rate equations limit, for calculating the steady-state density matrix and for formulating the optical Bloch equations in the weak probe approximation are also described. Program summaryProgram Title: CoOMBECPC Library link to program files:https://doi.org/10.17632/5wsg9d52dk.1Developers' repository link:https://github.com/durham-qlm/CoOMBELicensing provisions: GPLv3Programming language: Fortran 90Nature of problem: The present programs can be used for the following operations: (1) Integrating the optical-Bloch equations within the rotating wave approximation for a multi-state atomic system. At the choice of the user, the calculation will return either the time-dependent density matrix at given times or the density matrix in the long time limit if the system evolves into a steady state in that limit. The calculation can be done with or without averaging over the thermal velocity distribution of the atoms. The number of atomic states which can be included in the calculation is limited only by the CPU time available and possibly by memory requirements. An arbitrarily large number of laser or microwave fields can be included in the calculation if these fields are all CW. This number is currently limited to one or two for fields that are not all CW. The calculation can be done in the weak probe approximation, or in the rate equations approximation, or without assuming either of these two approximations. Calculating refractive indexes, absorption coefficients and complex susceptibilities is also possible. (2) Integrating the 1D Maxwell-Bloch equations in the slowly varying envelope approximation for one or two fields co-propagating in a single-species atomic vapour. Although geared towards the case of atoms interacting with laser fields, this code can also be used for more general quantum systems with similar equations of motion (e.g., molecular systems, spin systems, etc.).Solution method: The Lindblad master equation is expressed as a system of homogeneous first order linear differential equations, which are transformed as required and solved to obtain the density matrix representing the state of the atomic system. A variety of methods are offered to this end. The same approach is also used in the calculation of the polarisation of the medium when integrating the Maxwell-Bloch equations. The latter are integrated over space using predictor-corrector methods. The library includes a general driving program making it possible to use these codes without additional program development. The distribution also includes examples of the use of a container for running these programs without a pre-installed Fortran compiler.

Regularity, synthesis, rigidity and analytic classification for linear ordinary differential equations of second order

We study second order linear homogeneous differential equations a(x)y'' + b(x)y' + c(x)y = 0 with analytic coefficients in a neighborhood of a regular singularity in the sense of Frobenius. These equations are model for a number of natural phenomena in sciences and applications in engineering. We address questions which can be divided in the following groups: (i) Regularity of solutions. (ii) Analytic classification of the differential equation. (iii) Formal and differentiable rigidity. (iv) Synthesis and uniqueness of ODEs with a prescribed solution. Our approach is inspired by elements from analytic theory of singularities and complex foliations, adapted to this framework. Our results also reinforce the connection between classical methods in second order analytic ODEs and (geometric) theory of singularities. Our results, though of a clear theoretical content, are important in justifying many procedures in the solution of such equations.

A NEW ALGORITHM FOR GENERATING POWER SERIES SOLUTIONS FOR A BROAD CLASS OF FRACTIONAL PDES: APPLICATIONS TO INTERESTING PROBLEMS

This research offers a precise analytical solution for initial value problems of both linear and nonlinear fractional order partial differential equations. A new approach called the limit residual function method is developed and utilized to generate a series solution for the equations. The key tools of this method are the concepts of power series, residual function, and the limit at zero. The new method is characterized by the speed of determining the series solution coefficients and the limited mathematical operations used. The study examines five significant applications of fractional partial differential equations using this new method. To validate the accuracy of the results, they are compared with exact solutions in the classical case for the discussed applications. We conclude that the proposed approach is straightforward, effective, and successful in solving linear and nonlinear fractional differential equations.

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.

Electromagnetic wave propagation in Eddington-inspired Born–Infeld gravity space-time with topological defects

In this study, we focus on examining the characteristics of electromagnetic fields within a curved space-time background under the framework of Eddington-inspired Born–Infeld (EiBI) gravity, in the presence of a global monopole. We derived Maxwell’s vacuum field equations in this curved spacetime and obtained a set of linear differential equations for the electric and magnetic fields. After decoupling these equations, we solved for the analytical solutions of both the electric and magnetic fields using special functions. We then extended our analysis to the same EiBI-gravity framework, this time incorporating a cosmic string. Following a similar approach, we derived the first-order differential equations governing the electric and magnetic fields and obtained their analytical solutions using special functions. Our findings demonstrate significant influences of the global monopole, cosmic string, and the Eddington parameters on the behavior of electromagnetic waves in this curved space-time configuration with topological defects, resulting in notable deviations from the Minkowski flat space case.

Flow and heat transfer of non-miscible micropolar and Newtonian fluid in porous channel sandwiched between parallel plates

An analytical approach has been considered to inspect the flow and heat transfer of immiscible Newtonian and micropolar fluid in a porous medium. This study aims to elucidate the Darcy effect on the model in which isotropic porous regions with different permeability are used. The model of the current problem explains that the flow region is divided into two regions: Newtonian fluid flows in the upper layer, and micropolar fluid flows in the lower layer. Different constant temperatures imposed at boundary walls, heat transfer does not affect the pressure gradient. The governing equations are solved analytically by applying a linear differential equation (LDE). The effect of associated physical parameters such as material parameter, Darcy number, Eckert number, Prandtl number, and viscosity ratio on velocity, micro-rotation, heat transfer, flow rate, heat transfer rate, wall shear stress, and Nusselt number have been inspected. The most significant finding of this research work is that increasing the Darcy number signifies enhanced permeability, resulting in a higher flow rate. The heat transfer rate at the top occurs maximum when the material parameter’s range is minimum while raising the viscosity ratio leads to an increasing heat transfer rate at the bottom. Enhancement in material parameter influences Nusselt number and decreases in nature. The findings of our study are verified with the previously established results. The present work has a setup that is useful in petroleum extraction, transport problems in reservoir rock of an oil field, improving nutrient transport and thermal regulation in tissue engineering, and designing more efficient drug delivery systems and biomedical devices.

New Two Parameter Integral Transform "MAHA Transform" and its Applications in Real Life

In this paper, an integral transform with two parameters was proposed, and this transform was applied to obtain the exact solutions for the linear ordinary differential equations with constant coefficients of higher orders. This integral transform was also applied and we called it Maha transform in nuclear physics and for medical applications.

A transparent experimental modelling method for linear multiple-input multiple-output systems

This paper presents a transparent technique that simulates processes using input and output measurement data without the use of any complex optimization or iterative algorithms. The method is very simple and easy to understand as it comprises only of linear ordinary differential equations, Laplace transformations and least squares technique. The solution can be expressed only in two linear matrix equations. The main aim is to provide bachelor students of engineering with a tool to formulate transfer functions.The method is validated with artificially generated erroneous measurement data as well as using measurements obtained from a practical application at the university. Inclusion of dead times and estimation of initial conditions make it ideal to be used for various range of applications such as stable and unstable systems with and without damping, open and closed loop systems. Over-integration, normalizing and/or zeroing of different coefficients add more degrees of freedom to improve the model quality.

Error Analysis for Empirical Risk Minimization Over Clipped ReLU Networks in Solving Linear Kolmogorov Partial Differential Equations

Error Analysis for Empirical Risk Minimization Over Clipped ReLU Networks in Solving Linear Kolmogorov Partial Differential Equations