Elliptic partial differential equation

In summary, the conversation is about solving exercise 6.3 using equations 6.12 and 6.15 to obtain 6.16. The main challenge is organizing the terms to get a lot of occurrences of ##\phi'_i-\phi_i##, and using the equation 2(ab-cd)=(a-c)(b+d)+(a+c)(b-d) to simplify terms like ##(\phi'_i\phi'_{i-1}-\phi_i\phi_{i-1})##.
  • #1
ATY
34
1
Hey guys, so my professor told me to take a look at an equation, because he thinks that there is a mistake. We are basically talking about exercise 6.3 (on last image). The pictures will show you the text, so that you have all the information, that I have
11568ce5b2.png

http://puu.sh/mrNDl/ec19cdff63.png
8d9fc16cc0.png

http://puu.sh/mrNF3/7461f97ad4.png

So... I should "just use" 6.12 and 6.15 to get 6.16.
The point is that I have no clue how to do this.
my attemp was
http://puu.sh/mJ1HR/08899e8813.png
but, this gets super ugly when I start to insert the phi' from 6.15 and calculate everything. Is this even the right idea ?
 
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  • #2
I would concentrate on collecting up terms so that you get a lot of occurrences of ##\phi'_i-\phi_i##, and likewise with i-1.
You will find terms like ##(\phi'_i\phi'_{i-1}-\phi_i\phi_{i-1})##. Here you can use 2(ab-cd)=(a-c)(b+d)+(a+c)(b-d).
 

Related to Elliptic partial differential equation

1. What is an elliptic partial differential equation (PDE)?

An elliptic PDE is a type of differential equation that involves a second-order derivative and is characterized by a constant coefficient of the highest-order derivative. It can be written in the form of a linear combination of partial derivatives with the unknown function and its derivatives. Elliptic PDEs are often used in mathematical modeling to describe physical systems, such as heat conduction and electrostatics.

2. What are some examples of elliptic PDEs?

Some common examples of elliptic PDEs include the Laplace equation, Poisson's equation, and the Helmholtz equation. These equations are used to describe phenomena such as the steady-state temperature distribution in a metal plate, the electric potential in a region with fixed charges, and the propagation of waves in a medium.

3. What makes elliptic PDEs different from other types of PDEs?

Unlike hyperbolic and parabolic PDEs, elliptic PDEs do not involve time-dependent behavior. This means that solutions to elliptic PDEs do not depend on time and only describe the steady-state behavior of a system. Additionally, elliptic PDEs are generally well-posed, meaning they have unique solutions for given boundary conditions.

4. How are elliptic PDEs typically solved?

Elliptic PDEs are typically solved using numerical methods, such as finite difference, finite element, and spectral methods. These methods involve discretizing the PDE into a system of algebraic equations, which can then be solved using computer algorithms. Analytical solutions for elliptic PDEs are rare and are typically only possible for simple cases.

5. What are some real-world applications of elliptic PDEs?

Elliptic PDEs have a wide range of applications in physics, engineering, and other scientific fields. They are used in the design of electronic circuits, the analysis of groundwater flow, and the study of diffusion processes. They are also used in image processing and computer vision to enhance and analyze images. Additionally, many mathematical models in economics, biology, and social sciences involve elliptic PDEs.

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