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June 14th, 2017, 10:57 PM   #1
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What are partial differential equations?

If the slope of the curve (derivative) at a given point is a number .

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June 15th, 2017, 04:35 AM   #2
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That's a very strange question. Yes, if y is a function of a single variable, say y= f(x), then the graph of y= f(x) is a one dimensional curve in the two dimensional xy-plane and the slope of the tangent line to the graph of y= f(x) at any point, $\displaystyle (x_0, f(x_0))$, $\displaystyle f'(x_0)$, is a single number, the derivative of the function at that x, and we can think of the derivative function as giving that derivative at any x. An equation involving that derivative function, in particular, an equation having that derivative function as "unknown", is an "ordinary differential equation".



But that has nothing to do with "partial differential equations". A function of several variables, say w= f(x, y, z), has "partial derivatives" because its graph would NOT be a curve in the xy-plane but a three dimensional surface in the four dimensional xyzw space. At each $\displaystyle (x_0, y_0, z_0, f(x_0, y_0, z_0))$ point, we could move in three different directions parallel to the three x, y, and z coordinate axes. There may be a different slope in each of those directions, the partial derivatives $\displaystyle f_x(x_0, y_0, z_0)$, $\displaystyle f_y(x_0, y_0, z_0)$, and $\displaystyle f_z(x_0, y_0, z_0)$. An equation that involves the derivatives of a function of several variables is a "partial differential equation".

Because our universe is a "four dimensional space-time continuum", physics problems tend to involve "partial differential equations" with independent variables, x, y, z, and t. Most famously, the "wave equation"
$\displaystyle \frac{\partial^2 \phi}{\partial x^2}+ \frac{\partial^2 \phi}{\partial y^2}+ \frac{\partial^2 \phi}{\partial z^2}= \frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2}$
and the "diffusion" or "heat equation"
$\displaystyle \frac{\partial^2 \phi}{\partial x^2}+ \frac{\partial^2 \phi}{\partial y^2}+ \frac{\partial^2 \phi}{\partial z^2}= \kappa \frac{\partial \phi}{\partial t}$.

Where in the world did you hear about "partial differential equations" without hearing what they, or partial derivatives, are?
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Last edited by skipjack; June 15th, 2017 at 06:21 AM.
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June 15th, 2017, 05:37 AM   #3
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Thanks for that leet reply. It looks a bit complicated to me . I have to reread your reply a couple of times in order to completely understand it.

As part of computer science syllabus, we had numerical analysis.
During those days, no quality books were available to learn it.

Later, I found a good book like,

Numerical Analysis. NINTH EDITION. Richard L. Burden

This is what I had to do with a programming language like C.

For that, I had to understand the binary operations inside the computer. I learned a few things like these.

x is binary
x is a quantity
x is a point mass
x is digital logic
x is electrons

Then I had to refresh my math with books like these.

https://2012books.lardbucket.org/boo...nning-algebra/

https://2012books.lardbucket.org/pdf...ng-algebra.pdf







Numerical Analysis. NINTH EDITION. Richard L. Burden

http://ins.sjtu.edu.cn/people/mtang/textbook.pdf


I also learned when numerical methods are used.

Quote:
for example, for a polynomial,

a solution of a polynomial equation is also called a root of the polynomial.

A value for the variable that makes the polynomial zero.

If you can't find an exact expression, then you can use numerical methods to get approximations.

With numerical methods, you can choose how close to zero you want, and it will give you a value that's at least that close
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The slope of the curve (derivative) at a given point is a number
Quote:
The (standard) calculus is broken into two pieces.

i) Differential calculus - which is looking at the instantaneous rates of change of objects with respect to some variables. We have the notion of the derivative of a function.
ii) Integral calculus - which is calculating the area under curves, calculating volumes and so on. This is all given in terms if the (indefinite or definite) integral of a function.

The two notions are tied together via the fundamental theorem of calculus. This says that the derivative and indefinite integral are basically mutual inverses (but not quite)
Quote:
An ordinary differential equation (ODE or just DE) is a system with the following ingredients:


An independent variable (usually t think ”time” or x think ”position”) that derivatives are taken with respect to.
A dependent variable, i.e. function of the independent variable, e.g. y = y(t) ”the variable y which is a function of t”.

A multi-variable function F that describes a relationship between the derivatives of the dependent variable (taken with respect to the independent variable)

F(t, y, dy/dt, d2y/dt2 , . . . ,dny/dtn ) = 0.
Quote:
An equation containing the derivatives of one or more dependent variables, with respect to one or more independent variables, is said to be a differential equation


I still have a hard time understanding a system with a lot of variables. I can't imagine anything like that except, three strings working together or something like that?



Is that ok to think about it that way?


Last edited by skipjack; June 15th, 2017 at 06:27 AM.
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June 16th, 2017, 06:47 AM   #4
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This is very puzzling and little concerning. You want to work with "partial differential equations", typically an upper class college course, but are asking questions about basic, eighth grade algebra! You are apparently trying to cover topics that would typically take eight to nine years very quickly. Go slowly and make sure you understand what you have learned before you go on to another topic.
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June 16th, 2017, 09:51 AM   #5
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OK , Thanks for the reply
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June 17th, 2017, 06:32 AM   #6
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Maybe I can try to understand this.

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A derivative simply specifies the rate at which a quantity changes. In math terms, the derivative of a function f(x), which is depicted as df(x)/dx, or more commonly in this book, as f'(x), indicates how f(x) is changing at any value of x. The function f(x) has to be continuous at a particular point for the derivative to exist at that point.
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A derivative simply specifies the rate at which a point mass quantity changes. In math terms, the derivative of a function f(x), which is depicted as df(x)/dx, or more commonly in this book, as f'(x), indicates how f(x) is changing at any value of x. The function f(x) has to be continuous at a particular point for the derivative to exist at that point.
Which one is the independent variable and the dependent variable here?

Last edited by skipjack; June 17th, 2017 at 06:58 AM.
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June 17th, 2017, 07:01 AM   #7
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The independent variable is x. The dependent variable is f(x). The given continuity condition is necessary, but not sufficient.
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June 17th, 2017, 07:04 AM   #8
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Thanks a lot , another question .

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If y is a function of a single variable, say y= f(x), then the graph of y= f(x) is a one dimensional curve in the two dimensional xy-plane and the slope of the tangent line to the graph of y= f(x) at any point, (x0,f(x0)), f′(x0), is a single number, the derivative of the function at that x, and we can think of the derivative function as giving that derivative at any x. An equation involving that derivative function, in particular, an equation having that derivative function as "unknown", is an "ordinary differential equation".



But that has nothing to do with "partial differential equations". A function of several variables, say w= f(x, y, z), has "partial derivatives" because its graph would NOT be a curve in the xy-plane but a three dimensional surface in the four dimensional xyzw space. At each (x0,y0,z0,f(x0,y0,z0)) point, we could move in three different directions parallel to the three x, y, and z coordinate axes. There may be a different slope in each of those directions, the partial derivatives fx(x0,y0,z0), fy(x0,y0,z0), and fz(x0,y0,z0). An equation that involves the derivatives of a function of several variables is a "partial differential equation"

In this ,



The first function is a function of several variables right ?
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June 17th, 2017, 09:12 AM   #9
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Yes, is a function of the two independent variables, x and t.
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June 17th, 2017, 11:53 AM   #10
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Yes,Q(x,t) is a function of the two independent variables, x and t.


Quote:
Partial derivatives are a kind of extension to functions of several variables such as f(x,y) . if you keep y constant, e.g. at Y1 then the partial derivative wrt x of f is the derivative of the now single variable function f(x,Y1)
I Don't understand this
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