Solving Diff. Eq: p & dp/dx Conditions, Mesh Creation

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In summary, you are looking for a paper that will help you solve the equation numerically. The paper you attached has a method for doing so. You need to understand the method before you can create the code.
  • #1
Mad_MechE
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I am trying to solve this differential equation: http://sites.google.com/site/theuntouchableproject/"
I believe i can come up with 2 boundary conditions for p and dp/dx.
i want to create a mesh such that i can solve this equation at different locations that i perscribe. What methods can i use to solve this equation... this is the first time i have ran into an equation of this form.
 
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  • #2
Hi there Mad MechE,

Your mention of a mesh makes me inclined to think you are asking about the most suitable numeric algorithm to solve the equation, is it the case?

In any case the ODE looks to me rather well behaved and depending on the accuracy you are looking for any standard method would do your purposes.

I attach a copy of the resulst given by the quickest of the alogorithms I know, i.e. just use MathCAD. But any home built algorithm would do the same.

Cheers

Muzialis
 

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  • #3
Why do you have partial derivative signs on the left side when "x" appears to be the only variable? If there is some other variable, say "y" or "t", since it does not appear in the equation, you can just ignore it.

We can immediately integrate both sides with respect to x to get
[tex]h^3\frac{dp}{dx}= 6U\mu h[/itex]
so
[tex]\frac{dp}{dx}= \frac{6U\mu}{h^2}+ C[/tex]
and, since [itex]h= x_0+ x^2/2R[/itex], that is
[tex]\frac{dp}{dx}= \frac{6U\mu}{(x_0+ x^2/2R)^2}[/tex]
and you just have to integrate, using, I would recommend, "partial fractions".
 
  • #4
I think i may have over simplified the problem... which is good because now i have a baseline for the data i want... later tonight i will write up the problem i am having and upload them!
Thank you all for your help!

MT
 
  • #5
Alright... i found this paper that has the ability to solve my problem but i don't know any real way about setting it up to do it numerically. I have attached that paper such that it can be referenced to. I read it several times and i don't think i fully comprehend the way the solution was set up. I was hoping that i could produce some MATLAB code to solve the problem but i need to understand it first. Could you please assist me in figuring out what it all means?

Thanks

MT
 

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  • Numerical solution.pdf
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Related to Solving Diff. Eq: p & dp/dx Conditions, Mesh Creation

1. What is a differential equation and why is it important in science?

A differential equation is a mathematical equation that describes how a system changes over time or space. It is important in science because it allows us to model and understand complex systems and phenomena, such as population growth, chemical reactions, and physical processes.

2. What are initial conditions and why do they matter in solving differential equations?

Initial conditions are the values of the dependent variable and its derivatives at a specific point in time or space. They matter in solving differential equations because they serve as the starting point for the solution and help determine the behavior of the system.

3. What is the significance of boundary conditions in solving differential equations?

Boundary conditions are constraints that specify the behavior of the system at the boundaries of the domain. They are important in solving differential equations because they help determine a unique solution and ensure that the solution is physically meaningful.

4. How is a mesh created and why is it important in solving differential equations?

A mesh is a discretization of the domain into smaller elements or cells. It is important in solving differential equations because it allows us to approximate the solution at discrete points and is necessary for numerical methods to solve the equations.

5. What are some common numerical methods for solving differential equations?

Some common numerical methods for solving differential equations include Euler's method, Runge-Kutta methods, and finite difference methods. These methods use a step-by-step approach to approximate the solution at discrete points, making it possible to solve complex differential equations that have no analytical solution.

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