scholarly journals The Generalized Solutions of the nth Order Cauchy–Euler Equation

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 932 ◽  
Author(s):  
Amornrat Sangsuwan ◽  
Kamsing Nonlaopon ◽  
Somsak Orankitjaroen ◽  
Ismail Mirumbe
Keyword(s):  
Laplace Transform ◽  
Euler Equation ◽  
Euler Equations ◽  
Generalized Solutions ◽  
Distributional Solutions ◽  
Laplace Transform Technique ◽  
The Laplace Transform

In this paper, we use the Laplace transform technique to examine the generalized solutions of the nth order Cauchy–Euler equations. By interpreting the equations in a distributional way, we found that whether their solution types are classical, weak or distributional solutions relies on the conditions of their coefficients. To illustrate our findings, some examples are exhibited.

scholarly journals Generalized Solutions of the Third-Order Cauchy-Euler Equation in the Space of Right-Sided Distributions via Laplace Transform

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 376 ◽  
Author(s):  
Seksan Jhanthanam ◽  
Kamsing Nonlaopon ◽  
Somsak Orankitjaroen
Keyword(s):  
Laplace Transform ◽  
Euler Equation ◽  
Weak Solutions ◽  
Southeast Asian ◽  
Generalized Solutions ◽  
Third Order ◽  
The Third ◽  
Distributional Solutions ◽  
Laplace Transform Technique ◽  
The Laplace Transform

Using the Laplace transform technique, we investigate the generalized solutions of the third-order Cauchy-Euler equation of the form t 3 y ′ ′ ′ ( t ) + a t 2 y ′ ′ ( t ) + b y ′ ( t ) + c y ( t ) = 0 , where a , b , and c ∈ Z and t ∈ R . We find that the types of solutions in the space of right-sided distributions, either distributional solutions or weak solutions, depend on the values of a, b, and c. At the end of the paper, we give some examples showing the types of solutions. Our work improves the result of Kananthai (Distribution solutions of the third order Euler equation. Southeast Asian Bull. Math. 1999, 23, 627–631).

The generalized solutions of a certain nth order Cauchy–Euler equation

2018 ◽  
Vol 13 (02) ◽  
pp. 2050047 ◽  
Author(s):  
Amornrat Sangsuwan ◽  
Kamsing Nonlaopon ◽  
Somsak Orankitjaroen
Keyword(s):  
Laplace Transform ◽  
Euler Equation ◽  
Weak Solutions ◽  
Generalized Solutions ◽  
Distributional Solutions ◽  
Laplace Transform Technique

In this paper, we propose the generalized solutions of a certain [Formula: see text]th order Cauchy–Euler equation of the form [Formula: see text], where [Formula: see text] and [Formula: see text] are any integers with [Formula: see text], and [Formula: see text] using Laplace transform technique. We find that the types of solutions either distributional solutions or weak solutions, depend on the values of [Formula: see text]. At the end of the paper, we give some examples showing the types of solutions.

Analysis of the neutron spin structure function g1n by using the Laplace transform technique

2018 ◽  
Vol 27 (08) ◽  
pp. 1850071
Author(s):  
F. Teimoury Azadbakht ◽  
G. R. Boroun ◽  
B. Rezaei
Keyword(s):  
Laplace Transform ◽  
Structure Function ◽  
Spin Structure ◽  
Neutron Spin ◽  
Spin Structure Function ◽  
Transform Method ◽  
Laplace Transform Technique ◽  
The Laplace Transform ◽  
Convolution Approach ◽  
Neutron Spin Structure

In this paper, the polarized neutron structure function [Formula: see text] in the [Formula: see text] nucleus is investigated and an analytical solution based on the Laplace transform method for [Formula: see text] is presented. It is shown that the neutron spin structure function can be extracted directly from the polarized nuclear structure function of [Formula: see text]. The nuclear corrections due to the Fermi motion of the nucleons as well as the binding energy considerations are taken into account within the framework of the convolution approach and the polarized structure function of [Formula: see text] nucleus is expressed in terms of the spin structure functions of nucleons and the light-cone momentum distribution of the constituent nucleons. Then, the numerical results for [Formula: see text] are compared with experimental data of the SMC and HERMES collaborations. We found that there is an overall good agreement between the theory and experiments.

Study of Transient Coupled Thermoelastic Problems With Relaxation Times

1995 ◽  
Vol 62 (1) ◽  
pp. 208-215 ◽  
Author(s):  
Han-Taw Chen ◽  
Hou-Jee Lin
Keyword(s):  
Boundary Condition ◽  
Laplace Transform ◽  
Temperature Stress ◽  
Control Volume ◽  
Relaxation Times ◽  
Dimensionless Temperature ◽  
Radiation Boundary Condition ◽  
Laplace Transform Technique ◽  
The Laplace Transform ◽  
Thermoelastic Problems

A new hybrid numerical method based on the Laplace transform and control volume methods is proposed to analyze transient coupled thermoelastic problems with relaxation times involving a nonlinear radiation boundary condition. The dynamic thermoelastic model of Green and Lindsay is selected for the present study. The following computational procedure is followed for the solution of the present problem. The nonlinear term in the boundary condition is linearized by using the Taylor’s series approximation. Afterward, the time-dependent terms in the linearized equations are removed by the Laplace transform technique, and then the transformed field equations are discretized using the control volume method with suitable shape functions. The nodal dimensionless temperature and displacement in the transform domain are inverted to obtain the actual physical quantities, using the numerical inversion of the Laplace transform method. It is seen from various illustrative problems that the present method has good accuracy and efficiency in predicting the wave propagations of temperature, stress, and displacement. However, it should be noted that the distributions of temperature, stress, and displacement can experience steep jumps at their wavefronts. In the present study, the effects of the relaxation times on these thermoelastic waves are also investigated.

Thermodiffusion with two time delays and Kernel functions

2016 ◽  
Vol 23 (2) ◽  
pp. 195-208 ◽  
Author(s):  
Ahmed S El-Karamany ◽  
Magdy A Ezzat ◽  
Alaa A El-Bary
Keyword(s):  
Laplace Transform ◽  
Kernel Functions ◽  
Transform Domain ◽  
Thermoelastic Diffusion ◽  
One Dimensional ◽  
Laplace Transform Technique ◽  
The Laplace Transform ◽  
Isotropic Elastic Medium ◽  
Time Variations ◽  
With Memory

The present work is concerned with the investigation of disturbances in a homogeneous, isotropic elastic medium with memory-dependent derivatives (MDDs). A one-dimensional problem is considered for a half-space whose surface is traction free and subjected to the effects of thermodiffusion. For treatment of time variations, the Laplace-transform technique is utilized. The theories of coupled and of generalized thermoelastic diffusion with one relaxation time follow as limit cases. A direct approach is introduced to obtain the solutions in the Laplace transform domain for different forms of kernel functions and time delay of MDDs, which can be arbitrarily chosen. Numerical inversion is carried out to obtain the distributions of the considered variables in the physical domain and illustrated graphically. Some comparisons are made and shown in figures to estimate the effects of MDD parameters on all studied fields.

Thermoelasticity When the Material Coupling Parameter Equals Unity

1965 ◽  
Vol 32 (2) ◽  
pp. 378-382 ◽  
Author(s):  
O. W. Dillon
Keyword(s):  
Laplace Transform ◽  
Constant Velocity ◽  
Coupling Parameter ◽  
Step Function ◽  
Surface Strain ◽  
Coupled Thermoelasticity ◽  
Velocity Impact ◽  
One Dimensional ◽  
Laplace Transform Technique ◽  
The Laplace Transform

Analytical solutions of three problems in coupled thermoelasticity are presented for the case when the material coupling parameter equals unity. The problems considered are: (a) Danilovskaya’s problem of a step function in temperature at the surface; (b) a step function in surface strain; and (c) constant velocity impact. Solutions are presented for the case of thin bars (one-dimensional stress) and are obtained by the Laplace-transform technique. There is great simplification in the equations when the material coupling parameter equals unity which permits the straightforward inversion of the transformed solutions. The results demonstrate significant deviations from the corresponding uncoupled solutions.

scholarly journals On applications of Caputo k-fractional derivatives

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Ghulam Farid ◽  
Naveed Latif ◽  
Matloob Anwar ◽  
Ali Imran ◽  
Muhammad Ozair ◽  
...  
Keyword(s):  
Laplace Transform ◽  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Fractional Integral ◽  
Integral Inequalities ◽  
Chebyshev Inequality ◽  
Absolutely Continuous ◽  
Laplace Transform Technique ◽  
The Laplace Transform ◽  
Type Variable

Abstract This research explores Caputo k-fractional integral inequalities for functions whose nth order derivatives are absolutely continuous and possess Grüss type variable bounds. Using Chebyshev inequality (Waheed et al. in IEEE Access 7:32137–32145, 2019) for Caputo k-fractional derivatives, several integral inequalities are derived. Further, Laplace transform of Caputo k-fractional derivative is presented and Caputo k-fractional derivative and Riemann–Liouville k-fractional integral of an extended generalized Mittag-Leffler function are calculated. Moreover, using the extended generalized Mittag-Leffler function, Caputo k-fractional differential equations are presented and their solutions are proposed by applying the Laplace transform technique.

Sign in / Sign up

Export Citation Format

Share Document