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What are the differences between mathematical and physical equations (partial differential equations), Cauchy problem and initial-boundary value problem? Why discuss their uniqueness separately?
Cauchy problem is a definite solution problem with only initial conditions and no boundary conditions in partial differential equations.

Cauchy problem is a definite solution problem with only initial conditions and no boundary conditions in partial differential equations. Equation of Mathematical Physics, Li Mingqi Tian Taixin University of Electronic Technology Press, 40 pages: Initial Value Problem (or Cauchy Problem)

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