The linear differential equations with complex constant coefficients and Schrödinger equations

Jung, Soon-Mo; Roh, Jaiok

doi:10.1016/j.aml.2016.11.003

Cited by 8 publications

(9 citation statements)

References 11 publications

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“…Finally, compared to the results in [14][15][16] we used very classical integral method and have better error estimation than theirs. While we can find an approximate solution generally with very small error near c for any desired point c ∈ R, in their estimates this is only possible in special cases.…”

Section: Discussionmentioning

confidence: 99%

“…Since α 2 − 4β < 0, we have u 1 (x) = e λx cos µx and u 2 (x) = e λx sin µx. By the definition (14) of n(x), we know that…”

Section: Approximation Propertiesmentioning

confidence: 99%

“…is called the homogeneous differential equation corresponding to (4). With various additional conditions, the Hyers-Ulam stability of Equation (4) was proved in [12][13][14][15][16][17]. In this paper, with weaker conditions, we will study the Hyers-Ulam stability of Equation (4) in three different cases with respect to the constant coefficients.…”

Section: Introductionmentioning

confidence: 99%

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Some Properties of Approximate Solutions of Linear Differential Equations

2019

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In this paper, we will consider the Hyers-Ulam stability for the second order inhomogeneous linear differential equation, u (x) + αu (x) + βu(x) = r(x), with constant coefficients. More precisely, we study the properties of the approximate solutions of the above differential equation in the class of twice continuously differentiable functions with suitable conditions and compare them with the solutions of the homogeneous differential equation u (x) + αu (x) + βu(x) = 0. Several mathematicians have studied the approximate solutions of such differential equation and they obtained good results. In this paper, we use the classical integral method, via the Wronskian, to establish the stability of the second order inhomogeneous linear differential equation with constant coefficients and we will compare our result with previous ones. Specially, for any desired point c ∈ R we can have a good approximate solution near c with very small error estimation.

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Section: Discussionmentioning

confidence: 99%

“…Since α 2 − 4β < 0, we have u 1 (x) = e λx cos µx and u 2 (x) = e λx sin µx. By the definition (14) of n(x), we know that…”

Section: Approximation Propertiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Some Properties of Approximate Solutions of Linear Differential Equations

2019

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“…irty years ago, in the community of DEs, systems of DEs and integrable dynamical systems, Prelle and Singer [8] had made a particularly powerful approach for solving a kind of first-order nonlinear ODEs in the real line, and it obtains a solution which consists of elementary functions on the real line. In the present work, the authors are moving a step forward to generalize one particular method, namely, Prelle-Singer method, and present a related procedure on the complex plane; the authors have generalized a class of nonlinear ODEs [9], in which it has a real interval of definition. en, they construed a class of (NLOCDE (Nonlinear Ordinary Complex Differential Equations (NLOCDE, where the general solution to the mentioned))) is an algebraic combination of complex elementary functions that are analytic on a particular region in the complex plane.…”

Section: Introductionmentioning

confidence: 99%

An Investigation of Solving Third-Order Nonlinear Ordinary Differential Equation in Complex Domain by Generalising Prelle–Singer Method

Joohy

Al-Juaifri

Mechee

2020

International Journal of Differential Equations

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A method to solve a family of third-order nonlinear ordinary complex differential equations (NLOCDEs) —nonlinear ODEs in the complex plane—by generalizing Prelle–Singer has been developed. The approach that the authors generalized is a procedure of obtaining a solution to a kind of second-order nonlinear ODEs in the real line. Some theoretical work has been illustrated and applied to several examples. Also, an extended technique of generating second and third motion integrals in the complex domain has been introduced, which is conceptually an analog to the motion in the real line. Moreover, the procedures of the method mentioned above have been verified.

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“…Since then, Hyers-Ulam stability and Hyers-Ulam-Rassias stability of various classes of differential equations and differential operators were explored by using a wide spectrum of approaches; see [2,4,5,9,10,12,14,16,[24][25][26][27] and the references cited there.…”

Section: Introductionmentioning

confidence: 99%