Higher Order Linear Differential Equations Research Articles

Effective Method for Solving Higher-Order Fractional Differential Equations Using Spline Functions

In this paper, appropriate spline functions in polynomial form are outlined and applied to solve higher-order linear fractional differential equations (H-OLFDEs). Several techniques have been proposed to solve this type of equations. A description of the projected methodology is first introduced. The method involves transforming the H-OLFDE of order α, with given initial conditions, into a system of fractional differential equations with , denoting the order of the Caputo fractional derivative for each equation i in the transformed system, where the number of equations in the resulting system is equal to that of initial conditions. The numerical results obtained using this strategy demonstrate substantial agreement with the precise analytical solutions. The results in this paper affirm the robustness and ease of application of the proposed technique.

Special function models of indecomposable sl(2) representations: the Laguerre case

In this paper, we point out connections between certain types of indecomposable representations of sl(2) and generalizations of well-known orthogonal polynomials. Those representations take the form of infinite dimensional chains of weight or generalised weight spaces, for which the Cartan generator acts in a diagonal way or via Jordan blocks. The other generators of the Lie algebras sl(2) act as raising and lowering operators but are now allowed to relate the different chains as well. In addition, we construct generating functions, we calculate the action of the Casimir invariant and present relations to systems of non-homogeneous second-order coupled differential equations. We present different properties as higher-order linear differential equations for building blocks taking the form of one variable polynomials. We also present insight into the zeroes and recurrence relations.

Growth and value distribution of linear difference polynomials generated by meromorphic solutions of higher-order linear difference equations

In this article, we investigate the relationship between growth and value distribution of meromorphic solutions for the higher-order complex linear difference equations $$ A_n(z)f(z+n)+\dots+A_1(z)f(z+1)+A_0(z)f(z)=0 \quad \text{and } =F(z), $$ and for the linear difference polynomial $$ g(z)=\alpha_n(z)f(z+n)+\dots+\alpha_1(z)f(z+1)+\alpha_0(z)f(z) $$ generated by \(f(z)\) where \(A_j(z)\), \(\alpha_j(z)\) (\(j=0,1,\ldots,n\)), \(F(z)\) \((\not\equiv0)\) are meromorphic functions. We improve some previous results due to Belaidi, Chen and Zheng and others.
 For more information see https://ejde.math.txstate.edu/Volumes/2023/84/abstr.html

The Elzaki transform and its applications for Hyers -Ulam stability of higher order differential equations using Covid 19 method

The Elzaki transform is being used to present the generic solution and prove the Hyers-Ulam stability of higher-order linear differential equations. In addition, the stability of homogeneous and non-homogeneous linear differential equations is discussed using the Mittag-Leffler-Hyers-Ulam method. Furthermore, such equations’ stability constants are determined.

Stability of forced higher-order continuous-time Lur'e systems: a behavioural input-output perspective

We consider a class of forced continuous-time Lur'e systems obtained by applying nonlinear feedback to a higher-order linear differential equation which defines an input-output system in the sense of behavioural systems theory. This linear system directly relates the input and output signals and does not involve any internal, latent or state variables. A stability theory subsuming results of circle criterion type is developed, including criteria for input-to-output stability and strong integral input-to-output stability, concepts which are very much reminiscent of input-to-state stability and strong integral input-to-state stability, respectively. The methods used in the paper combine ideas from the behavioural approach to systems and control, absolute stability theory and input-to-state stability theory.

Applications of the Tarig Transform and Hyers–Ulam Stability to Linear Differential Equations

In this manuscript, we discuss the Tarig transform for homogeneous and non-homogeneous linear differential equations. Using this Tarig integral transform, we resolve higher-order linear differential equations, and we produce the conditions required for Hyers–Ulam stability. This is the first attempt to use the Tarig transform to show that linear and nonlinear differential equations are stable. This study also demonstrates that the Tarig transform method is more effective for analyzing the stability issue for differential equations with constant coefficients. A discussion of applications follows, to illustrate our approach. This research also presents a novel approach to studying the stability of differential equations. Furthermore, this study demonstrates that Tarig transform analysis is more practical for examining stability issues in linear differential equations with constant coefficients. In addition, we examine some applications of linear, nonlinear, and fractional differential equations, by using the Tarig integral transform.

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.

Growth properties of solutions of complex differential equations with entire coefficients of finite (alpha,beta,gamma)-order

In this article, we investigate the complex higher order linear differential equations in which the coefficients are entire functions of (α, β, γ)-order and obtain some results which improve and generalize some previous results of Tu et al. [29] as well as Belaidi [1, 2, 3].

Solving System of Higher-Order Linear Differential Equations On The Level of Operators

In this paper, I present a method for solving the system of higher-order linear differential equations (HLDEs) with inhomogeneous initial conditions on the level of operators. Using proposed method, we compute the matrix Green’s operator as well as the vector Green’s function of a fully-inhomogeneous initial value problems (IVPs). Examples are discussed to demonstrate the proposed method.

Complex Oscillation of Solutions and Their Arbitrary-Order Derivatives of Linear Differential Equations With Analytic Coefficients of [p, q]-Order in the Unit Disc

Throughout this article, we investigate the growth and fixed points of solutions of complex higher order linear differential equations in which the coefficients are analytic functions of [p, q]−order in the unit disc. This work improves some results of Belaïdi [3–5], which is a generalization of recent results from Chen et al. [9].

Shehu Integral Transform and Hyers-Ulam Stability of [formula omitted] order Linear Differential Equations

In this paper, we establish the Shehu transform expression for homogeneous and non-homogeneous linear differential equations. With the help of this new integral transform, we solve higher order linear differential equations in the Shehu sense. Moreover, this paper introduced a new concept to find the Hyers-Ulam stability of the differential equation. This is the first attempt to use the Shehu transform to prove the Hyers-Ulam stability of the differential equation. Finally, the applications and remarks are discussed to demonstrate our strategy. Applications of the Shehu transform to fractional differential equations, Newton’s law of cooling, and free undamped motion are also discussed.

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.

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.

Growth and fixed points of solutions and their arbitrary-order derivatives of higher-order linear differential equations in the unit disc

In this paper, we investigate the growth and fixed points of solutions of higher-order linear differential equations in the unit disc. We extend the coefficient conditions to a type of one-constant-control coefficient comparison and obtain the same estimates of iterated order of solutions. We also obtain better estimates by providing a precise value of iterated order of solution instead of a range of that in the case of coefficient characteristic function comparison. Moreover, we utilize iteration to investigate and estimate the fixed points of solutions’ arbitrary-order derivatives with higher-order equations f^{(k)}+A_{k-1}(z)f^{(k-1)}+{cdots }+A_{1}(z)f'+A_{0}(z)f=0 and provide a concise method to judge if the items generated by the iteration do not vanish identically and ensure the iteration proceeds. Our results are an improvement over those by B. Belaïdi, T. B. Cao, G. W. Zhang and A. Chen.

The hyper order and fixed points of solutions of a class of linear differential equations

In this paper, we precise the hyper order of solutions for a class of higher-order linear differential equations and investigate the exponents of convergence of the fixed points of solutions and their first derivatives for the second-order case. These results generalize those of Nan Li and Lianzhong Yang and of Chen and Shon.

Fast growing solutions of linear differential equations with analytic coefficients in the unit disc

In this paper, we investigate the growth of solutions of higher order linear differential equations with analytic coefficients of ϕ-order in the unit disc. We introduce new definitions of the lower order and the type related to the ϕ-order concepts to generalise and extend previous results due to Chyzhykov-Semochko [6], Semochko [14], Belaïdi [1,2,3], Hu-Zheng [12].

Numerical solutions of the randall-wilkins and one trap one recombination models for first order kinetic

Randall-Wilkins and One Trap One Recombination (otor) models have been proposed to explain thermoluminescence emission and it should be emphasized that each model has its own allowed charge carrier transitions, trapping parameters and differential equations set. The equations are generally first or higher order linear differential equations with constant coefficients and their numerical solutions are an initial value problem. From this point on, numerical solutions of the thermoluminescence equations have been effectively used. In this paper the models were solved, numerically by using Euler and Runge-Kutta methods on Mathematica 8.0. In this work, although the fastest result calculated by Explicit Euler method, the most accurate results were calculated Linearly Implicit Euler method.

Correctness conditions for high-order differential equations with unbounded coefficients

We give some sufficient conditions for the existence and uniqueness of the solution of a higher-order linear differential equation with unbounded coefficients in the Hilbert space. We obtain some estimates for the weighted norms of the solution and its derivatives. Using these estimates, we show the conditions for the compactness of some integral operators associated with the resolvent.

Study of the growth properties of meromorphic solutions of higher-order linear difference equations

In this paper, we investigate the growth of meromorphic solutions of homogeneous and non-homogeneous linear difference equations Ak(z)f(z+ck)+⋯+A1(z)f(z+c1)+A0(z)f(z)=0,Ak(z)f(z+ck)+⋯+A1(z)f(z+c1)+A0(z)f(z)=F,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} A_{k}(z)f(z+c_{k})+\\cdots +A_{1}(z)f(z+c_{1})+A_{0}(z)f(z)= & {} 0, \\\\ A_{k}(z)f(z+c_{k})+\\cdots +A_{1}(z)f(z+c_{1})+A_{0}(z)f(z)= & {} F, \\end{aligned}$$\\end{document}where A_{k}left( zright) ,ldots ,A_{0}left( zright) ,Fleft( zright) are meromorphic functions and c_{j}left( 1,ldots ,kright) are non-zero distinct complex numbers. Under some conditions on the coefficients, we extend early results due to Zhou and Zheng, Belaïdi and Benkarouba.

Numerical technique for higher-order differential equation with variable coefficients via Haar wavelet method

Lately, Haar wavelet has been established as a mathematical implement for several problems. In this article, the numerical method is founded on operational matrices and Haar wavelet collocation points on a new-found interval [0, Ω]. For a chosen domain from 0 to Ω, we have established the method of integration the Haar wavelets functions. The operational matrices of integration and product are used to change the higher order differential equations containing variable coefficients into system of algebraic equations that can be solved via MATLAB. Three numerical cases have be presented which including higher order linear differential equations involving variable coefficients. The numerical solution completely converge in comparison with accurate results.