Closed-form solutions for one-dimensional inhomogeneous anisotropic medium in a special case. I. Direct scattering problem

1997 ◽  
Vol 45 (6) ◽  
pp. 936-941 ◽  
Author(s):  
Tie Jun Cui ◽  
Chang Hong Liang ◽  
W. Wiesbeck
Keyword(s):  
Closed Form ◽  
Anisotropic Medium ◽  
Scattering Problem ◽  
One Dimensional ◽  
Closed Form Solutions ◽  
Inhomogeneous Anisotropic Medium ◽  
Direct Scattering ◽  
Special Case

scholarly journals Transient Solutions for One-Dimensional Problems With Strain Softening

1987 ◽  
Vol 54 (3) ◽  
pp. 513-518 ◽  
Author(s):  
T. Belytschko ◽  
Xiao-Jun Wang ◽  
Z. P. Bazant ◽  
Y. Hyun
Keyword(s):  
Strain Rate ◽  
Closed Form ◽  
Transient Response ◽  
Softening Point ◽  
Strain Softening ◽  
Stress Strain ◽  
One Dimensional ◽  
Closed Form Solutions ◽  
Transient Solutions

Closed-form solutions are presented for the transient response of rods in which strain softening occurs and the stress-strain laws exhibit nonvanishing stresses after the strain-softening regime. It is found that the appearance of any strain softening results in an infinite strain rate if the material is inviscid. For a stress-strain law with a monotonically decreasing stress the strains are infinite also. If the stress increases after the strain-softening portion, the strains remain finite and the strain-softening point moves through the rod.

scholarly journals COMPARISON OF CLOSED-FORM SOLUTIONS FOR THE LUCAS-UZAWA MODEL VIA THE PARTIAL HAMILTONIAN APPROACH AND THE CLASSICAL APPROACH

2017 ◽  
Vol 22 (4) ◽  
pp. 464-483 ◽  
Author(s):  
Rehana Naz ◽  
Azam Chaudhry
Keyword(s):  
Closed Form ◽  
First Integrals ◽  
Hamiltonian Approach ◽  
Growth Path ◽  
Three Solutions ◽  
Balanced Growth ◽  
Closed Form Solutions ◽  
Long Run ◽  
Balanced Growth Path ◽  
Special Case

In this paper we derive the closed-form solutions for the Lucas-Uzawa growth model with the aid of the partial Hamiltonian approach and then compare our results with those derived in literature. The partial Hamiltonian approach provides two first integrals [9] in the case where there are no parameter restrictions and these two first integrals are utilized to construct three sets of closed form solutions for all the variables in the model. We begin by using the two first integrals to find two closed form solutions, one of which is new to the literature. We then use only one of the first integrals to derive a third solution that is the same as that found in the previous literature. We continue by analyzing the newly derived solution in detail also show that all three solutions converge to the same long run balanced growth path. The special case when the share of capital is equal to the inverse of the intertemporal elasticity of substitution is also investigated in detail.

The Use of the Milne-Eddington Absorption Coefficient for Radiative Heat Transfer in Combustion Systems

1977 ◽  
Vol 99 (3) ◽  
pp. 458-465 ◽  
Author(s):  
J. D. Felske ◽  
C. L. Tien
Keyword(s):  
Heat Transfer ◽  
Absorption Coefficient ◽  
Closed Form ◽  
Radiative Heat Transfer ◽  
Radiative Heat ◽  
Wide Band ◽  
One Dimensional ◽  
Soot Particles ◽  
Closed Form Solutions ◽  
Combustion Systems

The applicability of the Milne-Eddington absorption coefficient approximation is discussed in relation to the calculation of radiative transport involving the two distinct types of species produced in combustion systems—gases and soot particles. The approximation is found to apply well to hydrocarbon soot particles and as a result analytical closed-form solutions are derived for the radiative heat transfer inside one-dimensional slab shaped soot clouds. (The applicability of the gray approximation to soot is also discussed.) For the calculation of total band radiation from gases, however, the Milne-Eddington approximation is found to be questionable. The meaning of its assumption is discussed in light of an established Curtis-Godson wide band scaling approximation. Its usefulness for real gases is then assessed through the calculation and comparison of slab radiation by both techniques.

Joint and Cross Acceptances for Cross-Flow-Induced Vibration—Part I: Theoretical and Finite Element Formulations

2000 ◽  
Vol 122 (3) ◽  
pp. 349-354 ◽  
Author(s):  
M. K. Au-Yang
Keyword(s):  
Finite Element ◽  
Closed Form ◽  
Cross Flow ◽  
Theoretical Development ◽  
Element Formulation ◽  
Flow Induced Vibration ◽  
Arbitrary Boundary ◽  
One Dimensional ◽  
Closed Form Solutions ◽  
Special Cases

The theoretical development of the acceptance integral method to estimate the random vibration of structures subject to turbulent flow is critically reviewed and put onto a firm mathematical basis. Closed-form solutions for the joint acceptances for cross-flow-induced vibration of one-dimensional structures are derived for two special cases of spring-supported and simply supported beams. These are used to check results from a finite element formulation of the acceptance integrals for one-dimensional structures with arbitrary boundary conditions, and for arbitrary correlation lengths. Agreements between the finite element and closed-form solutions are excellent. [S0094-9930(00)02303-9]

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