Fractional Calculus Model of the Semilunar Heart Valve Vibrations

2003 ◽  
Author(s):  
Moustafa El-Shahed
Keyword(s):  
Differential Equation ◽  
Experimental Data ◽  
Numerical Methods ◽  
Fractional Calculus ◽  
Closed Form ◽  
Heart Valve ◽  
Laplace Transformation ◽  
Closed Form Solution ◽  
Equation Of Motion ◽  
Form Solution

The objective of this paper is to solve the equation of motion of semilunar heart valve vibrations. The vibrations of the closed semilunar valves were modeled with a Caputo hactional derivative of order α. With the help of Laplace transformation, closed-form solution is obtained for the equation of motion in terms of Mittag-Leffler function. An alternative Method for Semi-differential equation, when α = 0.5, is examined using MATHEMATICA. The simplicity of these solutions makes them ideal for testing the accuracy of numerical methods. This solution can be of some interest for a better fit of experimental data.

Validation of Nitinol SMA Characteristics Using Finite Element Analysis and Closed Form Solutions

2013 ◽  
Vol 856 ◽  
pp. 147-152
Author(s):  
S.H. Adarsh ◽  
U.S. Mallikarjun
Keyword(s):  
Experimental Data ◽  
Finite Element Analysis ◽  
Finite Element ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Fe Analysis ◽  
Element Analysis ◽  
Complex Behaviour ◽  
Closed Form Solutions

Shape Memory Alloys (SMA) are promising materials for actuation in space applications, because of the relatively large deformations and forces that they offer. However, their complex behaviour and interaction of several physical domains (electrical, thermal and mechanical), the study of SMA behaviour is a challenging field. Present work aims at correlating the Finite Element (FE) analysis of SMA with closed form solutions and experimental data. Though sufficient literature is available on closed form solution of SMA, not much detail is available on the Finite element Analysis. In the present work an attempt is made for characterization of SMA through solving the governing equations by established closed form solution, and finally correlating FE results with these data. Extensive experiments were conducted on 0.3mm diameter NiTinol SMA wire at various temperatures and stress conditions and these results were compared with FE analysis conducted using MSC.Marc. A comparison of results from finite element analysis with the experimental data exhibits fairly good agreement.

scholarly journals Exact Solution of Ambartsumian Delay Differential Equation and Comparison with Daftardar-Gejji and Jafari Approximate Method

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 331 ◽  
Author(s):  
Huda Bakodah ◽  
Abdelhalim Ebaid
Keyword(s):  
Differential Equation ◽  
Closed Form ◽  
Closed Form Solution ◽  
Approximate Solutions ◽  
Form Solution ◽  
Proportional Delay ◽  
Delay Term ◽  
The Laplace Transform ◽  
Linear Differential ◽  
Delay Differential

The Ambartsumian equation, a linear differential equation involving a proportional delay term, is used in the theory of surface brightness in the Milky Way. In this paper, the Laplace-transform was first applied to this equation, and then the decomposition method was implemented to establish a closed-form solution. The present closed-form solution is reported for the first time for the Ambartsumian equation. Numerically, the calculations have demonstrated a rapid rate of convergence of the obtained approximate solutions, which are displayed in several graphs. It has also been shown that only a few terms of the new approximate solution were sufficient to achieve extremely accurate numerical results. Furthermore, comparisons of the present results with the existing methods in the literature were introduced.

Pascal's recurrence relation and even-aged-forest stand dynamics

1992 ◽  
Vol 22 (12) ◽  
pp. 1996-1999
Author(s):  
Rolfe A. Leary ◽  
Hien Phan ◽  
Kevin Nimerfro
Keyword(s):  
Differential Equation ◽  
Relative Density ◽  
Closed Form ◽  
Initial Conditions ◽  
Closed Form Solution ◽  
Integral Form ◽  
Forest Stand ◽  
Form Solution ◽  
Density Change ◽  
Stand Dynamics

A common method of modelling forest stand dynamics is to use permanent growth plot remeasurements to calibrate a whole-stand growth model expressed as an ordinary differential equation. To obtain an estimate of future conditions, either the differential equation is integrated numerically or, if analytic, the differential equation is solved in closed form. In the latter case, a future condition is obtained simply by evaluating the integral form for the age of interest, subject to appropriate initial conditions. An older method of modelling forest stand dynamics was to use a normal or near-normal yield table as a density standard and calibrate a relative density change equation from permanent plot remeasurements. An estimate of a future stand property could be obtained by iterating from a known initial relative density. In this paper we show that when the relative density change equation has a particular form, the historical method also has a closed form solution, given by a sequence of polynomials with coefficients from successive rows of Pascal's arithmetic triangle.

Calculation of Discharge Current Waveforms in High Voltage Spark Sources. II. Extensions and Limitations of Closed-Form Solution

1982 ◽  
Vol 36 (1) ◽  
pp. 25-29 ◽  
Author(s):  
Alexander Scheeline ◽  
T. V. Tran
Keyword(s):  
Exact Solution ◽  
Numerical Methods ◽  
Closed Form ◽  
High Voltage ◽  
Discharge Current ◽  
Closed Form Solution ◽  
Approximate Model ◽  
Form Solution ◽  
Dynamic Impedance ◽  
Numerical Integration Methods

Simulation of gap breakdown and dynamic impedance effects in high voltage spark sources is performed using an algebraically exact solution to an approximate model of source behavior. The importance of diode shunt capacitance in determining gap breakdown behavior is shown. Limitations in generality and implicit use of numerical methods in dynamic situations lead naturally to consideration of numerical integration methods. Comparisons to hardware sources are made.

scholarly journals Closed form solution of the SIR model for the COVID-19 outbreak in Italy

2020 ◽  
Author(s):  
Riccardo Giubilei
Keyword(s):  
Experimental Data ◽  
Closed Form ◽  
Closed Form Solution ◽  
Sir Model ◽  
Form Solution ◽  
Model Equations

AbstractThe CODIV-19 outbreak in early 2020 generated a tremendous effort of epidemiologists and researchers to fit the experimental data with the solutions of the SIR model equations [1] or with more sophisticated models. In this paper we show that under same hypotheses, a closed form solution exists that reasonably fits the experimental data for Italy, and the results can be extended to any other area.

scholarly journals Closed-Form Solution of Large Deflected Cantilever Beam a gainst Follower Loading Using Complex Analysis

2017 ◽  
Vol 11 (12) ◽  
pp. 12 ◽  
Author(s):  
Ibrahim Mousa Abu-Alshaikh
Keyword(s):  
Numerical Methods ◽  
Closed Form ◽  
Cantilever Beam ◽  
Complex Analysis ◽  
Numerical Solutions ◽  
Closed Form Solution ◽  
Form Solution ◽  
Beam Deflection ◽  
Integral Approach ◽  
Loading Conditions

The literature reveals that the non-conservative deflection of an elastic cantilever beam caused by applying follower tip loading was investigated and solved by various numerical methods like: Runge Kutta, iterative shooting, finite element, finite difference, direct iterative and non-iterative numerical methods. This is due to the fact that the Euler–Bernoulli nonlinear differential equation governing the problem contains the “slope at the free end”, this slope however needs special numerical treatment. On the other hand, some of these methods fail to find numerical solutions for extremely large loading conditions. Hence, this paper is aimed to obtain a closed-form solution for solving the large deflection of a cantilever beam opposed to a concentrated point follower load at its free end. This closed-form solution when compared with other conventional numerical approaches is characterized by simplicity, stability and straightforwardness in getting the beam deflection and slopes even for extremely large loading conditions. The closed-form solution is obtained by applying complex analysis along with elliptic-integral approach. Very good results were obtained when the elastica of the beam compared with that of various numerical methods which are used in analyzing similar problem.

Dynamic Stresses in a Driven Pile During Installation-Classical Wave Equation Model Solution Using Partial Differential Equations

2014 ◽  
Author(s):  
Syed Muhammad Mohsin Jafri ◽  
Phayak Takkabutr
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Wave Equation ◽  
Closed Form Solution ◽  
Equation Of Motion ◽  
Form Solution ◽  
Design Parameters ◽  
Partial Differential ◽  
Equation Analysis ◽  
Driven Pile

This paper derives and solves the governing dynamic wave equation of motion of a driven pile during the installation phase, when the driven pile is subjected to hammer blows. The pile is assumed as an elastic solid body. The equation of motion is a partial differential equation in space (axial coordinate) and time. The governing partial differential equation of motion is solved for installation boundary conditions, and simplified soil resistance models. The solution of the governing equation yields important design parameters, such as stress variation at any cross-section along the pile length with respect to time, and propagating wave speed. The resulting closed-form solution can be easily implemented using a standard spreadsheet or an engineering calculation program. This approach is compared with conventional wave equation analysis (WEAP) used in industry practice. The conventional wave equation analysis is based on discretization of the pile into mass-spring-damper elements (lumped parameter approach), rather than continuous modeling. The models and solutions from these two approaches are compared.

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