Read about Initial Value Problem and Boundary Value Problem from wikipedia or google around:
https://en.wikipedia.org/wiki/Initial_value_problem
https://en.wikipedia.org/wiki/Boundary_value_problem
If you want to see a wider perspective of these two classes of mathematical problems, get an access to the famous book
Numerical Recipes. The Art of Scientific Computing, 3rd Edition, 2007, ISBN 0-521-88068-8
or any former edition of it. Read the chapters about Partial Differential Equations. In addition to discussing initial value problems and boundary value problems, it also brings a lot of intuitions and knowledge on the Courant stability condition, numerical schemes for time integration and numerical dissipation. A lot to learn if you are interested in computational science.
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