The solution of Euler-Cauchy equation using Laplace transform

Ghil, Byungmoon; Kim, Hwajoon

doi:10.12988/ijma.2015.59225

Cited by 10 publications

(6 citation statements)

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“…Remark 1.1. It was mentioned in [5], that this observation justifies the proposal of Ghil and Kim given in [6], in which the inverse Laplace transform of C is given by t −1 , where C is a constant; although their choice C = −1 cannot be justified.…”

Section: Introductionsupporting

confidence: 61%

Solution of Euler’s Differential Equation and AC-Laplace Transform of Inverse Power Functions and Their Pseudofunctions, in Nonstandard Analysis

Morita

2021

JAMCS

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It is shown that the index law of the Riemann-Liouville fractional derivative is recovered when nonstandard analysis is applied, and then the solutions of Euler’s differential equation are obtained in nonstandard analysis, where infinitesimal number appears. They are given in the form, from which the solutions in distribution theory are obtained. In the derivation, the AC-Laplace transforms of functions tν and tν(loge t) m for complex number ν and positive integer m, are used. By using these formulas, the AC-Laplace transforms of functions t− n + andt− n +(loge t) m for positive integers n and m, and their pseudofunctions are obtained with the aid of nonstandard analysis.

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Section: Introductionsupporting

confidence: 61%

Solution of Euler’s Differential Equation and AC-Laplace Transform of Inverse Power Functions and Their Pseudofunctions, in Nonstandard Analysis

Morita

2021

JAMCS

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“…Remark 13. In Reference [22], Ghil and Kim adopt that the inverse Laplace transform of s n−1 for n ∈ Z >0 gives Ct −n , which is justified by the present study, where we obtain solution Ct −n and the Laplace transform s n−1 for α k = −n, where C is a constant, with the aid of nonstandard analysis.…”

Section: Euler's Equation Studied In Nonstandard Analysismentioning

confidence: 53%

Solution of Euler’s Differential Equation in Terms of Distribution Theory and Fractional Calculus

Morita

Sato²

2020

Mathematics

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For Euler’s differential equation of order n, a theorem is presented to give n solutions, by modifying a theorem given in a recent paper of the present authors in J. Adv. Math. Comput. Sci. 2018; 28(3): 1–15, and then the corresponding theorem in distribution theory is given. The latter theorem is compared with recent studies on Euler’s differential equation in distribution theory. A supplementary argument is provided on the solutions expressed by nonregular distributions, on the basis of nonstandard analysis and Laplace transform.

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“…In this section, we introduce the concept of one-dimensional differential transform and review some basic fundamental theorems [8][9][10][11][12][13][14][15][16][17]. We assume that the function f(x) ∈ C ∞ (I), and x 0 be any point of I.…”

Section: The Differential Transform Methodsmentioning

confidence: 99%

“…Note that, the inverse differential transform of F(k) is defined by [1][2][3][4][5][6][7][8][9][10][11][13][14][15][16][17]). Let f(x) and g(x) be analytic functions, with differential transforms F(k) and G(k), respectively, then for σ and β:…”

Section: The Differential Transform Methodsmentioning

confidence: 99%

“…There are more than one method to solve the Euler-Cauchy equation such as the Laplace transform, the variation of parameters method, and the method of reduction of the order. Kim [13] applied the Laplace transform to find the solution of a homogeneous Euler-Cauchy equation of the second order ODE, Abualrab [1], found a solution of a special case of nonhomogeneous Euler-Cauchy equation using the variation of parameters, and Haarsa and Pothat [14], found the solution of a second order Euler-Cauchy equation with bulge function using the reduction of the order method. The differential transform method presented by Pukhov [12] in 1982 and the concept of differential transform proposed first by Zhou [23] in 1986 when applied to solve linear and nonlinear initial value problems in electric circuit analysis.…”

Section: Introductionmentioning

confidence: 99%

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The particular solutions of some types of Euler-Cauchy ODE using the differential transform method

Alesemi¹,

El-Moneam²,

Bader³

et al. 2018

J. Nonlinear Sci. Appl.

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In this paper, we apply the differential transform method to find the particular solutions of some types of Euler-Cauchy ordinary differential equations. The first model is a special case of the nonhomogeneous n th order ordinary differential equations of Euler-Cauchy equation. The second model under consideration in this paper is the nonhomogeneous second order differential equation of Euler-Cauchy equation with a bulge function. This study showed that this method is powerful and efficient in finding the particular solution for Euler-Cauchy ODE and capable of reducing the size of calculations comparing with other methods.

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The solution of Euler-Cauchy equation using Laplace transform

Cited by 10 publications

References 0 publications

Solution of Euler’s Differential Equation and AC-Laplace Transform of Inverse Power Functions and Their Pseudofunctions, in Nonstandard Analysis

Solution of Euler’s Differential Equation and AC-Laplace Transform of Inverse Power Functions and Their Pseudofunctions, in Nonstandard Analysis

Solution of Euler’s Differential Equation in Terms of Distribution Theory and Fractional Calculus

The particular solutions of some types of Euler-Cauchy ODE using the differential transform method

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