PDE Linear Equation Q: Homogeneous vs Nonhomogeneous

In summary, there is a confusion regarding the equation 2.2.4 in the document. The highlighted green section states that if f is equal to zero, the equation is homogeneous, but the underlined orange section states that the equation is nonhomogeneous. The explanation is that when the function u is set to zero, the equation becomes f(x,t)=0, which makes it non-homogeneous. However, if f(x,t) is not present in the equation, it becomes L(0)=0 and the equation is then homogeneous.
  • #1
Miike012
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My questions concerns the information in the document.
I highlighted the portion that is confusing me and a sample problem at the bottom.

Question:
Look at the equation 2.2.4 in the document.
When I set the function u equal to zero the equation becomes
0 = 0 + 0 + f(x,t) or f(x,t) = 0.

Now if you look in the document once more at the underlined green section is says that if f is equal to zero then the equation is homogeneous. However if you read the underlined orange section it says equation 2.2.4 is a nonhomogeneous equation...

This seems a bit contradictory. Some one please help me understand what they are saying.
 

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  • #2
You have shown that L(0) = f(x,t). Since that isn't zero, the equation is non-homogeneous. If that f(x,t) hadn't been in the equation, you would have gotten L(0)=0 and the equation would have been homogeneous.
 

1. What is the difference between a homogeneous and nonhomogeneous PDE linear equation?

A homogeneous PDE linear equation is one in which all of the terms contain the dependent variable and its derivatives. In other words, the equation is equal to zero. A nonhomogeneous PDE linear equation, on the other hand, has at least one term that does not contain the dependent variable or its derivatives.

2. How do you determine if a PDE linear equation is homogeneous or nonhomogeneous?

To determine if a PDE linear equation is homogeneous or nonhomogeneous, you can look at the highest order derivatives of the dependent variable. If all of the terms contain the same order of derivatives, the equation is homogeneous. If there is at least one term with a different order of derivatives, the equation is nonhomogeneous.

3. What are some examples of homogeneous PDE linear equations?

Some examples of homogeneous PDE linear equations include the Laplace equation, the heat equation, and the wave equation. These are all equations that have the same order of derivatives for all terms and are equal to zero.

4. How can you solve a homogeneous PDE linear equation?

To solve a homogeneous PDE linear equation, you can use separation of variables, Fourier series, or other methods such as the method of characteristics. These methods involve breaking down the equation into simpler forms and finding a particular solution that satisfies the equation.

5. What is the significance of homogeneous and nonhomogeneous PDE linear equations in science?

In science, both homogeneous and nonhomogeneous PDE linear equations are used to model physical phenomena and solve real-world problems. Homogeneous equations are often used to describe systems in equilibrium, while nonhomogeneous equations are used to model systems with external forces or sources. Understanding the difference between these two types of equations is important in accurately representing and predicting behavior in various scientific fields.

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