# Initial Boundary Value Problem (IBVP)

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Phenomena in science, technology and nature like wave propagation, heat distribution are varying in space and time. The description of these phenomena in a physical domain requires the knowledge of their boundaries of the domain as well as the initial status or asymptotic behavior in the case of unbounded domains. Problems that model such properties are called Initial Boundary Value Problem (IBVP).

The governing equations of fluid flow are second order partial differential equations. The mathematical character of these equations dictate the numerical solution technique, the number of initial conditions as well as the boundary conditions. A classificiation of these equations is applied to the linearized form of the Navier-Stokes equation, which include:

• Hyperbolic Equation
• Parabolic Equation
• Elliptic Equation

Relevant for initial boundary value problems are parabolic equations which describe time dependent problems involving significant amount of diffusion. The governing equation for instance describes transient heat conduction in a plane wall:

\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2}

For this problem the initial condition at time t = 0 is needed and both ends of the walls need a boundary condition at time t > 0. This is a typical example for an Initial Boundary Value Problem (IBVP).

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