Our physical intuition suggests that the steady state

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Unformatted text preview: equation is an equation which contains partial derivatives, such as the equation ∂u ∂2u = , ∂t ∂x2 in which u is regarded as a function of x and t. Unlike the theory of ordinary differential equations which centers upon one key theorem—the fundamental existence and uniqueness theorem—there is no real unified theory of partial differential equations. Instead, each type of partial differential equations exhibits its own special features, which usually mirror the physical phenomena which the equation was first used to model. Many of the foundational theories of physics and engineering are expressed by means of systems of partial differential equations. The reader may have heard some of these equations mentioned in previous courses in physics. Fluid mechanics is often formulated by the Euler equations of motion or the so-called Navier-Stokes equations, electricity and magnetism by Maxwell’s equations, general relativity by Einstein’s field equations. It is therefore important to dev...
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