scholarly journals Some New Approximate Solutions in Closed-Form to Problems of Nanobars

2021 ◽  
Vol 5 (4) ◽  
pp. 161-167
Author(s):  
Uğurcan EROĞLU
Keyword(s):  
Closed Form ◽  
Approximate Solutions

scholarly journals On the Convergence of LHAM and its Application to Fractional Generalised Boussinesq Equations for Closed Form Solutions

2021 ◽  
pp. 25-47
Author(s):  
S. O. Ajibola ◽  
E. O. Oghre ◽  
A. G. Ariwayo ◽  
P. O. Olatunji
Keyword(s):  
Differential Equations ◽  
Closed Form ◽  
Fractional Differential Equations ◽  
Closed Form Solution ◽  
Boussinesq Equations ◽  
Approximate Solutions ◽  
Form Solution ◽  
Restrictive Assumption ◽  
Nonlinear Fractional Differential Equations ◽  
Fractional Differential

By fractional generalised Boussinesq equations we mean equations of the form \begin{equation} \Delta\equiv D_{t}^{2\alpha}-[\mathcal{N}(u)]_{xx}-u_{xxxx}=0, \: 0<\alpha\le1,\label{main}\nonumber \end{equation} where $\mathcal{N}(u)$ is a differentiable function and $\mathcal{N}_{uu}\ne0$ (to ensure nonlinearity). In this paper we lay emphasis on the cubic Boussinesq and Boussinesq-like equations of fractional order and we apply the Laplace homotopy analysis method (LHAM) for their rational and solitary wave solutions respectively. It is true that nonlinear fractional differential equations are often difficult to solve for their {\em exact} solutions and this single reason has prompted researchers over the years to come up with different methods and approach for their {\em analytic approximate} solutions. Most of these methods require huge computations which are sometimes complicated and a very good knowledge of computer aided softwares (CAS) are usually needed. To bridge this gap, we propose a method that requires no linearization, perturbation or any particularly restrictive assumption that can be easily used to solve strongly nonlinear fractional differential equations by hand and simple computer computations with a very quick run time. For the closed form solution, we set $\alpha =1$ for each of the solutions and our results coincides with those of others in the literature.

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.

Closed-Form Approximations to the Solution of V-Belt Force and Slip Equations

1985 ◽  
Vol 107 (2) ◽  
pp. 292-300 ◽  
Author(s):  
J. P. Dolan ◽  
W. S. Worley
Keyword(s):  
Differential Equations ◽  
Closed Form ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Design Practice ◽  
Tension Distribution ◽  
Analytical Approximations ◽  
Current Design

A method for generating accurate numerical solutions of the exact differential equations describing tension distribution and radial penetration of a flexible V-belt on driveN and driveR sheaves is presented and results are compared with approximate solutions reported in the literature. Analytical approximations for these solutions of higher accuracy than any previously published have been found and are presented. They suggest important modifications of current design practice for belt tensioning and life appraisal.

Relaxation of Thick-Walled Cylinders and Spheres

1982 ◽  
Vol 49 (3) ◽  
pp. 487-491 ◽  
Author(s):  
N. S. Ottosen
Keyword(s):  
Closed Form ◽  
Plane Strain ◽  
Approximate Solutions ◽  
Nonlinear Creep ◽  
Closed Form Solutions ◽  
Cylinders And Spheres

Using the nonlinear creep law proposed by Soderberg, closed-form solutions are derived for the relaxation of incompressible thick-walled spheres and cylinders in plane strain. These solutions involve series expressions which, however, converge very quickly. By simply ignoring these series expressions, extremely simple approximate solutions are obtained. Despite their simplicity these approximations possess an accuracy that is superior to approximations currently in use. Finally, several physical aspects related to the relaxation of cylinders and spheres are discussed.

Eshelby's tensor for embedded inclusions and the Elasto-Capillary phenomenon

2016 ◽  
Vol 01 (03n04) ◽  
pp. 1630002 ◽  
Author(s):  
Shengyou Yang ◽  
Pradeep Sharma
Keyword(s):  
Closed Form ◽  
Solid Mechanics ◽  
Transformation Strain ◽  
Approximate Solutions ◽  
Inclusion Problem ◽  
Physical Sciences ◽  
Classical Paper ◽  
Eshelby’S Tensor ◽  
The Subject ◽  
Curvature Sphere

The elastic state of an embedded inclusion undergoing a stress-free transformation strain was the subject of John Douglas Eshelby's now classical paper in 1957. This paper, the subject of which is now widely known as “Eshelby's inclusion problem”, is arguably one of the most cited papers in solid mechanics and several other branches of physical sciences. Applications have ranged from geophysics, quantum dots to composites. Over the past two decades, due to an interest in all things “small”, attempts have been made to extend Eshelby's elastic analysis to the nanoscale by incorporating capillary or surface energy effects. In this note, we revisit a particular formulation that derives a very general expression for the elasto-capillary state of an embedded inclusion. This approach, that closely mimics that of Eshelby's original paper, appears to have the advantage that it can be readily used for inclusions of arbitrary shape (for numerical calculations) and provides a facile route for approximate solutions when closed-form expressions are not possible. Specifically, in the case of inclusions of constant curvature (sphere, cylinder) subject to some simplifications, closed-form expressions are obtained.

Nongray Particulate Radiation in an Isothermal Cylindrical Medium

1981 ◽  
Vol 103 (1) ◽  
pp. 121-126 ◽  
Author(s):  
J. D. Felske ◽  
K. M. Lee
Keyword(s):  
Heat Flux ◽  
Exact Solution ◽  
Closed Form ◽  
Radiative Heat ◽  
Exact Result ◽  
Approximate Solutions ◽  
Small Particles ◽  
Dimensionless Heat Flux ◽  
Dimensionless Heat ◽  
Good Agreement

The radial radiative heat flux and its divergence are determined both exactly and approximately for homogeneous suspensions of small particles. Scattering is assumed to be small compared to absorption and the absorption coefficient is taken to be inversely proportional to wavelength. The exact solution is reduced to an infinite series of single integrals. The optically thin and the next higher order behavior appear in closed form as the first two terms in the series. Two approximate solutions are also developed. One is in good agreement with the exact solution while the other is not. Finally, a closed form approximate relation is derived for the dimensionless heat flux at the surface. This expression, which also gives the emissivity or absorptivity of the medium, is in excellent agreement with the exact result.

Macroscopic permeability of doubly porous solids with spheroidal macropores: closed-form approximate solutions of the longitudinal permeability

2020 ◽  
Vol 476 (2244) ◽  
pp. 20200224
Author(s):  
Vincent monchiet
Keyword(s):  
Closed Form ◽  
Volume Fraction ◽  
Stokes Problem ◽  
Approximate Solutions ◽  
Interface Condition ◽  
Porous Solids ◽  
Pressure Fields ◽  
Effective Transport ◽  
Analytic Expressions ◽  
Rigorous Bounds

The presence of macropores and fractures significantly affects the effective transport properties of porous solids such as concrete and rocks. The dimensions of the fractures are generally large behind that of the initial porosity, so that the problem contains two porosities. The influence of the macroporosity is studied in the homogenization framework by solving at the intermediate scale, that of the macropores, a coupled Darcy/Stokes problem with the Beavers–Joseph–Saffman (BJS) interface condition. We derive analytic expressions of the macroscopic permeability in the case of an isotropic permeable matrix containing spheroidal-shaped macropores. To this aim, we consider a representative volume element (RVE) on which uniform boundary conditions are considered for the velocity and pressure fields. The local problem is written as minimum principles; kinematic and static approaches are developed to derive rigorous bounds for the macroscopic permeability. Closed-form expressions of the longitudinal permeability (along the revolution axe of the spheroid) are determined by considering a simplified RVE constituted of two confocal spheroids. They depend on the volume fraction and the eccentricity of the spheroidal macropores, the scale factor between the two porosities and the slip coefficient of the BJS model. Illustrations show the influence of these parameters.

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