# Linear partial differnetial equations (PDE's)

• Niles
In summary, the authors argue that in case (ii), where the boundary conditions are given a single point, the solution is given by -3y+g(x^2+y^2) subject to the condition g(1)=2.
Niles

## Homework Statement

Please take a look at the example at the bottom (at eq. 18:18, at page 619):

Q1: In case (ii), why do they add g(x^2+y^2)?
Q2: Why do they not add it in case (i)?
Q3: In case (ii), if I chose f(p) = p + 1 instead of f(p) = 2p, then the solution would be u(x,y) = 1+ x^2 +y^2 + g(x^2 + y^2), right?

I hope you can help me; I find this really hard to understand, and I've spent hours trying to find out, but I find that the book is poorly written. They don't emphasize the important things at all, and the reader is left behind with so many questions.

Last edited:
The difference between (ii) and (i) is that in (i) it is a function, but in (ii) it is a single point. Although I am not sure about this?

Can you guys give me a hint? I still can't see the system in it.

The difference between i) and ii) is that in i) the boundary conditions are given along a line y=0 and in ii) they are specified only a a single point. In the first case that's enough information to specify a unique solution. In the second case you can add a general function that solves the homogeneous equation (g(x^2+y^2)) subject to only one condition. I don't see why they are messing around with splitting g into two parts. That is confusing. I would just write the solution as -3y+g(x^2+y^2) subject to the condition g(1)=2. Your other form for Q3 is fine, but you forgot to put the -3y in it.

Thank you for taking the time to help me.

## What are linear partial differential equations?

Linear partial differential equations (PDE's) are mathematical equations that describe the relationship between a function and its partial derivatives. They are commonly used to model physical phenomena such as heat transfer, fluid dynamics, and electromagnetic fields.

## What is the difference between a linear and a non-linear PDE?

A linear PDE is one in which the unknown function and its derivatives appear in a linear fashion, meaning that they are only raised to the first power and do not appear inside of any other functions. In contrast, a non-linear PDE has terms that involve the function and its derivatives raised to different powers or inside of other functions.

## What is the importance of boundary conditions in solving PDE's?

Boundary conditions are necessary in solving PDE's because they provide information about the behavior of the function at the boundaries of the domain. These conditions, along with the PDE itself, help to determine a unique solution to the problem.

## How are PDE's solved?

PDE's can be solved analytically or numerically. Analytical solutions involve finding an exact expression for the unknown function, while numerical solutions involve approximating the solution using computational methods. The choice of method depends on the complexity of the problem and the desired level of accuracy.

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

PDE's are used to model a wide range of physical phenomena, including heat transfer in materials, fluid flow in pipes and channels, and the propagation of electromagnetic waves. They are also used in financial modeling, image processing, and machine learning.

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