Fourth-order fractional diffusion model of thermal grooving: integral approach to approximate closed form solution of the Mullins model

2018 ◽  
Vol 13 (1) ◽  
pp. 6 ◽  
Author(s):  
Jordan Hristov
Keyword(s):  
Integration Method ◽  
Closed Form Solution ◽  
Multiple Integral ◽  
Fractional Diffusion ◽  
Form Solution ◽  
Balance Method ◽  
Integral Approach ◽  
Integration Technique ◽  
Multiple Integration ◽  
Space Coordinate

A multiple integration technique of the integral-balance method allowing solving high-order subdiffusion diffusion equations is presented in this article. The new method termed multiple-integral balance method (MIM) is based on multiple integration procedures with respect to the space coordinate. MIM is a generalization of the widely applied Heat-balance integral method of Goodman and the double integration method of Volkov. The method is demonstrated by a solution of the linear subdiffusion model of Mullins for thermal grooving by surface diffusion.

scholarly journals Multiple integral-balance method: Basic idea and an example with Mullin’s model of thermal grooving

2017 ◽  
Vol 21 (3) ◽  
pp. 1555-1560 ◽  
Author(s):  
Jordan Hristov
Keyword(s):  
Heat Balance ◽  
Integral Method ◽  
Integration Method ◽  
Multiple Integral ◽  
Balance Method ◽  
Diffusion Equations ◽  
Integration Technique ◽  
Linear Diffusion ◽  
Multiple Integration ◽  
Heat Balance Integral Method

A multiple integration technique of the integral-balance method allowing solving high-order diffusion equations is conceived in this note. The new method termed multiple-integral balance method is based on multiple integration procedures with respect to the space co-ordinate and is generalization of the widely applied heat-balance integral method of Goodman and the double integration method of Volkov. The method is demonstrated by a solution of the linear diffusion models of Mullins for thermal grooving.

scholarly journals Diffusion models of heat and momentum with weakly singular kernels in the fading memories: How the integral-balance method can be applied?

2015 ◽  
Vol 19 (3) ◽  
pp. 947-957 ◽  
Author(s):  
Jordan Hristov
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Diffusion Models ◽  
Integral Approach ◽  
Fading Memory ◽  
Weakly Singular Kernels ◽  
Weakly Singular ◽  
Liouville Derivative ◽  
Singular Kernels

This work presents an attempt to apply the heat-balance integral approach to diffusion models with fading memories with weakly singular kernels resulting in closed-form solutions. The main steps are exemplified by solutions where the fading memory is represented by Volterra integrals and by a time-fractional Riemann-Liouville derivative. The examples address sole elastic (damping) effects and cases where the viscous diffusivity should be taken into account. As examples polynomial approximation is applied, demonstrating how to avoid problems in determination of the exponent of the general parabolic profile, but without freedom to optimize the final closed-form solution. In general, this is a new implementation of an old idea and related methods to new models and we hope the demonstrated technique could be useful in solutions of practical problems.

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.

scholarly journals Nonlinear Oscillations of CNT Nano-resonator Based on Nonlocal Elasticity: The Energy Balance Method

2021 ◽  
Vol 2 (1) ◽  
pp. 41-50
Author(s):  
Masoud Goharimanesh ◽  
◽  
Ali Koochi ◽  
Keyword(s):  
Energy Balance ◽  
Closed Form ◽  
Nonlocal Elasticity ◽  
Casimir Force ◽  
Nonlocal Parameter ◽  
Closed Form Solution ◽  
Form Solution ◽  
Balance Method ◽  
Energy Balance Method ◽  
The Impact

This paper deals with investigating the nonlinear oscillation of carbon nanotube manufactured nano-resonator. The governing equation of the nano-resonator is extracted in the context of the nonlocal elasticity. The impact of the Casimir force is also incorporated in the developed model. A closed-form solution based on the energy balance method is presented for investigating the oscillations of the nano-resonator. The proposed closed-form solution is compared with the numerical solution. The impact of influential parameters including applied voltage, Casimir force, geometrical and nonlocal parameters on the nano resonator’s vibration and frequency are investigated. The obtained results demonstrated that the Casimir force reduces the nano-resonator frequency. However, the nonlocal parameter has a hardening effect and enhances the system’s frequency.

Interaction Between Two Bodies Translating in an Inviscid Fluid

1991 ◽  
Vol 35 (01) ◽  
pp. 1-8
Author(s):  
L. Landweber ◽  
A. T. Chwang ◽  
Z. Guo
Keyword(s):  
Translational Motion ◽  
Closed Form Solution ◽  
Added Mass ◽  
Form Solution ◽  
Circular Cylinders ◽  
Integral Approach ◽  
Inviscid Fluid ◽  
Added Masses ◽  
Two Bodies ◽  
Good Agreement

The equations of motion of two bodies in translational motion in an inviscid fluid at rest at infinity are expressed in Lagrangian form. For the case of one body stationary and the other approaching it in a uniform stream, an exact, closed-form solution in terms of added masses is obtained, yielding simple expressions for the velocity of the moving body as a function of its relative position and for the interaction forces. This solution is applied to the case of a rectangular cylinder approaching a cylindrical one, for which the added-mass coefficients had been previously obtained in a companion paper by an integral-equation procedure. In order to compare results with those in the literature, and to evaluate the accuracy of the present procedures, results were calculated for a pair of circular cylinders by these methods as well as by successive images. Very good agreement was found. Comparison with published results showed good agreement with the added mass but very poor agreement on the forces, including disagreement as to whether the forces were repulsive or attractive. The discrepancy is believed to be due to the omission of terms in the Bernoulli equation which was used to obtain the pressure distribution and then the force on a body. The Lagrangian formulation is believed to be preferable to the pressure-integral approach because it yields the hydrodynamic force directly in terms of the added masses and their derivatives, thus requiring the calculation of many fewer coefficients.

scholarly journals A BACKWARD IDENTIFICATION PROBLEM FOR AN AXIS-SYMMETRIC FRACTIONAL DIFFUSION EQUATION

2017 ◽  
Vol 22 (3) ◽  
pp. 311-320 ◽  
Author(s):  
Xiangtuan Xiong ◽  
Xiaojun Ma
Keyword(s):  
Diffusion Equation ◽  
Numerical Stability ◽  
Closed Form Solution ◽  
Identification Problem ◽  
Fractional Diffusion ◽  
Form Solution ◽  
Fractional Diffusion Equation ◽  
Polar Coordinates ◽  
Regularization Methods ◽  
Ill Posed

We consider a backward ill-posed problem for an axis-symmetric fractional diffusion equation which is described in polar coordinates. A closed form solution of the inverse problem is obtained. However, this solution blows up. For numerical stability, a general regularization principle is presented for constructing regularization methods. Several numerical examples are conducted for showing the validity and effectiveness of the proposed methods.

On a Closed-Form Solution of Prandtl's System of Equations

2013 ◽  
Vol 40 (2) ◽  
pp. 106-114
Author(s):  
J. Venetis ◽  
Aimilios (Preferred name Emilios) Sideridis
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
System Of Equations

Plane Wave Resonance in the Tire Air Cavity as a Vehicle Interior Noise Source

1995 ◽  
Vol 23 (1) ◽  
pp. 2-10 ◽  
Author(s):  
J. K. Thompson
Keyword(s):  
Closed Form Solution ◽  
Form Solution ◽  
Interior Noise ◽  
Cavity Resonance ◽  
Test Results ◽  
Vehicle Interior ◽  
Air Cavity ◽  
Vehicle Interior Noise ◽  
Wave Resonance ◽  
Resonance Frequencies

Abstract Vehicle interior noise is the result of numerous sources of excitation. One source involving tire pavement interaction is the tire air cavity resonance and the forcing it provides to the vehicle spindle: This paper applies fundamental principles combined with experimental verification to describe the tire cavity resonance. A closed form solution is developed to predict the resonance frequencies from geometric data. Tire test results are used to examine the accuracy of predictions of undeflected and deflected tire resonances. Errors in predicted and actual frequencies are shown to be less than 2%. The nature of the forcing this resonance as it applies to the vehicle spindle is also examined.

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