Techniques in Partial Differential Equations |
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Page 69
... function ( 3-5 ) harmonic in a region R. An immediate consequence of it is the following theorem . Theorem 3-2 If a harmonic function vanishes ... FUNCTIONS IN BOUNDED REGIONS 69 3-4 Properties of Harmonic Functions in Bounded Regions.
... function ( 3-5 ) harmonic in a region R. An immediate consequence of it is the following theorem . Theorem 3-2 If a harmonic function vanishes ... FUNCTIONS IN BOUNDED REGIONS 69 3-4 Properties of Harmonic Functions in Bounded Regions.
Page 82
... function need not be harmonic . Solutions of equations other than the potential equation are frequently called Green's functions provided they have the " picking - out " or delta - function property and provided they satisfy appropriate ...
... function need not be harmonic . Solutions of equations other than the potential equation are frequently called Green's functions provided they have the " picking - out " or delta - function property and provided they satisfy appropriate ...
Page 90
... harmonic function is the real part of an analytic function . The harmonic function that solves a given problem - Dirichlet , Neumann , or otherwise -is necessarily single - valued . Hence the real part of the corresponding analytic function ...
... harmonic function is the real part of an analytic function . The harmonic function that solves a given problem - Dirichlet , Neumann , or otherwise -is necessarily single - valued . Hence the real part of the corresponding analytic function ...
Common terms and phrases
analytic arbitrary function assume axis called canonical form Cauchy problem Chap characteristic curves characteristic directions characteristic equations coefficients complete integral cone constant coordinates defined DEFINITION depend derivatives determine dimensions Dirichlet problem domain dual integral equations elliptic end point Euler's equation family of characteristics first-order PDE function u(x given Green's function Green's identity Hamilton-Jacobi equation Hamilton's equations harmonic function Hence homogeneous hyperbolic equation independent variables infinity initial curve integral equation integral surface Laplace transform Math method Neumann problem nonhomogeneous nonlinear obtain ODEs one-parameter family P₁ parameter Partial Differential Equations piston Poisson's potential equation prescribed quasi-linear region satisfy second-order linear solution solve space spacelike Suppose Theorem timelike V²u vanishes vector velocity wave equation xy plane zero