1.4. Steps to Formulate Boundary Value Problems

First is we have to establish a region of interest? So, where are we going to apply that boundary very prominent we have to determine for example, a region of interest would be a wave tank or tank where the water flows?

The first step in formulating a boundary value problem is to determine the specific area or domain in which you are interested. This can be any location where the phenomena you want to analyze occurs, such as a wave tank where water flows. Identifying this region is crucial because it sets the boundaries and conditions under which you will apply mathematical equations.

Think of setting up a science experiment in school. Before you start, you need to choose where you'll conduct the experiment—like a lab bench or classroom desk. In the same way, specifying the region of interest is like choosing where the 'action' of your problem will happen.

Secondly, we have to specify a differential equation that must be satisfied within the region.

Next, you need to specify a mathematical equation, usually a differential equation, that describes how the physical quantities of interest change within your established region. This equation represents the laws of physics governing the situation, such as fluid dynamics or heat transfer.

Consider this step similar to writing a recipe for a cake. The recipe tells you how the ingredients (like flour, sugar, and eggs) interact with each other to create a cake, just like the differential equation describes how the different properties of the fluid interact with each other in your analysis.

Thirdly, we have to select 1 or more solutions out of infinite number of solutions which are relevant to the physical problem under investigation.

After specifying the differential equation, you explore the potential solutions that satisfy this equation. Due to the nature of differential equations, there may be an infinite number of solutions. Therefore, you need to select those solutions that directly relate to the physical scenario you are investigating, ensuring they comply with the physical laws and conditions of your problem.

Imagine you're going to a library to find a book. There are thousands of books (potential solutions), but you’re only interested in those that cover a specific topic (relevant solutions). You sift through to find the few that will help you understand your particular subject better.

We must provide a boundary condition; a boundary condition like that the water entering here as a velocity let us say 3 meters per second here there is a wall, you know, this is open boundary, we can either specify a top surface or even a wall here.

The fourth step involves applying boundary conditions, which are constraints that delineate how the field variables (like velocity and pressure) behave at the boundaries of your region. These conditions are crucial because they narrow down the infinite solutions to just a few that can actually be realized based on the physical context. They dictate how the system interacts with its surroundings.

Think of boundary conditions like the rules of a game. Just as rules dictate how players can interact with each other and move in the game space, boundary conditions tell how the flow behaves at the edges of your designated area—like specifying that the water enters a tank at a certain speed.

In addition to the spatial boundary condition as I said, these are called the spatial boundary conditions or geometrical boundary condition, but there will also be a temporal boundary condition.

Beyond spatial conditions (which pertain to the physical space), temporal boundary conditions are essential for dealing with time-dependent problems. These specify states of the system at a starting time (initial conditions), which help define how the system evolves over time.

When you bake a cake, you might start with all ingredients mixed together (initial condition) at time zero and track how the batter rises and cooks over time (temporal conditions). Similarly, in fluid dynamics, you must establish initial states to understand how your system will behave as time progresses.

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