Linear, nonlinear, homogeneous, nonhomogeneous PDE's

In summary, we have a series of equations that we need to classify as either nonlinear, linear inhomogeneous, or linear homogeneous. We determine this by looking at the order of the equation and whether it contains any nonlinear terms. After analyzing each equation, we find that (a) is linear homogeneous, (b) is linear inhomogeneous, (c) is linear homogeneous, (d) is nonlinear, (e) is linear inhomogeneous, (f) is nonlinear, (g) is linear inhomogeneous, and (h) is nonlinear. However, there is some discrepancy with (c) as the book classifies it as just linear, which may be due to different interpretations of what constitutes a nonlinear term.
  • #1
roldy
237
2
1. For each of the following equations, state the order and whether it is nonlinear, linear inhomogeneous, or linear homogeneous; provide reasons.

(a) ut-uxx+1=0
(b) ut-uxx+xu=0
(c) ut-uxxt+uux=0
(d) utt-uxx+x2=0
(e) iut-uxx+u/x=0
(f) ux(1+u2x)-1/2+uy(1+u2y)-1/2=0
(g) ux+eyuy=0
(h) ut+uxxxx+(1+u)-1/2=0


tial&space;x^{2}}-\frac{\partial&space;v^{2}}{\partial&space;x^{2}}+u+v=L(u)+L(v).gif

there it is linear
\partial&space;t}&space;-\frac{\partial^2&space;u}{\partial&space;x^2}+1\right&space;)=k(u).gif

therefore it is nonhomogeneous

I think I have the method down but I tried (c) and the book gave a different answer...

(c) linear homogeneous
the book says it's just linear, this is where I don't understand.
 
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  • #2
(a) linear homogeneous(b) linear inhomogeneous(c) linear homogeneous(d) nonlinear(e) linear inhomogeneous(f) nonlinear(g) linear inhomogeneous(h) nonlinear
 

Related to Linear, nonlinear, homogeneous, nonhomogeneous PDE's

1. What is the difference between linear and nonlinear PDE's?

Linear PDE's are those where the dependent variable and its derivatives appear only in a linear form. Nonlinear PDE's, on the other hand, have terms that involve products or powers of the dependent variable and its derivatives.

2. How do we determine if a PDE is homogeneous or nonhomogeneous?

A PDE is homogeneous if all of its terms have the same degree with respect to the dependent variable and its derivatives. If the terms have different degrees, the PDE is nonhomogeneous.

3. What is the general solution for a homogeneous PDE?

The general solution for a homogeneous PDE is a solution that satisfies the PDE for all possible values of the independent variables. It contains an arbitrary function of one or more variables, which can be determined by applying any necessary boundary or initial conditions.

4. How do we solve a nonhomogeneous PDE?

To solve a nonhomogeneous PDE, we first need to find the general solution of the corresponding homogeneous PDE. Then, we use the method of undetermined coefficients to find a particular solution for the nonhomogeneous PDE. The general solution is the sum of the particular solution and the general solution of the homogeneous PDE.

5. What are some real-world applications of PDE's?

PDE's have many applications in various fields of science and engineering. Some examples include modeling heat transfer in materials, predicting the spread of diseases, and simulating the behavior of fluids and gases. They are also used in image and signal processing, financial modeling, and many other areas.

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