Techniques in Partial Differential Equations |
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Page 32
... spacelike . More precisely , an initial curve is called spacelike with respect to a second- order hyperbolic equation in two independent variables if both character- istics when traversed in the direction of increasing t point to the ...
... spacelike . More precisely , an initial curve is called spacelike with respect to a second- order hyperbolic equation in two independent variables if both character- istics when traversed in the direction of increasing t point to the ...
Page 235
... spacelike at P. If one character- istic when traced in the positive direction on it leaves I from one side but the ... spacelike for the wave equation , while the t axis is timelike for it . Timelike arc Figure 9-3 Spacelike arc x It ...
... spacelike at P. If one character- istic when traced in the positive direction on it leaves I from one side but the ... spacelike for the wave equation , while the t axis is timelike for it . Timelike arc Figure 9-3 Spacelike arc x It ...
Page 236
... spacelike and timelike portions because there are always characteristics connecting the space- and timelike parts . If two data are prescribed on a spacelike part of an initial curve , only one datum can be prescribed on a timelike part ...
... spacelike and timelike portions because there are always characteristics connecting the space- and timelike parts . If two data are prescribed on a spacelike part of an initial curve , only one datum can be prescribed on a timelike part ...
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