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1. What is the difference between boundary conditions and initial conditions?
2. Existence and uniqueness of solutions for initial value problems depends only on the equation. For boundary value value problems, it depends on the equation and the boundary condition.

3. (Original post by Fenugreek)
What is the difference between boundary conditions and initial conditions?
Wikipedia answers this quite well. "A boundary value problem has conditions specified at the extremes ("boundaries" of the independent variable in the equation whereas an initial value problem has all of the conditions specified at the same value of the independent variable (and that value is at the lower boundary of the domain, thus the term "initial" value)."

Where one should note that "independent variable" is usually taken to be multi-dimensional for PDEs.

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