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Initial and Boundary-Value Problems

As it was mentioned above the solution of PDEs involve arbitrary functions. So, in order to solve the system in question completly, additional conditions are needed. These conditions can be given in the form of initial or boundary conditions. Initial conditions define the values of the dependent variables at the initial stage (e.g. at ), whereas the boundary conditions give the information about the value of the dependent valiable or its derivative on the boundary of the area of interest. Typically, one distinguishes

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Drichlet conditions specify the values of the dependent variables of the boundary points.

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Neumann conditions specify the values of the normal gradients of the boundary.

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Robin conditions defines a linear combination of the Drichlet and Neumann conditions.

From the computational point of view it is useful to classify the PDE in question in terms of initial value or boundary value problem.

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Initial value problem: PDE in question describes time evolution, i.e., the initial space-distribution id given; the goal is to find how the dependent variable propagates in time. Example: The diffusion equation.

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Boundary value problem: A static solution of the problem should be found in some region-and the dependent variable is specified on its boundary. Example: The Laplace equation.

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Next: Finite difference method Up: Intorduction Previous: Linear Second-Order PDEs