Order Homogeneous Linear Differential Equations Research Articles

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.

Integrability by quadratures of the problem of rolling motion of a heavy homogeneous ball on a surface of revolution of the second order

In this paper, we consider the problem of rolling of a heavy homogeneous ball on a perfectly rough surface of revolution. Usually, when considering this problem, it is convenient to specify explicitly the surface along which the center of the ball moves during rolling, instead of the surface along which the ball rolls. The surface on which the center of the ball moves is equidistant to the surface on which the ball is rolling. It is well known, that the considered problem is reduced to the integration the second order linear homogeneous differential equation. In this paper we assume, that the surface along which the center of the ball moves is a non - degenerate surface of revolution of the second order. Using the Kovacic algorithm we prove that the general solution of the corresponding linear differential equation can be found explicitly. This means, that in this case the problem of rolling of a ball on a surface of revolution can be integrated by quadratures.

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.

Infinite Order of The Solutions of Higher Order Homogeneous Linear Differential Equations with Entire Coefficients Having Completely Regular Growth Property and Bounded Components Fatou Set

The homogeneous higher order complex linear differential equations (n-thCLDEs) with entire functions is considered in this paper. We investigated some conditions that can be put on the coefficients which guarantee that any nonzero solution of such equations has infinite order. The conditions we stated are the completely regular growth (CRG), the characteristic function of some coefficients is approximately equals to the logarithm of its maximum modulus and the Denjoy’s conjecture (DC) property.

Stieltjes analytic functions and higher order linear differential equations

In this work we develop a theory of Stieltjes-analytic functions. We first define the Stieltjes monomials and polynomials and we study them exhaustively. Then, we introduce the g-analytic functions locally, as an infinite series of these Stieltjes monomials and we study their properties in depth and how they relate to higher order Stieltjes differentiation. We define the exponential series and prove that it solves the first order linear problem. Finally, we apply the theory to solve higher order linear homogeneous Stieltjes differential equations with constant coefficients.

Analytical Proof of the Solution to Second Order Linear Homogeneous Differential Equation

Analytical Proof of the Solution to Second Order Linear Homogeneous Differential Equation

Effect of the Arbitrary Coefficients on the convergence of numerical solution of General Second Order Linear Homogeneous Partial Differential Equation

Effect of the Arbitrary Coefficients on the convergence of numerical solution of General Second Order Linear Homogeneous Partial Differential Equation

On the Growth of Solutions of Higher Order Linear Homogeneous and Non-homogeneous Differential Equations

In this paper, we discuss the growth of meromorphic solutions of some classes of homogeneous and nonhomogeneous differential equations. We will prove that if meromorphic functions F, A_j, D_j and polynomials P_j with degree nge 1(j=0,1,cdots ,k-1) satisfy some conditions, then the equation f^{(k)}+(A_{k-1}e^{P_{k-1}}+D_{k-1})f^{(k-1)}+cdots +(A_1e^{P_1}+D_1)f'+(A_0e^{P_0}+D_0)f=F(z) (kge 2) when Fequiv 0, all solutions fnot equiv 0 have infinite order and hyper order sigma _2(f)ge n. This is a continue work of [Gan and Sun in Adv. Math. 36(1), 51–60 (2007)] and [Chen and Xu in Electron. J. Qual. Theory Differ. Equ. 1: 1–13 (2009)]

Generalized Hukuhara Derivatives of Solving of Second Order Linear Homogeneous ODEs by Generalized Triangular Fuzzy Number Using Fuzzy Laplace Transform

Second order linear homogeneous ordinary differential equations are solved using fuzzy Laplace transform method with generalized Hukuhara differentiability concept. The results obtained is in this study is in the form of generalized triangular fuzzy number. Existence and uniqueness of solution are also obtained. However, based on the cases presented, the results have shown that relationship existed between the FLT of second order and its kth derivative for k ≥1.

Concept of gH Differentiability in Solving Second Order Linear Homogeneous ODEs Based on the Relation between FLT and Its kth Derivative for the Case in Fuzzy Environment

In this study, a fuzzy Laplace transform is used to solve second order linear homogeneous ordinary differential equations. The solution obtained is based on the concept of gH differentiability and the relation between the fuzzy Laplace transform and its derivative for is obtained. Examples are constructed for the existence and uniqueness of solutions of second order FODE.

PENGEMBANGAN WELCOME DRINK BERBAHAN DASAR TEMULAWAK (Curcuma xanthorrhiza roxb) BERBASIS PEMODELAN MATEMATIKA DAN EKSPERIMEN

Welcome drink is an added value that enhances the Indonesian hospitality in the hotel industry, especially if it is made with spices such as curcuma, cinnamon, lemon, and brown sugar however industries that make typical curcuma beverage are often found to use excessive sugar and tend to turn off the distinctive taste of curcuma, besides that it also has an adverse effect on health.This research is an applied research which aims to make a beverage from curcuma and cinnamon with a mathematical approach at the end of the making process, a mathematical model can be simulated by numerical computation so that information on sugar levels are normal and reasonable for consumption in the welcome drink that has been made. Result of the research obtained 1st order homogeneous linear differential equation and simulation results using 40-80 gram sugar obtained a constant rate of change in sugar substances of or equivalent to 9-120 brix

О коградиентности и контрградиентности линейных дифференциальных уравнений и систем

The Siegel-Shidlovskii method remains one of the basic methods in the theory of transcendental numbers. By using this method, it is possible to prove the transcendence and algebraic independence of the values of entire functions of a certain class (so-called E-functions). A necessary condition for applying this method is that all of the considered functions must constitute a solution of a system of linear differential equations and were algebraically independent over . The question about algebraic independence of the solutions of linear differential equations and systems of such equations is of great importance in differential algebra, analytical theory of differential equations, theory of special functions, and calculus (in the broad sense of the word). As is shown in papers by E. Kolchin, F. Beukers, W.D. Brownawell, and G. Heckman, this question boils down in many instances to verification of the cogredience and contragredience condition. Two systems of 1st order linear homogeneous differential equations with coefficients from are said to be cogredient (or, respectively, contragredient), if for arbitrary fundamental matrices and Ψ of these systems one of the equations Φ=gBΨC, Φ (ΨC)^*=gB, is fulfilled, where C∈GL(C), B∈GL(C(z)), g=g(z) is a function with the condition g^'/g∈C(z), and A^* is the matrix transposed to . The notions of cogredience and contragredience for linear homogeneous differential equations of arbitrary order are defined similarly. Another, more restricted definitions of cogredience and contragredience were in fact used in some papers of the author, devoted to generalized hypergeometric functions. According to these definitions, the function in the presented equalities is the product of a power function and an exponential function of some kind. The conditions for equivalence of these definitions are found.

Stability index of linear random dynamical systems

Given a homogeneous linear discrete or continuous dynamical system, its stability index is given by the dimension of the stable manifold of the zero solution. In particular, for the $n$ dimensional case, the zero solution is globally asymptotically stable if and only if this stability index is $n.$ Fixed $n,$ let $X$ be the random variable that assigns to each linear random dynamical system its stability index, and let $p_k$ with $k=0,1,\ldots,n,$ denote the probabilities that $P(X=k)$. In this paper we obtain either the exact values $p_k,$ or their estimations by combining the Monte Carlo method with a least square approach that uses some affine relations among the values $p_k,k=0,1,\ldots,n.$ The particular case of $n$-order homogeneous linear random differential or difference equations is also studied in detail.

Extraction of Some New Solutions of Linear Differential Equations by Dual Space

In this study, the solutions of second order linear homogeneous ordinary differential equations being variable and constant coefficients have been obtained by using power series method which is used frequently in dual space. The solutions of second order homogeneous ordinary differential equations being constant and variable coefficients in dual space by using power series have given and some applications have implemented.

Bank–Laine functions, the Liouville transformation and the Eremenko–Lyubich class

The Bank–Laine conjecture concerning the oscillation of solutions of second order homogeneous linear differential equations has recently been disproved by Bergweiler and Eremenko. It is shown here, however, that the conjecture is true if the set of finite critical and asymptotic values of the coefficient function is bounded. It is also shown that if E is a Bank–Laine function of finite order with infinitely many zeros, all real and positive, then its zeros must have exponent of convergence at least 3/2, and an example is constructed via quasiconformal surgery to demonstrate that this result is sharp.

On the invertibility of solutions of first order linear homogeneous differential equations in Banach algebras

This work is devoted to the study of some properties of linear homogeneous differential equations of the first order in Banach algebras. It is found (for some types of Banach algebras), at what right-hand side of such an equation, from the invertibility of the initial condition it follows the invertibility of its solution at any given time. Associative Banach algebras over the field of real or complex numbers are considered. The right parts of the studied equations have the form [F(t)](x(t)), where {F(t)} is a family of bounded operators on the algebra, continuous with respect to t∈R. The problem is to find all continuous families of bounded operators on algebra, preserving the invertibility of elements from it, for a given Banach algebra. In the proposed article, this problem is solved for only three cases. In the first case, the algebra consists of all square matrices of a given order. For this algebra, it is shown that all continuous families of operators, preserving the invertibility of elements from the algebra at zero must be of the form [F(t)](y)=a(t)⋅y+y⋅b(t), where the families {a(t)} and {b(t)} are also continuous. In the second case, the algebra consists of all continuous functions on the segment. For this case, it is shown that all families of operators, preserving the invertibility of elements from the algebra at any time must be of the form [F(t)](y)=a(t)⋅y, where the family {a(t)} is also continuous. The third case concerns those Banach algebras in which all nonzero elements are invertible. For example, the algebra of complex numbers and the algebra of quaternions have this property. In this case, any continuous families of bounded operators preserves the invertibility of the elements from the algebra at any time. The proposed study is in contact with the research of the foundations of quantum mechanics. The dynamics of quantum observables is described by the Heisenberg equation. The obtained results are an indirect argument in favor of the fact, that the known form of the Heisenberg equation is the only correct one.

On Liouvillian Solutions of Third Order Homogeneous Linear Differential Equations

In this article we will consider third order homogeneous differential equations:  L(y)=y'''+a_1y'+a_0y(a_0,a_1 ∈k) whose Galois group G（L） is imprimitive. This case is characterised by the fact that the third symmetric power equation L ^ⓢ3(y)=0 has an exponential solution whose square is rational (Singer & Ulmer 1993). If L(y)=0 has a Liouvillian solution z whose logarithmic derivative u=z'/z  is algebraic over a differential field (k,') ,we will give an algorithm to find the relation between a_0, a_1 , the semi-invariant S=Y_1Y_2Y_3 which is unique up to multiplication by a constant, the coefficients C_0, C_1 of the minimal polynomial P(u) of u  and their derivatives. The aim of this work is to diminutize the number of constants C_m  stated in the algorithm of Singer & Ulmer (Singer & Ulmer 1993 Algorithm p. 31) whose determination is not easy to do, and we will achieve this by using Groebner Basis.

Numerical simulation of the first order and second order linear homogeneous and non-homogeneous ordinary differential equation

Numerical simulation of the first order and second order linear homogeneous and non-homogeneous ordinary differential equation

Hydropower system operation stability considering the coupling effect of water potential energy in surge tank and power grid

This paper aims to study the hydropower system operation stability considering the coupling effect of water potential energy in surge tank and power grid. Firstly, the mathematical formulation for hydropower system with surge tank that connected to power grid is established. Then, the hydropower system operation stability is investigated. The coupling effect mechanism of surge tank and power grid on stability is revealed. Finally, the formula for critical stable sectional area (CSSA) of surge tank considering power grid is derived and analyzed. The results indicate that the free vibration equation for hydropower system with surge tank that connected to power grid under step load disturbance is a ninth order linear homogeneous differential equation. The domain at the bottom left corner of the critical line on the Proportional-Integral plane is the stable domain. The proposed novel analytic solution of stable domain has a high precision and is feasible for practical applications. The formula for CSSA of surge tank considering power grid realizes the unity of the coupling effect of water potential energy in surge tank and power grid. The CSSA contains four terms, i.e. headrace tunnel term, penstock term, governor term and power grid term.

Initial Value Problem of Second Order Intuitionistic Fuzzy Ordinary Differential Equations

In this research paper, different cases of solution of the second order linear Intuitionistic Fuzzy Ordinary Differential Equations (IFODEs) are discussed. The initial conditions and the coefficients of differential equations are taken as the Generalized Trapezoidal Intuitionistic Fuzzy Numbers (GTrIFNs). The concept for finding the solution of second order linear homogeneous intuitionistic fuzzy ordinary differential equations is discussed in detail.