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June 14th, 2017, 11:57 PM  #1 
Member Joined: Jun 2017 From: India Posts: 65 Thanks: 3  What are partial differential equations?
If the slope of the curve (derivative) at a given point is a number . 
June 15th, 2017, 05:35 AM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,953 Thanks: 800 
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 xyplane 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 xyplane 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 spacetime 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? Last edited by skipjack; June 15th, 2017 at 07:21 AM. 
June 15th, 2017, 06:37 AM  #3  
Member Joined: Jun 2017 From: India Posts: 65 Thanks: 3 
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...nningalgebra/ https://2012books.lardbucket.org/pdf...ngalgebra.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:
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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 07:27 AM.  
June 16th, 2017, 07:47 AM  #4 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,953 Thanks: 800 
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.

June 16th, 2017, 10:51 AM  #5 
Member Joined: Jun 2017 From: India Posts: 65 Thanks: 3 
OK , Thanks for the reply

June 17th, 2017, 07:32 AM  #6  
Member Joined: Jun 2017 From: India Posts: 65 Thanks: 3 
Maybe I can try to understand this. Quote:
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Last edited by skipjack; June 17th, 2017 at 07:58 AM.  
June 17th, 2017, 08:01 AM  #7 
Global Moderator Joined: Dec 2006 Posts: 18,586 Thanks: 1489 
The independent variable is x. The dependent variable is f(x). The given continuity condition is necessary, but not sufficient.

June 17th, 2017, 08:04 AM  #8  
Member Joined: Jun 2017 From: India Posts: 65 Thanks: 3 
Thanks a lot , another question . Quote:
In this , The first function is a function of several variables right ?  
June 17th, 2017, 10:12 AM  #9 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,953 Thanks: 800 
Yes, is a function of the two independent variables, x and t.

June 17th, 2017, 12:53 PM  #10  
Member Joined: Jun 2017 From: India Posts: 65 Thanks: 3  Yes,Q(x,t) is a function of the two independent variables, x and t. Quote:
 

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