Second-order Linear Differential Equations Research Articles

Solving Second-Order Homogeneous Linear Differential Equations in Terms of the Tri-Confluent Heun’s Function

In this paper, we state an algorithm that checks whether a given second-order linear differential equation can be reduced to the tri-confluent Heun’s equation. The algorithm provides a method for finding solutions of the form exp(∫r(x)dx)·HeunT(q,α,γ,δ,ϵ,f(x)), where the parameters α,β,λ∈C, the functions r,f∈C(x), and f are not constant.

Anisotropic functionally graded nano-beam models and closed-form solutions in plane gradient elasticity

This study delves into the investigation of exact analytical solutions for the plane stress and displacement fields within linear homogeneous anisotropic nano-beam models of gradient elasticity. It focuses on solving the Helmholtz equation, which encompasses a second-order non-homogeneous linear partial differential equation in plane gradient elasticity theory, utilizing polynomial series-type solutions. The analysis centers on the utilization of gradient Airy stress functions to derive stress fields in the gradient theory. The research yields closed-form analytical solutions for Airy stress functions, stress, and strain fields in both classical and gradient theories. The study considers five distinct types of two-dimensional functionally graded cantilever beams with various boundary conditions: a cantilever anisotropic nano-beam subjected to a concentrated force at the free end, a cantilever anisotropic nano-beam under a uniform load, a simple anisotropic nano-beam under a uniform load, a propped cantilever anisotropic nano-beam under a uniform load, and a fixed end cantilever anisotropic nano-beam under a uniform load. General analytical solutions for the gradient stress and displacement fields of two-dimensional and one-dimensional anisotropic nano-beams under different boundary conditions are provided. The study showcases significant strain gradient size effects at the nano-scale through the derived analytical solutions for anisotropic beams. Additionally, it demonstrates that the strain gradient theory results for the limit case of the gradient coefficient c precisely align with results for isotropic and anisotropic materials in elasticity theory and classical theory. Furthermore, real-world applications are discussed, considering stress and displacement fields in real anisotropic materials such as TiSi2 single crystals and orthotropic materials, which are special cases of anisotropic materials like wood and epoxy as documented in the literature.

Background of Second-Order Linear Differential Equations

The differential equations, usually the goal is to find a solution. In other words, we want to find a function (or functions) that satisfies the differential equation. The technique we use to find these solutions varies, depending on the form of the differential equation with which we are working. Second-order differential equations have several important characteristics that can help us determine which solution method to use. In this section, we examine some of these characteristics and the associated terminology. An equation containing the derivatives of one or more dependent variables, with respect to one or more independent variables, is said to be a differential equation (DE).

A systematic approach to obtain the analytical solution for coupled linear second-order ordinary differential equations: Part II

A systematic approach to obtain the analytical solution for coupled linear second-order ordinary differential equations: Part II

A note on Laplace Adomian decomposition method on a linear problem

In last two decades, Laplace Adomian Decomposition Method (LADM) is vastly used to solve non-linear (or even fractional order) differential equations. The method approaches the solution with the partial sums of function series. However, it is not easy to show that the limit of the function series is the exact solution of the problem. In this article, we consider a simple problem such as homogenous second-order linear ordinary differential equation with constant coefficients. We prove analytically that the LADM gives the right exact solution to the considered problem.

Expressing Solutions of Bessel Differential Equation in terms of Hypergeometric Functions

The BDE (Bessel differential equation) is a second-order linear ordinary differential equation (ODE), and it is considered one of the most significant differential equations because of its extensive applications. The solutions of this differential equation are called Bessel functions. These solutions can be expressed in terms of hypergeometric functions, and in this study, the Bessel functions of the first kind are worked. Hypergeometric functions are the sum of a hypergeometric series. A series ∑▒u_n is known as hypergeometric when the ratio u_(n+1)⁄u_n is a rational function of n. BDE has coefficients with variables; therefore, it is solved by the Frobenius approach. We implement some mathematical steps based on the definition of hypergeometric functions to express the solutions in terms of hypergeometric function. The result shows that the solution of the BDE in terms of hypergeometric function is f(x)=a_0 x^p (_1^ )F_2 (1;1+(p+ρ)/2,1+(p-ρ)/2;x^2⁄4) and the two linearly independent solutions for the BDE of order ρ in terms of hypergeometric function are 〖f_1 (x)=a〗_0 x^ρ (_0^ )F_1 (1+ρ;x^2⁄4) and〖〖 f〗_2 (x)=a〗_0 x^ρ (_0^ )F_1 (1-ρ;x^2⁄4).

Algorithmic implementation of an exact three-point difference scheme for a certain class of singular Sturm–Liouville problems

In this article, we present a new algorithmic implementation of exact three-point difference schemes for a certain class of singular Sturm–Liouville problems. We demonstrate that computing the coefficients of the exact scheme at any grid node xj requires solving two auxiliary Cauchy problems for the second-order linear ordinary differential equations: one problem on the interval [xj−1,xj] (forward) and one problem on the interval [xj,xj+1] (backward). We have also proven the coefficient stability theorem for the exact three-point difference scheme.

Construction of Polynomial Eigenfunctions of a Second-Order Linear Differential Equation

A system of third-order recurrence relations for the coefficients of polynomial eigenfunctions (PEFs) of a differential equation is solved. A recurrence relation for three consecutive PEFs and a formula for differentiating PEFs are obtained. We consider differential equations one of which generalizes the Hermite and Laguerre differential equations and the other is a generalization of the Jacobi differential equation. For these equations, we construct functions bringing them to self-adjoint form and find conditions under which these functions become weight functions. Examples are given where the PEFs for nonweight functions do not have real zeros.

Effect of the drag coefficient on the performance of vertical porous baffles in a sloshing tank

Purpose The present work involves analytical and experimental investigation of sloshing in a two-dimensional rectangular tank including the effect of porous baffles to control and/or reduce the wave motion in the sloshing tank. The purpose of this study is to assess the analytical solutions of the drag coefficient effect on porous baffles performance to track free surface motion variation in the sloshing tank by comparison with experimental shake table tests under a range of sway excitation. Design/methodology/approach The linear second-order ordinary differential equations for liquid sloshing in the rectangular tank were solved using Newmark’s beta method and obtained the analytical solutions for liquid sloshing with dual vertical porous baffles of full submergence depths in a sway-oscillated rectangular tank following the methodology similar to Warnitchai and Pinkaew (1998) and Tait (2008). Findings The porous baffles significantly reduce wave elevation in the varying filled levels of the tank compared to the baffle-free tank under the range of excitation frequencies. It is observed that the Reynolds number-dependent drag coefficient for porous baffles in the tank can significantly reduce the sloshing elevations and is found to be effective to achieve higher damping compared to the porosity-dependent drag coefficient for porous baffles in the sloshing tank. The analytical model’s response to free surface elevation variations in the sloshing tank was compared with the experiment’s test results. The analytical results matched with shake table test results with a quantitative difference near the first resonant frequency. Research limitations/implications The scope of the study is limited to porous baffles performance under range sway motion and three different filling levels in the tank. The porous baffle performance includes Reynolds number dependent drag coefficient to explore the damping effect in the sloshing tank. Originality/value The porous baffles with low-level porosities in the sloshing tank have many engineering applications where the first resonant mode of sloshing in the tank is more important. The porous baffle drag coefficient is an important parameter to study the baffle’s damping effect in sloshing tanks. Hence, obtained analytical solution for liquid sloshing in the rectangular tank with Reynolds number as well as porosity-dependent drag coefficient (model 1) and porosity-dependent drag coefficient porous baffles (model 2) performance is discussed. The model’s test results were validated using a series of shake table sloshing experiments for three fill levels in the tank with sway motion at various excitation frequencies covering the first four sloshing resonant modes.

An Operator Method for Investigation of the Stability of Time-Dependent Source Identification Telegraph Type Differential Problems

This article is devoted to the study of the stability of time-dependent source identification telegraph type differential problems with dependent coefficients. Time-dependent source identification problems (SIPs) for telegraph differential equations (TDEs) with constant coefficients can be solved by classical integral-transform methods. However, these classical methods can be used, basically, in cases where the differential equation has constant coefficients. We establish the basic theorem of the stability of the time-dependent SIPs for the second-order linear differential equation (DE) in a Hilbert space with a self-adjoint positive definite operator (SAPDO) and damping term. In practice, stability estimates for the solution of the three types of SIPs for one-dimensional and for multidimensional TDEs with dependent coefficients and classic and non-classic conditions are obtained.

Positivity of oscillatory integrals and Hankel transforms

ABSTRACT In consideration of the integral transform whose kernel arises as an oscillatory solution of certain second-order linear differential equation, its positivity is investigated on the basis of Sturm's theory. As applications, positivity criteria are obtained for Hankel transforms as well as trigonometric integrals defined on the positive real line.

Computing indefinite integrals by difference equations

In teaching mathematics to first-year undergraduates, and thus in the appropriate calculus textbooks, the task of calculating an integral that satisfies a specific first-order or second-order recurrence relation often appears. These relations are obtained mainly by applying the method of integration by parts. Calculating such integrals is usually tedious, especially for an integer n > 2, time-consuming, and presents the possibility of making a large number of errors when computing involves multiple iterative steps. In [1], it is shown that in two cases (Theorems 2.1. and 2.3), the process of calculating integrals satisfying first-order recurrence relations can be performed quickly using easily memorised closed-form formulas for corresponding primitive functions. The question can rightly be asked whether there is a faster way to calculate other integrals of this type. In this paper, our goal is to give an affirmative answer to such a question, though without convering all situations. Since each recurrence relation is equivalent to a difference equation of the same order, the calculation of integrals mentioned above can be reduced to solving the corresponding difference equations. Since every first-order or second-order linear difference equation is solvable, it follows that for every integral which can be reduced to a first- order or second-order recurrence formula, it is possible to find corresponding primitive functions directly. Sometimes such a procedure is much faster than iterative solving of the integral. Closed-form formulas for the integrals discussed in the following sections are not unknown (see [2]). However, here our goal is to present the idea of computing indefinite integrals using difference equations. We will discuss it in more detail in Section 2. In Section 3, we discuss the application of the results obtained to calculate several improper integrals and the application of some of them in different sciences. An exciting example of such an application is the integral , which in the case n = 1 is used in the kinetic theory of gases, particularly in the Maxwell-Boltzmann distribution of gas molecules by energies (see Remark 4). Also, we compare the formulas obtained by the method of difference equations with the formulas obtained using Wolfram Alpha software (see Remark 5).

Static spherically symmetric black holes of Brans–Dicke theory of gravity

This paper investigates the static spherically symmetric vacuum solutions of Brans–Dicke Theory of Gravity. In this paper, we systematically derive the static spherically symmetric solutions by starting with the general static spherically symmetric anzats, discovering the nonzero Christoffel symbols, nonzero Ricci tensor components and the Ricci Scalar. These results allow us to rewrite the four Brans–Dicke gravitational field equations explicitly in terms of the metric components, the scalar field and its scalar potential. This paper systematically explains and shows how to modify these linear second-order ordinary differential equations in order to be able to find the solution of the scalar field, the metric components and the scalar potential analytically. We successfully find the most general solution of the static spherically symmetric vacuum solutions of Brans–Dicke gravity and the well-known Schwarzschild solution is recovered as the simplest solution.

Mathematical Model for Growth and Rifampicin-Dependent Killing Kinetics of Escherichia coli Cells.

Antibiotic resistance is a global health threat. We urgently need better strategies to improve antibiotic use to combat antibiotic resistance. Currently, there are a limited number of antibiotics in the treatment repertoire of existing bacterial infections. Among them, rifampicin is a broad-spectrum antibiotic against various bacterial pathogens. However, during rifampicin exposure, the appearance of persisters or resisters decreases its efficacy. Hence, to benefit more from rifampicin, its current standard dosage might be reconsidered and explored using both computational tools and experimental or clinical studies. In this study, we present the mathematical relationship between the concentration of rifampicin and the growth and killing kinetics of Escherichia coli cells. We generated time-killing curves of E. coli cells in the presence of 4, 16, and 32 μg/mL rifampicin exposures. We specifically focused on the oscillations with decreasing amplitude over time in the growth and killing kinetics of rifampicin-exposed E. coli cells. We propose the solution form of a second-order linear differential equation for a damped oscillator to represent the mathematical relationship. We applied a nonlinear curve fitting solver to time-killing curve data to obtain the model parameters. The results show a high fitting accuracy.

Multi-phase iterative learning control for high-order systems with arbitrary initial shifts

Aiming at the second-order tracking system with arbitrary initial shifts, this paper presents a multi-phase iterative learning control strategy. Firstly, utilizing the form of solution of the second-order non-homogeneous linear differential equation with constant coefficients and the initial shifts, we can select the appropriate control gain to ensure that the second-order systems are stable and reach the stable output after a fixed time. Secondly, on the premise that the second-order systems have reached the fixed output, two methods are proposed for rectifying the fixed shift, namely, shifts rectifying control and varied trajectory control. Theoretical analysis shows that the multi-stage iterative learning control strategy proposed in this paper can ensure that the second-order systems achieve complete tracking in the specified interval. Finally, the simulation examples affirm the validation of the designed algorithms.

Fast Generalized Sliding Sinusoidal Transforms

Discrete cosine and sine transforms closely approximate the Karhunen–Loeve transform for first-order Markov stationary signals with high and low correlation coefficients, respectively. Discrete sinusoidal transforms can be used in data compression, digital filtering, spectral analysis and pattern recognition. Short-time transforms based on discrete sinusoidal transforms are suitable for the adaptive processing and time–frequency analysis of quasi-stationary data. The generalized sliding discrete transform is a type of short-time transform, that is, a fixed-length windowed transform that slides over a signal with an arbitrary integer step. In this paper, eight fast algorithms for calculating various sliding sinusoidal transforms based on a generalized solution of a second-order linear nonhomogeneous difference equation and pruned discrete sine transforms are proposed. The performances of the algorithms in terms of computational complexity and execution time were compared with those of recursive sliding and fast discrete sinusoidal algorithms. The low complexity of the proposed algorithms resulted in significant time savings.

Some Approaches for Solving Multiplicative Second-Order Linear Differential Equations with Variable Exponentials and Multiplicative Airy’s Equation

This paper offers several approaches for solving multiplicative second-order linear differential equations with variable exponentials, such as normalization and reduction to Riccati equations. In addition, in this paper, the multiplicative version of the Airy equation, which emerges in fluid mechanics, geophysics, and atomic physics, is solved using the multiplicative power series solution method.

Construction of Polynomial Eigenfunctions of a Second-Order Linear Differential Equation

Construction of Polynomial Eigenfunctions of a Second-Order Linear Differential Equation

Gradient elasticity solutions of 2D nano-beams

In this study, the exact analytical solutions of a two-dimensional linear homogeneous isotropic nano-beam in gradient elasticity are studied. Four different types of two-dimensional cantilever beams and related boundary conditions are considered. The cases are a cantilever beam under a concentrated force at the end, a cantilever beam under a uniform load, a propped cantilever beam under a uniform load, and a fixed-end beam under a uniform load. The two-dimensional stress gradient fields are investigated and obtained from the analytical solutions of a linear second-order partial differential equation written in terms of the classical and the gradient Airy stress functions. Additionally, the micro-size effects in the displacement components for different loads and support conditions for the two-dimensional cantilever beams by using strain gradient elasticity theory are investigated. Furthermore, for one-dimensional Euler–Bernoulli beam model, the associated stress and strain elasticity solutions are obtained from two-dimensional analytical solutions. The graphical presentations of the exact closed-form solutions are provided and discussed.

Nested invariant tori foliating a vector field and its curl: Toward MHD equilibria and steady Euler flows in toroidal domains without continuous Euclidean isometries

This paper studies the problem of finding a three-dimensional solenoidal vector field such that both the vector field and its curl are tangential to a given family of toroidal surfaces. We show that this question can be translated into the problem of determining a periodic solution with periodic derivatives of a two-dimensional linear elliptic second-order partial differential equation on each toroidal surface and prove the existence of smooth solutions. Examples of smooth solutions foliated by toroidal surfaces that are not invariant under continuous Euclidean isometries are also constructed explicitly, and they are identified as equilibria of anisotropic magnetohydrodynamics. The problem examined here represents a weaker version of a fundamental mathematical problem that arises in the context of magnetohydrodynamics and fluid mechanics concerning the existence of regular equilibrium magnetic fields and steady Euler flows in bounded domains without continuous Euclidean isometries. The existence of such configurations represents a key theoretical issue for the design of the confining magnetic field in nuclear fusion reactors known as stellarators.