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Partial Differential Equations

Part of the Mathematical Engineering Book Series

DOI: 10.1007/978-3-319-55212-5

ISBN: 978-3-319-55211-8

Publisher: Springer International Publishing - Springer Journals

Apr 30, 2017

1 - 10 of 11 Chapters

LitStream

Vector Fields and Ordinary Differential Equations

Epstein, Marcelo

Pages: 3 –24

[Although the theory of partial differential equations (PDEs) is not a mere generalization of the theory of ordinary differential equations (ODEs), there are many points of contact between both theories.]
LitStream

Partial Differential Equations in Engineering

Epstein, Marcelo

Pages: 25 –47

[Many of the PDEs used in Engineering and Physics are the result of applying physical laws of conservation or balance to systems involving fields, that is, quantities defined over a continuous background of two or more dimensions, such as space and time.]
LitStream

The Single First-Order Quasi-linear PDE

Epstein, Marcelo

Pages: 51 –74

[Remarkably, the theory of linear and quasi-linear first-order PDEs can be entirely reduced to finding the integral curves of a vector field associated with the coefficients defining the PDE. This idea is the basis for a solution technique known as the method of characteristics. It can be used...
LitStream

Shock Waves

Epstein, Marcelo

Pages: 75 –88

[When the projected characteristics of a quasi-linear first-order PDE intersect within the domain of interest, the solution ceases to exist as a well-defined function. In the case of PDEs derived from an integral balance equation, however, it is possible to relax the requirement of continuity and...
LitStream

The Genuinely Nonlinear First-Order Equation

Epstein, Marcelo

Pages: 89 –112

[The general non-linear first-order partial differential equation requires a deeper analysis than its linear and quasi-linear counterparts. Instead of a field of characteristic directions, the non-linear equation delivers a one-parameter family of directions at each point of the space of...
LitStream

The Second-Order Quasi-linear Equation

Epstein, Marcelo

Pages: 115 –130

[A careful analysis of the single quasi-linear second-order equation is the gateway into the world of higher-order partial differential equations and systems. One of the most important aspects of this analysis is the distinction between hyperbolic, parabolic and elliptic types. From the physical...
LitStream

Systems of Equations

Epstein, Marcelo

Pages: 131 –153

[Quasi-linear equations of order higher than 1 and systems of quasi-linear equations of any order can be subjected to a classification akin to that of the single second-order equation.]
LitStream

The One-Dimensional Wave Equation

Epstein, Marcelo

Pages: 157 –182

[The archetypal hyperbolic equation is the wave equation in one spatial dimension. It governs phenomena such as the propagation of longitudinal waves in pipes and the free transverse vibrations of a taut string. Its relative simplicity lends itself to investigation in terms of exact solutions of...
LitStream

Standing Waves and Separation of Variables

Epstein, Marcelo

Pages: 183 –208

[In spite of being limited to the solution of only certain types of PDEs, the method of separation of variables provides often an avenue of approach to large classes of problems and affords important physical insights. An example of this kind is provided by the analysis of vibration problems in...
LitStream

The Diffusion Equation

Epstein, Marcelo

Pages: 209 –238

[The archetypal parabolic equation is the diffusion equation, or heat equation, in one spatial dimension. Because it involves a time derivative of odd order, it is essentially irreversible in time, in sharp distinction with the wave equation. In physical terms one may say that the diffusion...

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