1.2. EXAMPLES 3 NOTATION. We write “PDE” as an abbreviation for both the singular “partial differential equation” and the plural “partial differential equations”. 1.2. EXAMPLES There is no general theory known concerning the solvability of all partial differential equations. Such a theory is extremely unlikely to exist, given the rich variety of physical, geometric, and probabilistic phenomena which can be modeled by PDE. Instead, research focuses on various particular partial differential equations that are important for applications within and outside of mathematics, with the hope that insight from the origins of these PDE can give clues as to their solutions. Following is a list of many specific partial differential equations of in- terest in current research. This listing is intended merely to familiarize the reader with the names and forms of various famous PDE. To display most clearly the mathematical structure of these equations, we have mostly set relevant physical constants to unity. We will later discuss the origin and interpretation of many of these PDE. Throughout x U, where U is an open subset of Rn, and t 0. Also Du = Dxu = (ux1 , . . . , uxn ) denotes the gradient of u with respect to the spatial variable x = (x1, . . . , xn). The variable t always denotes time. 1.2.1. Single partial differential equations. a. Linear equations. 1. Laplace’s equation Δu = n i=1 uxixi = 0. 2. Helmholtz’s (or eigenvalue) equation −Δu = λu. 3. Linear transport equation ut + n i=1 biuxi = 0. 4. Liouville’s equation ut n i=1 (biu)xi = 0.
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