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differential equation

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Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005453
EISBN: 978-1-62708-196-2
... Abstract This article describes the models of physical phenomena involving solution of differential equations such as ordinary or partial differential equations. It reviews the basic concepts of differential calculus and tabulates the expansion of functions into power series. A table of linear...
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005431
EISBN: 978-1-62708-196-2
... Abstract Several methods are developed for the numerical solution of partial differential equations, namely, meshed-solution methods such as the finite-element method (FEM), finite-difference method, and boundary-element method; and numerical algorithms consisting of so-called meshed-solution...
Series: ASM Handbook
Volume: 4C
Publisher: ASM International
Published: 09 June 2014
DOI: 10.31399/asm.hb.v04c.a0005877
EISBN: 978-1-62708-167-2
... Abstract Electromagnetic problem solutions are based on the macroscopic theory of the continuous model for the electromagnetic field (EMF). It is described by a system of integral or partial differential equations for five vector quantities, namely, electric field strength, electric flux...
Series: ASM Handbook
Volume: 13A
Publisher: ASM International
Published: 01 January 2003
DOI: 10.31399/asm.hb.v13a.a0003642
EISBN: 978-1-62708-182-5
... Abstract Corrosion modeling is an essential benchmarking element for the selection and life prediction associated with the introduction of new materials or processes. These models are most naturally expressed in terms of differential equations or in other nonexplicit forms of mathematics...
Series: ASM Handbook
Volume: 4F
Publisher: ASM International
Published: 01 February 2024
DOI: 10.31399/asm.hb.v4F.a0007008
EISBN: 978-1-62708-450-5
... facilities, quench system manufacturers, and quench fluid suppliers. It describes the governing differential equations for the fluid flow and heat-transfer phenomena during quenching. The article also discusses different modeling categories to determine a CFD methodology for quenching. computational...
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Published: 01 December 2009
Fig. 10 Structure of a chemical vapor deposition (CVD) reactor simulation. PDE, partial differential equation More
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Published: 01 January 2006
. If one assumes θ to be the fraction of abraded oxide volume that has reformed at any time, and (1−θ) to be the fraction not yet reformed, then the above differential equation can be used to model the transient response. Source: Ref 49 , 52 More
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Published: 01 June 2012
assumes θ to be the fraction of abraded oxide volume that has reformed at any time, and (1−θ) to be the fraction not yet reformed, then the above differential equation can be used to model the transient response. Source: Ref 49 , 52 More
Series: ASM Handbook
Volume: 6
Publisher: ASM International
Published: 01 January 1993
DOI: 10.31399/asm.hb.v06.a0001480
EISBN: 978-1-62708-173-3
... of the enthalpy, thermal flux, and a distributed heat-source term, S (J/m 3 · s), the energy equation in differential form is: (Eq 3) H − ∇ · q − S = 0 or, in terms of temperature, it is: (Eq 4) ρ C p T ˙ − ∇ · ( − κ ∇ T ) − S = 0 This is a parabolic partial...
Series: ASM Handbook
Volume: 6A
Publisher: ASM International
Published: 31 October 2011
DOI: 10.31399/asm.hb.v06a.a0005587
EISBN: 978-1-62708-174-0
... In terms of the enthalpy, thermal flux, and a distributed heat-source term, S (J/m 3 · s), the energy equation in differential form is: (Eq 3) H − ∇   ⋅ q − S = 0 or, in terms of temperature, it is: (Eq 4) ρ C p T − ∇ ⋅ ( − κ ∇ T ) − S = 0...
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005451
EISBN: 978-1-62708-196-2
... transforms. laplace transformations THE METHODS OF LAPLACE AND FOURIER TRANSFORMATION and the Heaviside operational calculus are essentially aspects of the same method. This method simplifies solutions of such problems as ordinary differential equations with constant coefficients, linear...
Series: ASM Handbook
Volume: 6A
Publisher: ASM International
Published: 31 October 2011
DOI: 10.31399/asm.hb.v06a.a0005573
EISBN: 978-1-62708-174-0
... distribution around moving heat sources. The analytical solutions to the differential heat flow equation under conditions applicable to fusion welding were first presented by Rosenthal ( Ref 9 , 10 ), but the theory was later extended and refined by a number of other investigators ( Ref 3 , 11 , 12 , 13...
Series: ASM Handbook
Volume: 20
Publisher: ASM International
Published: 01 January 1997
DOI: 10.31399/asm.hb.v20.a0002445
EISBN: 978-1-62708-194-8
... are considered. Analog simulators must evaluate continuous changes in values in continuous time. The analog simulation engine must solve systems of simultaneous (usually nonlinear) differential equations. There will be one equation for each voltage node or current branch in the circuit. The equations...
Series: ASM Handbook
Volume: 4C
Publisher: ASM International
Published: 09 June 2014
DOI: 10.31399/asm.hb.v04c.a0005876
EISBN: 978-1-62708-167-2
... model. Usually, the mathematical model consists of continuous partial differential equations valid everywhere over the domain of the problem, while the numerical solution represents a discretized distribution of the actual values of fields (the numerical solution is not continuous; it exists only...
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005456
EISBN: 978-1-62708-196-2
... of friction do not change the direction of principal axes or distort the deformation field. This results in homogeneous deformation, with plane sections remaining plane. These assumptions converted the partial differential equations of plasticity to ordinary differential equations of first order that can...
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005426
EISBN: 978-1-62708-196-2
... and process design Background and History Computational fluid dynamics has as its objective the numerical solution of fluid-flow equations. The calculus problem of solving a coupled system of nonlinear partial differential equations (PDEs) for the variables of interest (e.g., velocity, pressure...
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005450
EISBN: 978-1-62708-196-2
... A 2 If ρ is constant, then: (Eq 65) V 1 A 1 = V 2 A 2 Differential Formulation The differential formulation may be derived by applying the transport equation to a differential control volume. For Cartesian coordinates, the continuity equation is: (Eq...
Series: ASM Handbook
Volume: 4B
Publisher: ASM International
Published: 30 September 2014
DOI: 10.31399/asm.hb.v04b.a0005993
EISBN: 978-1-62708-166-5
... to as Fourier's law of heat conduction. Table 1 presents the general form related to various coordinate systems. From Fourier's law and the first law of thermodynamics, the differential equations for heat conduction in solids can be derived ( Table 2 ). To determine temperature distribution due to heat...
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005449
EISBN: 978-1-62708-196-2
..., and is referred to as Fourier's law of heat conduction. Table 1 presents the general form related to various coordinate systems. From Fourier's law and the first law of thermodynamics, the differential equations for heat conduction in solids can be derived ( Table 2 ). To determine temperature distribution due...