Please help to solve this problem


Please advise to find the value of x to substitute

 

It's a function- x is a value _YOU_ assign.

 

Look at this example:
https://www.ni.com/docs/en-US/bundl..._finite_difference_method_for_laplace_eq.html

 

View attachment 294303
Please help to solve this problem
View attachment 294304

Please advise to find the value of x to substitute

Hi there,

I have to ask who gave you this problem to solve.
I ask because the numerical derivative equation in the drawing you provided is just first order (with the left f'(x)) and also it is for just a single variable.
To start, to solve a PDE like this you would need a second order, two variable numerical derivative. That's because both derivatives in the Laplace Equation are second order and depend on two variables, x and y. This means you will have two different formulas one for Uxx and another for Uyy. The only difference between the two is which variable gets incremented by 'h' and/or 'k' and which is taken as a constant. You could try to derive them yourself or i guess just look them up. If you try to derive them yourself i think you will get more insight out of this. You can start by deriving the second order central means formula for one variable, then go to the second order central means formula for two variables, and then you would also know how to move to three or more variables.

Luckily the Laplace Equation is static so it's easier to solve. This problem looks like it might be a potential problem where the field distribution is to be found, or at least a working formula for that.
It's been a while since i used the finite difference method in this fashion though so i'd have to look it up too. Most of the time in the past i would use the same numerical derivatives but i would plot the entire field so i can get an eye on the structure.
The Laplace Equation is pretty interesting and simpler because there is no time variable.

Did you know you could find an article on this right on this website if you do a search.

 

Hello again,

One thing i forgot to mention.
Are the stated 'boundary' conditions for this problem correct?
The reason i ask is because it look like there are only two 'points' given, not entire boundaries.
Im not sure if you can solve Laplaces Equation with just two points given for boundaries.
Usually we see something like these:
U(0,y)=1, which would mean when x=0 the entire set of the function values along y are equal to 1, so that is along the entire side of the square.
U(x,0)=0, which would mean when y=0 the entire set of the function values along x are equal to 0, which is along the entire bottom of the square.
So we have one entire side and one entire bottom here to work with. We may even need the conditions for the other two sides also.
Maybe this problem is a degenerate case or something.

 

Hi there,

I have to ask who gave you this problem to solve.
I ask because the numerical derivative equation in the drawing you provided is just first order (with the left f'(x)) and also it is for just a single variable.
To start, to solve a PDE like this you would need a second order, two variable numerical derivative. That's because both derivatives in the Laplace Equation are second order and depend on two variables, x and y. This means you will have two different formulas one for Uxx and another for Uyy. The only difference between the two is which variable gets incremented by 'h' and/or 'k' and which is taken as a constant. You could try to derive them yourself or i guess just look them up. If you try to derive them yourself i think you will get more insight out of this. You can start by deriving the second order central means formula for one variable, then go to the second order central means formula for two variables, and then you would also know how to move to three or more variables.

Luckily the Laplace Equation is static so it's easier to solve. This problem looks like it might be a potential problem where the field distribution is to be found, or at least a working formula for that.
It's been a while since i used the finite difference method in this fashion though so i'd have to look it up too. Most of the time in the past i would use the same numerical derivatives but i would plot the entire field so i can get an eye on the structure.
The Laplace Equation is pretty interesting and simpler because there is no time variable.

Did you know you could find an article on this right on this website if you do a search.

This question from City & Guilds Level 5 exam paper
The hand written equation in my post add by me because I have not familiar the equation , Now it is very clear that it needs two variable equation

I am still studding how to solve this problem
Can you please provide me a YouTube video that helps to solve this kind of problem . I searched but I could not found

Thank you so much for detailed explanation and advise
Please help to me if you have free time
Thanks again and again

 

This question from City & Guilds Level 5 exam paper
The hand written equation in my post add by me because I have not familiar the equation , Now it is very clear that it needs two variable equation

I am still studding how to solve this problem
Can you please provide me a YouTube video that helps to solve this kind of problem . I searched but I could not found

Thank you so much for detailed explanation and advise
Please help to me if you have free time
Thanks again and again

Hi,

Hey that's a good idea look for a YouTube version.
I can show an example that is slightly different but shows how to proceed with these problems. It's not really that hard, but takes some care in the way you set up the equations and such.
One nice thing, they don't seem to want you to have to actually calculate any new field points
The example i can show will be more typical so you may have to search around the web for a possible solution using just two points instead of actual full boundaries. Maybe that just means more variables to solve for, but then again they do mention the quantity of variables is four, so it doesn't make sense to me yet.

I'll try to get back here later with an example. You should know though that a study of PDE's is much more than just solving one problem or even just one type of equation. If you really want to study them you will have quite a bit of work ahead of you. The good news is that they can describe a lot of physical situations.

 

This question from City & Guilds Level 5 exam paper
The hand written equation in my post add by me because I have not familiar the equation , Now it is very clear that it needs two variable equation

I am still studding how to solve this problem
Can you please provide me a YouTube video that helps to solve this kind of problem . I searched but I could not found

Thank you so much for detailed explanation and advise
Please help to me if you have free time
Thanks again and again

Hello again,

I was thinking maybe you want to try to come up with the second derivative for a single variable first. If not that's ok but you may find this interesting.
Attached is a drawing that will help figure this out.
First you can note that to get the central means formula you take one step right and one step left, then subtract, then divide by the total steps between the two samples, which would be 2*h as you know.
Well, to get the central means formula for the second derivative it would be the same, except you would do it with the derivatives you found already for the first derivatives at two different values of 'x'. So if we call the first samples y1 and y2, the math would look like so:
Dy=(y2-y1)/(2*h)

and now to get the second derivative we just use two of those:
Dyy=(Dy2-Dy1)/(2*h)

Dy is y/dx
Dyy is d^2y/dx^2

You can look at the diagram while you think about this. The horizontal is 'x' as usual, and the vertical is 'y' as usual.
You can see that we are after the second derivative at the red point in the center. The two gray points are the points where we calculated the two first derivatives. We use steps of 'h' each time we make a step. After that we make a simple simplification.

You only have to come up with the second derivative for one variable, so do you want to try this yourself first? It should be informative.