Order Linear Differential Equations Research Articles

Exact solvability of certain linear ODEs

AbstractWe study the existence of exact solutions either for linear systems of ODEs or for scalar second order linear differential equations with Dirichlet boundary value conditions.

On infinite spectra of oscillation exponents of third order linear differential equations

The research topic of this work is at the junction of the theory of Lyapunov exponents and oscillation theory. In this paper, we study the spectra (i.e., the sets of different values on nonzero solutions) of the exponents of oscillation of signs (strict and nonstrict), zeros, roots, and hyperroots of linear homogeneous differential equations with coefficients continuous on the positive semi-axis. In the first part of the paper, we build a third order linear differential equation with the following property: the spectra of all upper and lower strong and weak exponents of oscillation of strict and non-strict signs, zeros, roots and hyper roots contain a countable set of different essential values, both metrically and topologically. Moreover, all these values are implemented on the same sequence of solutions of the constructed equation, that is, for each solution from this sequence, all of the oscillation exponents coincide with each other. In the construction of the indicated equation and in the proof of the required results, we used analytical methods of the qualitative theory of differential equations and methods from the theory of perturbations of solutions of linear differential equations, in particular, the author’s technique for controlling the fundamental system of solutions of such equations in one special case. In the second part of the paper, the existence of a third order linear differential equation with continuum spectra of the oscillation exponents is established, wherein the spectra of all oscillation exponents fill the same segment of the number axis with predetermined arbitrary positive incommensurable ends. It turned out that for each solution of the constructed differential equation, all of the oscillation exponents coincide with each other. The obtained results are theoretical in nature, they expand our understanding of the possible spectra of oscillation exponents of linear homogeneous differential equations.

Existence of basic solutions of first order linear homogeneous set-valued differential equations

The paper presents various derivatives of set-valued mappings,their main properties and how they are related to each other.Next, we consider Cauchy problems with linear homogeneousset-valued differential equations with different types ofderivatives (Hukuhara derivative, PS-derivative andBG-derivative). It is known that such initial value problems withPS-derivative and BG-derivative have infinitely many solutions.Two of these solutions are called basic. These are solutions suchthat the diameter function of the solution section is amonotonically increasing (the first basic solution) or monotonicallydecreasing (the second basic solution) function. However, the secondbasic solution does not always exist. We provideconditions for the existence of basic solutions of such initialvalue problems. It is shown that their existence depends on thetype of derivative, the matrix of coefficients on the right-handand the type of the initial set. Model examples are considered.

Formulation of physical fields from systems of first order linear partial differential equations

We present a heuristic approach to formulate physical fields from systems of first order linear partial differential equations by considering the symmetric and antisymmetric properties of the matrices that represent the systems of equations. Particularly, we show that Maxwell and Dirac fields can be established by imposing different types of symmetric and antisymmetric conditions on their associated representing matrices.

Existence and linear independence theorem for linear fractional differential equations with constant coefficients

Abstract We consider the l-th order linear fractional differential equations with constant coefficients. Here l ∈ ℕ {l\in\mathbb{N}} is the ceiling for the highest derivative of order α, l - 1 < α ≤ l {l-1<\alpha\leq l} . If β i < α {\beta_{i}<\alpha} are the other derivatives, the existing theory requires α - max ⁡ { β i } ≥ l - 1 {\alpha-\max\{\beta_{i}\}\geq l-1} for the existence of l linearly independent solutions. Thus, at most one derivative may have order greater than one, but all other derivatives must be between zero and one. We remove this essential restriction and construct l linearly independent solutions. With this aim, we remodel the series approaches and elaborate the multi-sum fractional series method in order to obtain the existence and linear independence results. We consider both Riemann–Liouville or Caputo fractional derivatives.

B-spline method for second order RLC closed series circuit with small inductance value

In this article, we investigates the application of RLC closed series electric circuits. We form the second order linear differential equations for an electric circuit depends upon the Kirchhoffs Voltage laws. A circuit containing an inductance L or capacitor C and resistor R with current and voltage variable given by boundary value problem. Then we use the cubic B-spline functions on graded mesh to solve second order linear boundary value problems. This method gives the second ordered convergent. Numerical results (charge q) for different cases of L are presented and compared with exact solutions in the table form.

Frobenius methods for analytic second order linear partial differential equations

The main subject of this text is the study of analytic second order linear partial differential equations. We aim to solve the classical equations and some more, in the real or complex analytical case. This is done by introducing methods inspired by the Frobenius method for second order linear ordinary differential equations. We introduce a notion of Euler type partial differential equation. To such a PDE we associate an indicial conic, which is an affine plane curve of degree two. Then comes the concept of regular singularity and finally convergence theorems, which must necessarily take into account the type of PDE (parabolic, elliptical or hyperbolic) and a nonresonance condition. This condition gives a new geometric interpretation of the original condition between the roots of the original Frobenius theorem for second order ODEs. The interpretation is something like, a certain reticulate has or not vertices on the indexical conic. Finally, we retrieve the solution of all the classical PDEs by this method (heat diffusion, wave propagation and Laplace equation), and also increase the class of those that have explicit algorithmic solution to far beyond those admitting separable variables. The last part of the text is dedicated to the construction of PDE models for the classical ODEs like Airy, Legendre, Laguerre, Hermite and Chebyshev by two different means. One model is based on the requirement that the restriction of the PDE to lines through the origin must be the classical ODE model. The second is based on the idea of having symmetries on the PDE model and imitating the ODE model. We study these PDEs and obtain their solutions, obtaining for the framework of PDEs some of the classical results, like existence of polynomial solutions (Laguerre, Hermite and Chebyshev polynomials).

Genericity of trivial Lyapunov spectrum for Lp-cocycles derived from second order linear homogeneous differential equations

We consider a probability space M on which an ergodic flow φt:M→M is defined. We study a family of continuous-time linear cocycles, referred to as kinetic, that are associated with solutions of the second-order linear homogeneous differential equation x¨+α(φt(ω))x˙+β(φt(ω))x=0. Here, the parameters α and β evolve along the φt-orbit of ω∈M. Our main result states that for a generic subset of kinetic continuous-time linear cocycles, where generic means a Baire second category with respect to an Lp-like topology on the infinitesimal generator, the Lyapunov spectrum is trivial.

A novel analytical model for fiber reinforced cementitious matrix FRCM coupons subjected to tensile tests

A much enhanced -with respect to existing literature- analytical model for FRCM coupons subjected to standard tensile tests is proposed. One fourth of the coupon is idealized considering two layers, matrix and fiber, subjected to a uniaxial stress state, mutually interacting at the interface through tangential stresses. The matrix is assumed elastic-perfectly fragile and the fiber linear elastic. The shear stress-slip relationship at the interface is tri-linear, with the first phase elastic, the second exhibiting linear softening followed by a third phase at possible non-null residual strength. The longitudinal equilibrium equations written for the two layers, suitably re-arranged, allow deducing a field problem governed in the three phases by a single second order linear differential equation where the independent variable is the interface slip, with solution retrievable analytically. Since the location of the points where the interface is in the different phases is not a-priori known, a discretization with small length elements is adopted. For each element the solution of the field problem is known in closed form and the only variables to determine are the integration constants coming from the solution of the differential equation. After a standard assemblage, all constants are derived imposing the boundary conditions at the extremes of the elements, which depend on the state of cracking of the matrix layer. The model is validated against two sets of experimental data on coupons tested in two different University laboratories in Italy. A satisfactory predictivity of both the global and local behavior is found.

Charged dust in Einstein–Gauss–Bonnet models

We investigate the influence of the higher order curvature terms on the static configuration of a charged dust distribution in EGB gravity. The EGB field equations for such a fluid are generated in higher dimensions. The governing equation can be written as an Abel differential equation of the second kind, or a second order linear differential equation. Exact solutions are found to these equations in terms of special functions, series and polynomials. The Abel differential equation of the second kind is reducible to a canonical differential equation; three new families of solutions are found by constraining the coefficients of the canonical equation. The charged dust model is shown to be physically well behaved in a region at the centre, and dust spheres can be generated. The higher order curvature terms influence the dynamics of charged dust and the gravitational behaviour which is distinct from general relativity.

Developing the Third Order Linear and Nonlinear Fuzzy Ordinary Differential Equations in Hilbert Space by the Extension Principle.

In this study, a third nonlinear fuzzy ordinary differential equation is defined and established in Hilbert space using extension principle. The results obtained gives clear distinction between this work and the existing ones in the literature an example is formulated which shows that a closed subset of a Hilbert space is a Hilbert Space. It is recommended that future study should consider the development of system of third order linear and nonlinear fuzzy ordinary differential equations in Hilbert space by the extension principle.

Algorithmic realization of exact three‐point difference scheme for singular Sturm–Liouville problem

AbstractA new algorithmic realization of exact three‐point difference schemes for a class of singular Sturm–Liouville problems is presented. It is shown that to compute the coefficients of the exact scheme in an arbitrary grid node, it is necessary to solve two auxiliary initial value problems for the second order linear ordinary differential equations: one problem on the interval (forward) and one problem on the interval (backward). Finally, the coefficient stability of the exact three‐point difference scheme is proved.

System of Caputo Fractional Differential Equations with Applications to Predator and Prey Model

In this research article, we provide a methodology to solve the three systems of qth order linear Caputo fractional differential equations, where 0 < q < 1. Since the Caputo derivative is in the convolution form, we can apply the Laplace transform technique. The solution of the two linear system can be used as a tool to study the stability of the equilibrium solution of the Lotka-Volterra predator-prey model. We have referenced three system SIR model in this work. Due to the global nature of the Caputo derivative, the solution obtained is closer to the real data than the integer derivative.

The special solutions of two-dimensional drift-flux equations for the two-phase flow

This paper considers the special solutions of two-dimensional isentropic drift-flux equations for the two-phase flow. The nonlinear system is transformed into a second order linear partial differential equation by the hodograph and the polar coordinate transformations. Specific solutions (circulatory flow, radial flow, and spiral flow) are obtained for steady and irrotational systems. Moreover, the special solution of radial flow for the pseudo-steady two-phase flow system is obtained.

Second order linear differential equations with a basis of solutions having only real zeros

AbstractLet A be a transcendental entire function of finite order. We show that, if the differential equation w″ + Aw = 0 has two linearly independent solutions with only real zeros, then the order of A must be an odd integer or one half of an odd integer. Moreover, A has completely regular growth in the sense of Levin and Pfluger. These results follow from a more general geometric theorem, which classifies symmetric local homeomorphisms from the plane to the sphere for which all zeros and poles lie on the real axis, and which have only finitely many singularities over finite non-zero values.

Ulam Type Stabilities of n-th Order Linear Differential Equations Using Gronwall’s Inequality

Ulam Type Stabilities of n-th Order Linear Differential Equations Using Gronwall’s Inequality

Oscillation of Odd Order Linear Differential Equations with Deviating Arguments with Dominating Delay Part

ABSTRACT In this paper new oscillatory criteria for odd order linear functional differential equations of the type y ( n ) ( t ) + p ( t ) y ( τ ( t ) ) = 0 \[{{y}^{(n)}}(t)+p(t)y(\tau (t))=0\] have been established. Deviating argument τ(t) is supposed to have dominating delay part.

On a first order linear singular differential equation in the space K’

We propose in this work to describe all the generalized-function solutions of the non-homogeneous first-order linear singular differential equation with A, B two real numbers, s and p∈ N, n ≥ 1， q∈Z+, in the space of generalized functions K’. In the case of a second right-hand side consisting of an s-order derivative of the Dirac-delta function, we have completely investigated the considered equation when we look for the solution in the form of y(x)=∑k=0NCkδ(k)(x), with the unknown coefficients Ck which we have determined case by case, taking into account the relationship between the parameters inside. On the basis of what has been done, we focus our present research on applying the principle of superposition of the solutions that is conducting us to the awaited result when we also maintain the classical solutions of the homogeneous equation which remains the same.

On the General Solution of 2th Order Linear Differential Equation

We employ a method of factorization to obtain the general solution of the second order linear differential equation, which is an alternative procedure to the usual Variation of Parameters method of Lagrange. We consider that our approach can be adapted to linear differential equations of the third and fourth order.

Phase function methods for second order linear ordinary differential equations with turning points

Phase function methods for second order linear ordinary differential equations with turning points