scholarly journals A detailed analysis for the fundamental solution of fractional vibration equation

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Li-Li Liu ◽  
Jun-Sheng Duan
Keyword(s):  
Fundamental Solution ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Detailed Analysis ◽  
Caputo Fractional Derivative ◽  
Integral Formula ◽  
Damping Term ◽  
Vibration Equation ◽  
Number Of Zeros ◽  
The Laplace Transform

AbstractIn this paper, we investigate the solution of the fractional vibration equation, where the damping term is characterized by means of the Caputo fractional derivative with the order α satisfying 0 < α < 1 or 1 < α < 2. Detailed analysis for the fundamental solution y(t) is carried out through the Laplace transform and its complex inversion integral formula. We conclude that y(t) is ultimately positive, and ultimately decreases monotonically and approaches zero for the case of 0 < α < 1, while y(t) is ultimately negative, and ultimately increases monotonically and approaches zero for the case of 1 < α < 2. We also consider the number of zeros, the maximum zero and the maximum extreme point of the fundamental solution y(t) for specified values of the coefficients and fractional order.

State space solution of implicit fractional continuous time systems

2012 ◽  
Vol 15 (3) ◽  
Author(s):  
Djillali Bouagada ◽  
Paul Dooren
Keyword(s):  
Laplace Transform ◽  
Fractional Derivative ◽  
State Space ◽  
Caputo Fractional Derivative ◽  
Continuous Time ◽  
State Space Models ◽  
Continuous Time Systems ◽  
The Laplace Transform ◽  
Time Systems ◽  
Time Linear

AbstractIn this work we extend a result from the literature on fractional continuous-time linear systems to the case of implicit fractional continuous-time state space models, based on the Caputo fractional derivative. The solution of the problem is derived using the Laplace transform.

scholarly journals Approximation of linear one dimensional partial differential equations including fractional derivative with non-singular kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Raheel Kamal ◽  
Kamran ◽  
Gul Rahmat ◽  
Ali Ahmadian ◽  
Noreen Izza Arshad ◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Meshless Method ◽  
Inverse Laplace Transform ◽  
Contour Integral ◽  
Partial Differential ◽  
One Dimensional ◽  
The Laplace Transform

AbstractIn this article we propose a hybrid method based on a local meshless method and the Laplace transform for approximating the solution of linear one dimensional partial differential equations in the sense of the Caputo–Fabrizio fractional derivative. In our numerical scheme the Laplace transform is used to avoid the time stepping procedure, and the local meshless method is used to produce sparse differentiation matrices and avoid the ill conditioning issues resulting in global meshless methods. Our numerical method comprises three steps. In the first step we transform the given equation to an equivalent time independent equation. Secondly the reduced equation is solved via a local meshless method. Finally, the solution of the original equation is obtained via the inverse Laplace transform by representing it as a contour integral in the complex left half plane. The contour integral is then approximated using the trapezoidal rule. The stability and convergence of the method are discussed. The efficiency, efficacy, and accuracy of the proposed method are assessed using four different problems. Numerical approximations of these problems are obtained and validated against exact solutions. The obtained results show that the proposed method can solve such types of problems efficiently.

scholarly journals A fractional differential equation with multi-point strip boundary condition involving the Caputo fractional derivative and its Hyers–Ulam stability

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mehboob Alam ◽  
Akbar Zada ◽  
Ioan-Lucian Popa ◽  
Alireza Kheiryan ◽  
Shahram Rezapour ◽  
...  
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Caputo Fractional Derivative ◽  
Integral Boundary Conditions ◽  
Ulam Stability ◽  
Integral Boundary ◽  
Laplace Transform Technique ◽  
The Laplace Transform ◽  
Fractional Differential

AbstractIn this work, we investigate the existence, uniqueness, and stability of fractional differential equation with multi-point integral boundary conditions involving the Caputo fractional derivative. By utilizing the Laplace transform technique, the existence of solution is accomplished. By applying the Bielecki-norm and the classical fixed point theorem, the Ulam stability results of the studied system are presented. An illustrative example is provided at the last part to validate all our obtained theoretical results.

scholarly journals The inversion of a transform related to the Laplace transform and to heat conduction

1964 ◽  
Vol 4 (1) ◽  
pp. 1-14 ◽  
Author(s):  
David V. Widder
Keyword(s):  
Heat Conduction ◽  
Fundamental Solution ◽  
Heat Equation ◽  
Laplace Transform ◽  
Physical Interpretation ◽  
Integral Transform ◽  
The Laplace Transform ◽  
Image Position

In a recent paper [7] the author considered, among other things, the integral transform where is the fundamental solution of the heat equation There we gave a physical interpretation of the transform (1.1). Here we shall choose a slightly different interpretation, more convenient for our present purposes. If then u(O, t) = f(t). That is, the function f(t) defined by equation (1.1) is the temperature at the origin (x = 0) of an infinite bar along the x-axis t seconds after it was at a temperature defined by the equation .

scholarly journals Transient Analysis of Dynamic Crack Propagation With Boundary Effect

1995 ◽  
Vol 62 (4) ◽  
pp. 1029-1038 ◽  
Author(s):  
Chien-Ching Ma ◽  
Yi-Shyong Ing
Keyword(s):  
Fundamental Solution ◽  
Crack Propagation ◽  
Laplace Transform ◽  
Fundamental Problem ◽  
Boundary Effect ◽  
Dynamic Crack Propagation ◽  
Dynamic Crack ◽  
Transform Domain ◽  
Propagating Crack ◽  
The Laplace Transform

In this study, a dynamic antiplane crack propagation with constant velocity in a configuration with boundary is investigated in detail. The reflected cylindrical waves which are generated from the free boundary will interact with the propagating crack and make the problem extremely difficult to analyze. A useful fundamental solution is proposed in this study and the solution is determined by superposition of the fundamental solution in the Laplace transform domain. The proposed fundamental problem is the problem of applying exponentially distributed traction (in the Laplace transform domain) on the propagating crack faces. The Cagniard’s method for Laplace inversion is used to obtain the transient solution in time domain. Numerical results of dynamic stress intensity factors for the propagation crack are evaluated in detail.

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.

scholarly journals Analysis of Multiterm Initial Value Problems with Caputo–Fabrizio Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Mohammed Al-Refai ◽  
Muhammed Syam
Keyword(s):  
Laplace Transform ◽  
Fractional Derivative ◽  
Initial Value Problems ◽  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Initial Value ◽  
The Laplace Transform ◽  
Necessary And Sufficient

In this paper, we discuss the solvability of a class of multiterm initial value problems involving the Caputo–Fabrizio fractional derivative via the Laplace transform. We derive necessary and sufficient conditions to guarantee the existence of solutions to the problem. We also obtain the solutions in closed forms. We present two examples to illustrate the validity of the obtained results.

scholarly journals Dynamic Crack Propagation in a Layered Medium Under Antiplane Shear

1997 ◽  
Vol 64 (1) ◽  
pp. 66-72 ◽  
Author(s):  
Chien-Ching Ma ◽  
Yi-Shyong Ing
Keyword(s):  
Stress Intensity ◽  
Fundamental Solution ◽  
Laplace Transform ◽  
Dynamic Stress ◽  
Layered Medium ◽  
Transform Domain ◽  
Intensity Factors ◽  
Dynamic Stress Intensity Factors ◽  
Propagating Crack ◽  
The Laplace Transform

In this study, the transient analysis of dynamic antiplane crack propagation with a constant velocity in a layered medium is investigated. The individual layers are isotropic and homogeneous. Infinite numbers of reflected cylindrical waves, which are generated from the interface of the layered medium, will interact with the propagating crack and make the problem extremely difficult to analyze. A useful fundamental solution is proposed in this study, and the solution can be determined by superposition of the fundamental solution in the Laplace transform domain. The proposed fundamental problem is the problem of applying exponentially distributed traction (in the Laplace transform domain) on the propagating crack faces. The Cagniard’s method for Laplace inversion is used to obtain the transient solution in time domain. The exact closed-form transient solutions of dynamic stress intensity factors are expressed in compact formulations. These solutions are valid for an infinite length of time and have accounted for contributions from all the incident and reflected waves interaction with the moving crack tip. Numerical results of dynamic stress intensity factors for the propagation crack in layered medium are evaluated and discussed in detail.

scholarly journals Linear Fractionally Damped Oscillator

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Mark Naber
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
The Other ◽  
Damping Term ◽  
Damped Oscillator ◽  
Limiting Value

The linearly damped oscillator equation is considered with the damping term generalized to a Caputo fractional derivative. The order of the derivative being considered is . At the lower end the equation represents an undamped oscillator and at the upper end the ordinary linearly damped oscillator equation is recovered. A solution is found analytically, and a comparison with the ordinary linearly damped oscillator is made. It is found that there are nine distinct cases as opposed to the usual three for the ordinary equation (damped, over-damped, and critically damped). For three of these cases it is shown that the frequency of oscillation actually increases with increasing damping order before eventually falling to the limiting value given by the ordinary damped oscillator equation. For the other six cases the behavior is as expected, the frequency of oscillation decreases with increasing order of the derivative (damping term).

scholarly journals Solutions of Cattaneo-Hristov model of elastic heat diffusion with Caputo-Fabrizio and Atangana-Baleanu fractional derivatives

2017 ◽  
Vol 21 (6 Part A) ◽  
pp. 2299-2305 ◽  
Author(s):  
Ilknur Koca ◽  
Abdon Atangana
Keyword(s):  
Diffusion Equation ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Heat Diffusion ◽  
Relaxation Kernel ◽  
Extended Version ◽  
Heat Diffusion Equation ◽  
The Laplace Transform ◽  
Non Local

Recently Hristov using the concept of a relaxation kernel with no singularity developed a new model of elastic heat diffusion equation based on the Caputo-Fabrizio fractional derivative as an extended version of Cattaneo model of heat diffusion equation. In the present article, we solve exactly the Cattaneo-Hristov model and extend it by the concept of a derivative with non-local and non-singular kernel by using the new Atangana-Baleanu derivative. The Cattaneo-Hristov model with the extended derivative is solved analytically with the Laplace transform, and numerically using the Crank-Nicholson scheme.

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