Analysis of two colliding fractionally damped spherical shells in modelling blunt human head impacts

Open Physics ◽  
2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Yury Rossikhin ◽  
Marina Shitikova
Keyword(s):  
Fractional Derivative ◽  
Human Head ◽  
Contact Spot ◽  
Spherical Shells ◽  
Spherical Object ◽  
Frontal Impacts ◽  
Age Related ◽  
Laplace Transform Technique ◽  
The Laplace Transform ◽  
The Impact

AbstractThe collision of two elastic or viscoelastic spherical shells is investigated as a model for the dynamic response of a human head impacted by another head or by some spherical object. Determination of the impact force that is actually being transmitted to bone will require the model for the shock interaction of the impactor and human head. This model is indended to be used in simulating crash scenarios in frontal impacts, and provide an effective tool to estimate the severity of effect on the human head and to estimate brain injury risks. The model developed here suggests that after the moment of impact quasi-longitudinal and quasi-transverse shock waves are generated, which then propagate along the spherical shells. The solution behind the wave fronts is constructed with the help of the theory of discontinuities. It is assumed that the viscoelastic features of the shells are exhibited only in the contact domain, while the remaining parts retain their elastic properties. In this case, the contact spot is assumed to be a plane disk with constant radius, and the viscoelastic features of the shells are described by the fractional derivative standard linear solid model. In the case under consideration, the governing differential equations are solved analytically by the Laplace transform technique. It is shown that the fractional parameter of the fractional derivative model plays very important role, since its variation allows one to take into account the age-related changes in the mechanical properties of bone.

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 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 New analytical solutions of heat transfer flow of clay-water base nanoparticles with the application of novel hybrid fractional derivative

2020 ◽  
Vol 24 (Suppl. 1) ◽  
pp. 343-350
Author(s):  
Muhammad Asjad ◽  
Muhammad Ikram ◽  
Rizwan Ali ◽  
Dumitru Baleanu ◽  
Ali Alshomrani
Keyword(s):  
Fractional Derivative ◽  
Analytical Solutions ◽  
Engine Oil ◽  
Clay Nanoparticles ◽  
Flow Parameters ◽  
Laplace Transform Technique ◽  
Water Base ◽  
Mathcad Software ◽  
The Impact ◽  
Transfer Flow

Clay nanoparticles are hanging in three different based fluids (water, kerosene, and engine oil). The exact terminologies of Maxwell-Garnett and Brinkman for the current thermophysical properties of clay nanofluids are used, while the flow occurrence is directed by a set linear PDE with physical initial and boundary conditions. The classical governing equations are extended to non-integer order hybrid fractional derivative which is introduced in [33]. Analytical solutions for temperature and ve?locity fields are attained via Laplace transform technique. Some limiting solutions are also obtained from the existing literature and compared for different values of fractional parameter. To vision the impact of several flow parameters on the temperature and velocity some graphs are drawn using Mathcad software and designed in different figures. As a result, we found that hybrid fractional model is better in describing the decay behavior of temperature and velocity in comparison of classical derivatives. In comparison of nanofluid with different base fluids, it is concluded that water-based nanofluid has higher velocity than others.

scholarly journals MATHEMATICAL AND STATISTICAL ANALYSIS OF RL AND RC FRACTIONAL-ORDER CIRCUITS

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040030 ◽  
Author(s):  
NADEEM AHMAD SHEIKH ◽  
DENNIS LING CHUAN CHING ◽  
SAMI ULLAH ◽  
ILYAS KHAN
Keyword(s):  
Statistical Analysis ◽  
Fractional Derivative ◽  
Classical Model ◽  
Modern Concept ◽  
Inverse Transformation ◽  
Laplace Transform Technique ◽  
The Laplace Transform ◽  
Fractional Differential ◽  
Fractional Order Circuits ◽  
Fractional Parameter

The RL and RC circuits are analyzed in this research paper. The classical model of these circuits is generalized using the modern concept of fractional derivative with Mittag-Leffler function in its kernel. The fractional differential equations are solved for exact solutions using the Laplace transform technique and the inverse transformation. The obtained solutions are plotted and presented in tables to show the effect of resistance, inductance and fractional parameter on current and voltage. Furthermore, the statistical analysis is presented to predict the seasonal of time and other parameters on the current flowing in the circuit. The statistical analysis shows that the variation in current is insignificant with respect to time and is more significant with respect to other parameters.

scholarly journals Mathematical modeling for solute transport in aquifer

2015 ◽  
Vol 18 (3) ◽  
pp. 481-499 ◽  
Author(s):  
Mritunjay Kumar Singh ◽  
Vijay P. Singh ◽  
Pintu Das
Keyword(s):  
Mathematical Modeling ◽  
Solute Transport ◽  
Industrial Effluents ◽  
Numerical Dispersion ◽  
Predictive Tool ◽  
Laplace Transform Technique ◽  
The Laplace Transform ◽  
Explicit Finite Difference ◽  
The Impact ◽  
Source Concentration

Groundwater pollution may occur due to human activities, industrial effluents, cemeteries, mine spoils, etc. This paper deals with one-dimensional mathematical modeling of solute transport in finite aquifers. The governing equation for solute transport by unsteady groundwater flow is solved analytically by the Laplace transform technique. Initially, the aquifer is subjected to the spatially dependent source concentration with zero-order production. One end of the aquifer receives the source concentration and is represented by a mixed-type boundary condition in the splitting time domain. The concentration gradient at the other end of the porous media is assumed to be zero. The temporally dependent velocity and the dispersion coefficients are considered. A numerical solution is obtained by using an explicit finite difference scheme and compared with the analytical result. Accuracy of the solution is discussed by using the root mean square error method. Truncation error is also explored for the parameters like numerical dispersion and velocity terms. The impact of Peclet number is examined. For graphical interpretation, unsteady velocity expressions (i.e., such as exponential, sinusoidal, asymptotic, and algebraic sigmoid) are considered. The work may be used as a preliminary predictive tool for groundwater resource and management.

scholarly journals Exact Solution of Fractional Convective Casson Fluid Through an Accelerated Plate

CFD letters ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 15-25
Author(s):  
Muhammad Nazirul Shahrim ◽  
Ahmad Qushairi Mohamad ◽  
Lim Yeou Jiann ◽  
Muhamad Najib Zakaria ◽  
Sharidan Shafie ◽  
...  
Keyword(s):  
Fractional Derivative ◽  
Newtonian Fluid ◽  
Casson Fluid ◽  
Electrical Networks ◽  
Moving Plate ◽  
Transform Method ◽  
The Laplace Transform ◽  
Physical Phenomena ◽  
The Impact ◽  
Fractional Parameter

Fractional derivative has perfectly adopted to model few physical phenomena such as viscoelasticity of coiling polymers, traffic construction, fluid dynamics and electrical networks. However, the application of the fractional derivatives for describing the physical characteristics of non-Newtonian fluid over a moving plate is still rare. In the present study, the effect of the Caputo fractional derivative on the Casson fluid flow which is induced by an accelerated plate is analytically analysed. The governing equations are initially transformed into dimensionless expressions by using suitable dimensionless variables. Then the Laplace transform method is utilized to calculate the exact solutions for the fractional governing partial differential equations. The obtained solutions are validated by comparing the results for specific case with the existing solutions in the literature. The impact of fractional parameter, Prandtl number, and time on the velocity and temperature profiles are graphically showed and discussed. The results depict that the temperature and velocity increase with the increment of fractional parameter and time. Interestingly, the velocity decreases at region near the plate but is enhanced at the area far away from the plate when the Casson fluid parameter is increased. This study is essential in understanding the factional non-Newtonian fluid flows which is more realistic in nature.

Dynamics Response of a Fractionally Damped Spherical Shell Impacted by a Body of Finite Dimensions

2014 ◽  
Vol 595 ◽  
pp. 111-116
Author(s):  
T.K. Chang ◽  
Yury Rossikhin ◽  
Marina Shitikova ◽  
C.K. Chao
Keyword(s):  
Laplace Transform ◽  
Spherical Shell ◽  
Contact Region ◽  
Contact Spot ◽  
Low Velocity Impact ◽  
Strong Discontinuity ◽  
Ray Method ◽  
Laplace Transform Technique ◽  
The Laplace Transform ◽  
Contact Domain

In the present paper, the problem on normal low-velocity impact of a solid upon an isotropic spherical shell is studied without considering the changes in the geometrical dimensions of the contact domain. At the moment of impact, shock waves (surfaces of strong discontinuity) are generated in the target, which then propagate along the shell during the process of impact. Behind the wave fronts up to the boundary of the contact domain, the solution is constructed with the help of the theory of discontinuities and one-term ray expansions. The ray method is used outside the contact spot, but the Laplace transform method is applied within the contact region. It is assumed that the viscoelastic features of the shell are exhibited only in the contact domain, while the remaining part retains its elastic properties. In this case, the contact spot is assumed to be a plane disk with constant radius, and the viscoelastic features of the shell are described by the fractional derivative standard linear solid model. In the case under consideration, the governing differential equations are solved analytically by the Laplace transform technique. As a result, the exact solution of the contact force is determined as a function of time.

scholarly journals New analytical solutions of heat transfer flow of clay-water base nanoparticles with the application of novel hybrid fractional derivative

2020 ◽  
Vol 24 (Suppl. 1) ◽  
pp. 343-350
Author(s):  
Muhammad Asjad ◽  
Muhammad Ikram ◽  
Rizwan Ali ◽  
Dumitru Baleanu ◽  
Ali Alshomrani
Keyword(s):  
Fractional Derivative ◽  
Analytical Solutions ◽  
Engine Oil ◽  
Clay Nanoparticles ◽  
Flow Parameters ◽  
Laplace Transform Technique ◽  
Water Base ◽  
Mathcad Software ◽  
The Impact ◽  
Transfer Flow

Clay nanoparticles are hanging in three different based fluids (water, kerosene, and engine oil). The exact terminologies of Maxwell-Garnett and Brinkman for the current thermophysical properties of clay nanofluids are used, while the flow occurrence is directed by a set linear PDE with physical initial and boundary conditions. The classical governing equations are extended to non-integer order hybrid fractional derivative which is introduced in [33]. Analytical solutions for temperature and ve?locity fields are attained via Laplace transform technique. Some limiting solutions are also obtained from the existing literature and compared for different values of fractional parameter. To vision the impact of several flow parameters on the temperature and velocity some graphs are drawn using Mathcad software and designed in different figures. As a result, we found that hybrid fractional model is better in describing the decay behavior of temperature and velocity in comparison of classical derivatives. In comparison of nanofluid with different base fluids, it is concluded that water-based nanofluid has higher velocity than others.

The Role of Vascular Aging in Atherosclerotic Plaque Development and Vulnerability

2019 ◽  
Vol 25 (29) ◽  
pp. 3098-3111 ◽  
Author(s):  
Luca Liberale ◽  
Giovanni G. Camici
Keyword(s):  
Atherosclerotic Plaque ◽  
Molecular Mechanisms ◽  
Driving Forces ◽  
Plaque Burden ◽  
Vascular Aging ◽  
Age Related ◽  
Plaque Development ◽  
Increased Risk ◽  
The Impact ◽  
Over Time

Background: The ongoing demographical shift is leading to an unprecedented aging of the population. As a consequence, the prevalence of age-related diseases, such as atherosclerosis and its thrombotic complications is set to increase in the near future. Endothelial dysfunction and vascular stiffening characterize arterial aging and set the stage for the development of cardiovascular diseases. Atherosclerotic plaques evolve over time, the extent to which these changes might affect their stability and predispose to sudden complications remains to be determined. Recent advances in imaging technology will allow for longitudinal prospective studies following the progression of plaque burden aimed at better characterizing changes over time associated with plaque stability or rupture. Oxidative stress and inflammation, firmly established driving forces of age-related CV dysfunction, also play an important role in atherosclerotic plaque destabilization and rupture. Several genes involved in lifespan determination are known regulator of redox cellular balance and pre-clinical evidence underlines their pathophysiological roles in age-related cardiovascular dysfunction and atherosclerosis. Objective: The aim of this narrative review is to examine the impact of aging on arterial function and atherosclerotic plaque development. Furthermore, we report how molecular mechanisms of vascular aging might regulate age-related plaque modifications and how this may help to identify novel therapeutic targets to attenuate the increased risk of CV disease in elderly people.

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