Order Linear Differential Equations Research Articles

Infinite Order of Growth of Solutions of Second Order Linear Differential Equations

We consider the di?erential equation f'' +A(z)f' +B(z)f = 0, where A(z) and B(z) are entire complex functions. We improve various restrictions on coe?cients A(z) and B(z) and prove that all non-trivial solutions are of in?nite order.

Growth of solutions of complex differential equations with entire coefficients having a multiply-connected Fatou component

In this study, we show that all non-trivial solutions of $$f''+A(z)f'+B(z)f=0$$ have infinite order, provided that the entire coefficient A(z) has certain restrictions and B(z) has multiply-connected Fatou component. We also extend these results to higher order linear differential equations.

Sawi transform and Hyers-Ulam stability of n t h order linear differential equations

The use of the Sawi transform has increased in the light of recent events in different approaches. The Sawi transform is also seen as the easiest and most effective way among the other transforms. In line with this, the research deals with the Hyers-Ulam stability of \(n^{\rm th}\) order differential equations using the Sawi transform. The study aims at deriving a generalised Hyers-Ulam stability result for linear homogeneous and non-homogeneous differential equations.

On the Growth of Solutions of Nonhomogeneous Higher order Complex Linear Differential Equations

The nonhomogeneous higher order linear complex differential equation (HOLCDE) with meromorphic (or entire) functions is considered in this paper. The results are obtained by putting some conditions on the coefficients to prove that the hyper order of any nonzero solution of this equation equals the order of one of its coefficients in case the coefficients are meromorphic functions. In this case, the conditions were put are that the lower order of one of the coefficients dominates the maximum of the convergence exponent of the zeros sequence of it, the lower order of both of the other coefficients and the nonhomogeneous part and that the solution has infinite order. Whiles in case the coefficients are entire functions, any nonzero solution with finite order has hyper order equals to the lower order of one of its coefficients is proved. In this case, the condition that the lower order of one of the coefficients is greater than the maximum of the lower order of the other coefficients and the lower order of the nonhomogeneous part is assumed.

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.

Study of Complex Oscillation of Solutions of a Second Order Linear Differential Equation With Entire Coefficients of (α,β, γ )-Order

In this paper, we deal with the complex oscillation of solutions of linear differential equation. We mainly study the interaction between the growth, zeros of solutions with the coefficients of second order linear differential equations in terms of ( α, β, γ )-order and obtain some results in general form which considerably extend some results of [5], [18] and [21].

Determining a Particular Solution for n-th Order Linear Differential Equations with Constant Coefficients that Have the Free Term in the Following form P(x). ln x and the Characteristic Equation Has Distinct Real Roots

Abstract For n-th order linear differential equations with constant coefficients the problem of determining the general solution involves determining the general solution of the attached homogeneous equation and, on the other hand, determining a particular solution. This paper aims to analyze the conditions under which the general solution can be obtained for a n-th order linear differential equation with constant coefficients which have the free term in the following form P(x). ln x under the condition that the attached characteristic equation has real and distinct roots, meaning the order of multiplicity for each root is 1. For this we will use the method of variation of constants which leads us to a system whose solution we will gain by using the Upper incomplete Gamma function.

Determining a Particular Solution for n-th Order Linear Differential Equations with Constant Coefficients that Have the Free Term in the Following form P(x). ln x and the Characteristic Equation Has a Real Root of Order of Multiplicity n

Abstract For n-th order linear differential equations with constant coefficients the problem of determining the general solution involves determining the general solution of the attached homogeneous equation and, on the other hand, determining a particular solution. This paper aims to analyze the conditions under which the general solution can be obtained for a n-th order linear differential equation with constant coefficients which have the free term in the following form P(x). ln xunder the condition that the attached characteristic equation hassingle real root of order of multiplicity n. For this we will use the method of variation of constants which leads us to a system whose solution we will gain using the Upper incomplete Gamma function.

Cost Analysis of Solar Water Pumping System for Small Town Water Supply

This study investigates the economics of a solar water pumping system for usage in small towns. The system consists of two solar panels, an inverter, a water pumping mechanism, and two parallel tanks. Using the transition diagram, a system of first order linear ordinary differential equations is generated and used to generate a statement of system availability and profit. To validate the expressions of system availability and profit, numerical experiments in the MATLAB software in the form of tables and graphs are utilized. According to the findings, the system's profitability increases as the value of repair rates across all subsystems increases, while it decreases when the value of failure rates across all subsystems increases.

Exponential stability for a system of second and first order delay differential equations

Exponential stability of the second order linear delay differential equation in x and u-control ẍ(t)+a1(t)ẋ(h1(t))+a2(t)x(h2(t))+a3(t)u(h3(t))=0is studied, where indirect feedback control u̇(t)+b1(t)u(g1(t))+b2(t)x(g2(t))=0 connects u with the solution. Explicit sufficient conditions guarantee that both x and u decay exponentially.

Approximate solution of the special type differential equation of higher order using Taylor's series

We study the approximate solution of the special type \(n^{th}\) order linear differential equation by applying initial and boundary conditions using Taylor's series formula. That is, we prove the sufficient condition for the Mittag-Leffler-Hyers-Ulam stability and Mittag-Leffler-Hyers-Ulam-Rassias stability of the special type linear differential equation of higher order with initial and boundary conditions using Taylor's series formula.

English

The main purpose of this work is to obtain and describe the solutions of the non-homogeneous first order linear singular differential equation of the following type: with two real numbers, and in the space of generalized functions We look for the solution of this equation in the form of with the unknown coefficients to be determined. We also obtain and analyze the classical solutions of the homogeneous equation.

Generalized osculating curves of type (n-3) in the n-dimensional Euclidean space

In this paper, we give a generalization of the osculating curves to the $n$-dimensional Euclidean space. Based on the definition of an osculating curve in the 3 and 4 dimensional Euclidean spaces, a new type of osculating curve has been defined such that the curve is independent of the ( n − 3 ) (n−3) th binormal vector in the n-dimensional Euclidean space, which has been called ”a generalized osculating curve of type ( n − 3 ) (n−3) ”. We find the relationship between the curvatures for any unit speed curve to be congruent to this osculating curve in E n En . In particular, we characterize the osculating curves in E n En in terms of their curvature functions. Finally, we show that the ratio of the ( n − 1 ) (n−1) th and ( n − 2 ) (n−2) th curvatures of the osculating curve is the solution of an ( n − 2 ) (n−2) th order linear nonhomogeneous differential equation.

On a differential intermediate value property

Liouville closed $H$-fields are ordered differential fields whose ordering and derivation interact in a natural way and where every linear differential equation of order $1$ has a nontrivial solution. (The introduction gives a precise definition.) For a Liouville closed $H$-field $K$ with small derivation we show: $K$ has the Intermediate Value Property for differential polynomials iff $K$ is elementarily equivalent to the ordered differential field of transseries. We also indicate how this applies to Hardy fields.

A short note on the Painlevé XXV–Ermakov equation

Solutions of the second member of the Riccati chain and of the corresponding third order linear differential equation are related to solutions of the so-called Painlevé XXV–Ermakov equation via the Schwarzian derivative. The reduction to the generalised Ermakov equation is shown to arise naturally from the Painlevé XXV–Ermakov equation. Specifically, the first order system of ordinary differential equations, equivalent to the Painlevé XXV–Ermakov equation, is analysed by resolving points of indeterminacy of the vector field over P1×P1.

Soft variable structure control of linear fractional-order systems with actuators saturation

In this paper, a new method of soft variable structure control for Fractional-Order System (FOS) is proposed to achieve faster response while the control signal is continuous and satisfies actuators’ constraints. Our proposed method, by describing a desired commensurate FOS, with the help of a pole placement algorithm and applying an optimization routine, in the form of a procedure, leads the system to the desired character. The routine is done by solving an optimization problem subject to a control signal constraint qua we obtain the fastest response possible in the sense of stability region. The sufficient condition of stability of the control system is developed based on the stability theory of Fractional Order (FO) linear differential equations, and attributes of the Mittag-Leffler function. Finally, an example and corresponding numerical simulations are presented to show the efficiency of the proposed method. In the proposed method, a new control strategy for the FOS is presented, a complex problem is solved in a simple way, and it can exploit the benefits of using FOS in the modeling and control of complex physical phenomena.

Conditions for the Boundness of the Solution for Second Order Linear Delay Differential Equations

It is well known that a solution to a second order differential equation with roots of its characteristic equation in the right half plane is unbounded. If a delay of the right magnitude is introduced into the equation the solution is bounded. This paper investigates this phenomenon by imbedding a second order differential equation with first order delay by embedding the equation in a field of equivalent classes of quotients of polynomials in the modified differential and delay operator over the real field.

On first and second order linear Stieltjes differential equations

This work deals with the obtaining of solutions of first and second order Stieltjes differential equations. We define the notion of Stieltjes derivative on the whole domain of the functions involved, provide a notion of n-times continuously Stieltjes-differentiable functions and prove existence and uniqueness results of Stieltjes differential equations in the space of such functions. We also present the Green's functions associated to the different problems and an application to the Stieltjes harmonic oscillator.

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.

Oscillation of Solutions of LDE\u2019s in Domains Conformally Equivalent to Unit Disc

Oscillation of solutions of f^{(k)} + a_{k-2} f^{(k-2)} + cdots + a_1 f' +a_0 f = 0 is studied in domains conformally equivalent to the unit disc. The results are applied, for example, to Stolz angles, horodiscs, sectors, and strips. The method relies on a new conformal transformation of higher order linear differential equations. Information on the existence of zero-free solution bases is also obtained.