2. Classification of Physical Problems

The classification of the physical problems can be classified into equilibrium problems or propagation problems, the third is Eigen problems. So, any physical problems can be classified into 3 different forms.

Physical problems can be divided into three main categories: equilibrium problems, propagation problems, and Eigen problems. Understanding these categories helps to identify the approach and equations needed to solve these problems in various fields of science and engineering.

Think of it like having three different types of cooking techniques. Just as you might use boiling for pasta, frying for meat, and baking for cake, different physical problems require different methods to reach a solution depending on their nature.

Equilibrium problems are the steady state problems in closed domain. Steady state means there is no dependence on time; there is an equilibrium. An example is the Laplace equation of such type of problems.

Equilibrium problems are those that remain constant over time, meaning the system is in a stable state. The Laplace equation is a key example, used in scenarios such as heat distribution in a solid object where the temperature does not change with time. The solution of such problems is governed by an elliptic partial differential equation, which takes boundary conditions into account.

Consider a bathtub filled with water that is not being filled or drained. The water level stays constant (equilibrium), and the temperature of the water stabilizes as it reaches uniform warmth. The Laplace equation helps describe this condition mathematically.

Propagation problems are initial value problems in open domains. Open refers to one of the independent variables, for example, time, where the solution is progressed from an initial stage.

Propagation problems involve scenarios where a system evolves over time. These problems often require specifying initial conditions, such as the starting values at time t = 0, and understanding how these values change as time progresses. Examples include wave propagation and diffusion processes. The solutions to these problems depend on boundary conditions as well.

Imagine planting a seed in the ground. At time t = 0, there is just a seed, but as time passes, it sprouts and grows, representing how solutions to propagation problems evolve over time much like the development of the plant.

Eigen problems are those where the solution exists only for special values of the parameter of the problem. These special values are called Eigenvalues, requiring an additional step to determine them in the solution procedure.

Eigen problems are unique as they focus on specific scenarios where solutions are valid only for certain parameter values, termed Eigenvalues. This requires solving additional equations to determine these values before finding the solution to the main problem. Examples include problems in quantum mechanics and vibration analysis.

Think of a music note produced by a guitar string. The string vibrates at specific frequencies (Eigenvalues) to create particular notes. If you strum the string in a way that corresponds to its natural frequency, you will hear the beautiful sound; otherwise, the string might just vibrate without a clear note.

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