Why Is the Source Term Crucial in Differential Equations?

In summary, the conversation discusses the concept of a source term in ODE and PDE and its impact on the difficulty of solving the equations. It is mentioned that if the region of observation does not contain the source, the equation becomes homogeneous and easier to solve. However, it is also noted that in many cases, a source must exist somewhere to create a disturbance. The conversation then shifts to discussing electrostatics with point charges and how solving without the source term may be sufficient, but can become more complicated with multiple charges and the need for boundary conditions. Ultimately, it is concluded that the quantity of information to be considered remains the same regardless of whether the source term is present or not.
  • #1
fisico30
374
0
Hello everyone,

my question is regarding the source term in ODE and PDE.
If the region where the phenomenon (wave field, temperature,...) is observed is circumscribed to a volume not containing its source, then the differential equation becomes homogeneous (no source term) and easier.
So why solve the inhomogeneous eqn ever, unless we are inside the source, since our volume of observation can always omit the source?
Clearly, a source must exist somewhere to create the dusturbance.
thanks
 
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  • #2
Let's consider electrostatics with point charges.
Then you are right: solving without source term is all that is needed.
But you should realize that the domain to be considered will become more complicated if many charges are involved. And you will need to use boundary conditions around each of these charges. The simplification is a pure illusion. But there are indeed many methods to solve these problems, each with there specific advantages in specific situations.

In the end, the quantity of information to be taken into acount remains the same.
 
  • #3


The source term in differential equations represents the external influences or factors that contribute to the phenomenon being studied. It is an essential component in understanding and modeling real-world systems and processes. While it may be easier to solve a homogeneous equation with no source term, it is important to consider the impact of external factors on the phenomenon being studied. In many cases, the source term cannot be omitted as it plays a crucial role in the behavior and evolution of the system. Therefore, solving the inhomogeneous equation is necessary to accurately describe and predict the behavior of the phenomenon. Additionally, even if the source is not within the observed volume, its influence can still be felt and must be accounted for in the equation. In conclusion, while it may be tempting to simplify the equation by omitting the source term, it is important to recognize its significance and address it in order to fully understand and model the system.
 

FAQ: Why Is the Source Term Crucial in Differential Equations?

1. What is the source term in a differential equation?

The source term in a differential equation is a function that represents the external forces or inputs acting on a system. It is often denoted as f(x) and can depend on one or more independent variables.

2. How does the source term affect the solution of a differential equation?

The source term can significantly affect the behavior of the solution to a differential equation. It can introduce or remove energy from the system, causing the solution to change over time. In some cases, the source term can also determine the stability of the solution.

3. Can the source term be negative in a differential equation?

Yes, the source term can be negative in a differential equation. This indicates that the external forces are acting against the direction of the system's motion, resulting in a decrease in energy or a damping effect.

4. How do you determine the source term in a real-world problem?

Determining the source term in a real-world problem requires a thorough understanding of the physical system and the forces at play. It often involves experimental data, theoretical models, and mathematical analysis to accurately represent the external inputs in the differential equation.

5. Can the source term change over time in a differential equation?

Yes, the source term can change over time in a differential equation. In some cases, it may be a function of time or other variables that affect the system. This can make the problem more complex, but it is necessary to accurately model real-world systems that are subject to changing external forces.

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