An approximation of the analytic solution of the shock wave equation

Allan, Fathi M.; Al‐Khaled, Kamel

doi:10.1016/j.cam.2005.05.009

Cited by 35 publications

(20 citation statements)

References 7 publications

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“…In recent years, the decomposition method has emerged as an alternative method for solving a wide range of problems whose mathematical models involve algebraic, differential, integral, integro-differential, higher-order ordinary and partial differential equations [1][2][3]7,10,12,13,18,19].…”

Section: Adm Methodologymentioning

confidence: 99%

A new convergence proof of the Adomian decomposition method for a mixed hyperbolic elliptic system of conservation laws

Az-Zo’bi

Al‐Khaled

2010

Applied Mathematics and Computation

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Section: Adm Methodologymentioning

confidence: 99%

A new convergence proof of the Adomian decomposition method for a mixed hyperbolic elliptic system of conservation laws

Az-Zo’bi

Al‐Khaled

2010

Applied Mathematics and Computation

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“…Substituting the assumed form of head and velocity into the reduced set yields The reduced set in equation (20) is a first approximation to the attenuating wave represented in the full equations (18) and (19). Our analysis of this section shows that the head and velocity are dependent upon two timescales: (i) a wave oscillation timescale t that corresponds to τ = O( ) and (ii) a wave attenuation timescale at τ = O (1). Thus, the head and velocity represented in the full equations (18) and (19) may be written as H = H(X, t, τ ) and V = √ c 4 V (X, t, τ ).…”

Section: Dimensional Analysismentioning

confidence: 98%

“…The dimensionless parameters from the momentum equation, C 1 , C 2 , and C 3 , are treated as O (1). The convective term V V X is typically neglected (e.g.…”

Section: Dimensional Analysismentioning

confidence: 99%

“…The analysis of a limited number of 2-D unsteady velocity profiles is in [7], where flow stability is described with respect to the 'spectrum' of turbulence at certain times and locations during the water hammer event. Other semi-analytic solutions include Adomian decomposition, which was applied in [1] to the problem of shock-wave propagation through an unconfined gas. Standard eigenfunction methods have also been used, and in [16] this approach was applied to the pressure-wave transmission through a conduit between two pressure-vessels.…”

Section: Introductionmentioning

confidence: 99%

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Multiple scales analysis of water hammer attenuation

Han¹,

Hansen

Kember

2011

Quart. Appl. Math.

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Abstract.A multiple scales analysis is used to construct a uniformly accurate approximation to water hammer pressure wave attenuation that is initiated by a sudden valve opening. The method of analysis is well suited to the study of a water hammer that possesses several time scales and is applied to a mild generalization of the classical equations. It should prove useful for finding attenuation curves when effects such as unsteady friction and fluid-structure interaction are added. The analytical results are numerically verified using the method of characteristics. Introduction.The nonlinearity of the classical water hammer model and its various generalizations has led to the development of increasingly sophisticated numerical and semi-analytical methods of analysis that have proven useful in engineering applications. However, it remains of interest to find uniformly accurate approximations whenever possible that are not simply ad hoc. The development of a multiple-scale decomposition of the water hammer equations, one that may be capable of generalization to various boundary and initial conditions and to features such as unsteady friction, is the focus of this paper.Numerical and semi-analytic studies have appeared for a wide range of water hammer problems. A semi-analytic solution was developed in [17] based on the use of a wavetracking algorithm to describe hydraulic transients in thin-walled conduits. The analysis of a limited number of 2-D unsteady velocity profiles is in [7], where flow stability is

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“…However, it is very difficult to find explicit solutions of nonlinear partial differential equations generally. The Adomain decomposition method is the most transparent method for solutions of linear and nonlinear problem (see [3,5,7,16,17]); however, this method is involved in the calculation of complicated Adomain's polynomials which narrow down its application. Recently, many researchers and engineers have done excellent work, such as Laplace decomposition algorithm [12,18].…”

Section: Introductionmentioning

confidence: 99%