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linear differential equations

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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
... on the mathematical principles of solving differential equations. It also reviews linear structural problems to illustrate the concept of the FEMs. partial differential equation meshed-solution method finite-element method finite-difference method boundary-element method ENGINEERING MODELS are often...
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
... and quantities of the EMF, their usage for the determination of the EMF distribution is possible only in a limited number of cases characterized by linear properties and simple geometry. Thus, for the analysis of induction heating phenomena, the Maxwell's equations in differential form are used more frequently...
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: 4C
Publisher: ASM International
Published: 09 June 2014
DOI: 10.31399/asm.hb.v04c.a0005876
EISBN: 978-1-62708-167-2
.... As a result, the partial differential equation can be converted to a pair of ordinary differential equations, linked by a constant known as the separation constant, and so solved. A general solution then is taken that consists of a suitable linear combination of the products of the pairs of particular...
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: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005426
EISBN: 978-1-62708-196-2
... Abstract Computational fluid dynamics (CFD) is a computationally intensive three-dimensional simulation of thermal fluids systems where non-linear momentum transport plays an important role. This article presents the governing equations of fluid dynamics and an introduction to the CFD...
Series: ASM Handbook
Volume: 20
Publisher: ASM International
Published: 01 January 1997
DOI: 10.31399/asm.hb.v20.a0002444
EISBN: 978-1-62708-194-8
... design 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, and temperature) is transformed...
Book Chapter

Series: ASM Handbook
Volume: 18
Publisher: ASM International
Published: 31 December 2017
DOI: 10.31399/asm.hb.v18.a0006359
EISBN: 978-1-62708-192-4
... W = ∂ W ∂ N d N + ∂ W ∂ q d q Letting b = 1, and inserting Eq 25 into Eq 24 , the differential relationship becomes: (Eq 26) d W = W N d N + ( 3 c W q ) d q The above equation can then be integrated for particular...
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005439
EISBN: 978-1-62708-196-2
.... It is a linear differential equation for which there are several well-known solutions. Diffusion in Ternary and Higher-Order Multicomponent Alloys Binary alloys have only one independent variable, because the mole fraction or weight fraction of one component can always be used to calculate the fraction...
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
... Solution of Partial Differential Equations by the Finite Element Method , Cambridge Press , 1987 8. Hughes T.J.R. , The Finite Element Method: Linear and Static and Dynamic Finite Element Analysis , Prentice-Hall , 1987 9. Rosenthal D. , The Theory of Moving Sources of Heat...
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
... Differential Equations by the Finite Element Method , Cambridge Press , 1987 8. Hughes T.J.R. , The Finite Element Method: Linear and Static and Dynamic Finite Element Analysis , Prentice-Hall , 1987 9. Rosenthal D. , The Theory of Moving Sources of Heat and Its Application...
Series: ASM Handbook
Volume: 19
Publisher: ASM International
Published: 01 January 1996
DOI: 10.31399/asm.hb.v19.a0002383
EISBN: 978-1-62708-193-1
... modulus. The total strain consists of a linear and a nonlinear part: (Eq 6) ε = ε  elastic  + ε  plastic  Other common forms of the Ramberg-Osgood equations may be used instead. Equation 6 is used here to improve insight into the equations that follow. The total strain...
Series: ASM Handbook
Volume: 8
Publisher: ASM International
Published: 01 January 2000
DOI: 10.31399/asm.hb.v08.a0009218
EISBN: 978-1-62708-176-4
... involves the use of a creep equation (i.e., an analytical expression for creep strain as a function of time, stress, and temperature). Because these equations can generally be differentiated with respect to time to yield expressions for creep rate, they are more useful for design purposes than the other...
Book: Casting
Series: ASM Handbook
Volume: 15
Publisher: ASM International
Published: 01 December 2008
DOI: 10.31399/asm.hb.v15.a0005233
EISBN: 978-1-62708-187-0
... problem of solving a coupled system of nonlinear partial differential equations (PDEs) for the variables of interest (e.g., velocity, pressure, and temperature) is transformed into an algebra problem of solving a large system of simultaneous linear equations for discrete unknowns that represent the state...
Series: ASM Desk Editions
Publisher: ASM International
Published: 01 November 1995
DOI: 10.31399/asm.hb.emde.a0003024
EISBN: 978-1-62708-200-6
... are combined to eliminate ε M and ε VK to obtain a single second-order linear differential constitutive equation, which is solved under appropriate boundary conditions. As with the simpler mechanical analogs, the four-parameter element response to constant stress, strain rate, and instantaneous fixed...
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: 20
Publisher: ASM International
Published: 01 January 1997
DOI: 10.31399/asm.hb.v20.a0005752
EISBN: 978-1-62708-194-8
...DOI: 10.31399/asm.hb.v20.a0005752 Abbreviations and Symbols a crack length; thermal diffusivity of a product DAE differential algebraic equation lhc interlaminar fracture toughness (mode I, A ampere peel; mode II, shear; mode m, scissor dB decibel shear) A area; ratio of the alternating stress DBTT...
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005437
EISBN: 978-1-62708-196-2
... with respect to ζ were eliminated from the differential equation, the end conditions in Eq 24 cannot be satisfied. It is expected that there will be a small region near the boundary, called a boundary layer, where the solution is two-dimensional. In this region, the axial and radial conduction terms must...
Book: Casting
Series: ASM Handbook
Volume: 15
Publisher: ASM International
Published: 01 December 2008
DOI: 10.31399/asm.hb.v15.a0005208
EISBN: 978-1-62708-187-0
... ). The simplification to a one-dimensional problem came at a price, however. Since the derivatives with respect to ζ were eliminated from the differential equation, the end conditions in Eq 24 cannot be satisfied. It is expected that there will be a small region near the boundary, called a boundary layer, where...