What are partial differential equations?

In summary: 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 functionThe slope of the curve (derivative) at a given point is a number
  • #1
awholenumber
200
10
If the slope of the curve (derivative) at a given point is a number .
 
Physics news on Phys.org
  • #2
What is your question ?
 
  • #3
What are partial differential equations ?:cry:
 
  • #4
Do you want PF to play textbook for you ? What did you find so far and what is unclear ?
 
  • #5
I was wondering if anybody could explain this in this context
If the slope of the curve (derivative) at a given point is a number
 
  • #6
rosekidcute said:
If the slope of the curve (derivative) at a given point is a number .
This question has no relation to the title. Partial differentia equations are not the same thing as derivatives (even though they are related).
 
  • #7
I was hoping somebody would explain the relationship between the two
 
  • #8
Where are you in your curriculum ? You seem to know what a derivative is -- for a function of a single independent variable, e.g. ##f(x)##. Right ?

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 ##Y_1## then the partial derivative wrt x of ##f## is the derivative of the now single variable function ##f(x, Y_1)##.

But you have googled such things yourself already, so I ask again: what did you find so far and what is unclear ?

(and perhaps also: what textbook are you using ?)
 
  • #9
Thanks for the reply ,

I was going through this website here ,
https://17calculus.com/differential-equations/

From a pdf :

Basic Terminology
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.
Is it possible to understand partial differential equations in terms of a working example of something ?

string.jpg


This is the only thing i could think of .
 
  • #10
rosekidcute said:
Thanks for the reply ,
I was going through this website here ,
https://17calculus.com/differential-equations/
Wrong website. This is about differential equations. Before going through that you should become familiar with differentials. An introductory calulus textbook or website if the Wiki link I gave you is too abstract.

Is it possible to understand partial differential equations in terms of a working example of something ?
Sure. Once you're comfortable wit Algebra, Calculus I and II, there are plenty in Calculus III on partial derivatives

I'm not trying to discourage you: it's more a matter of finding out what foundations you have as a start for your adventure

Not clear what you aim at with
rosekidcute said:
This is the only thing i could think of .
I suspect you have some specific topic that interests you ?
 
  • #11
It is ok , i sort of understand it with my new avatar .
Some points i tried to narrow down .

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

Partial derivative - A derivative of a function of two or more variables with respect to one variable, the other(s) being treated as constant.
An equation that involves the derivatives of a function of several variables is a "partial differential equation"
 
  • Like
Likes BvU
  • #12
rosekidcute said:
It is ok , i sort of understand it with my new avatar .
Some points i tried to narrow down .

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

Partial derivative - A derivative of a function of two or more variables with respect to one variable, the other(s) being treated as constant.
An equation that involves the derivatives of a function of several variables is a "partial differential equation"
All you're doing here is giving definitions of a few terms, a very long way from being able to solve ordinary differential equations (involving single-variable functions) or partial differential equations (involving multi-variable functions).

BTW, what does having a new avatar have to do with understanding anything?
 
  • #13
All you're doing here is giving definitions of a few terms, a very long way from being able to solve ordinary differential equations (involving single-variable functions) or partial differential equations (involving multi-variable functions).

BTW, what does having a new avatar have to do with understanding anything?

Thanks ,

First you have a function f(x)
Then you can take the derivative of that function from first principles
Then you get f'(x) or dy/dx

That dy/dx , the slope of the curve (derivative) at a given point is a number .

I have been unnecessarily thinking about the cause behind why differential equations have dy/dx or f'(x) in them .I guess there is no need to think about it that way .
 
  • #14
In a sense, these are equations where the " input data" is given in terms of partial derivatives: you are given a numerical relation involving an unknown function and some of its partial derivatives and your goal is to identify the function(s) satisfying said relations, e.g., you are given that

## a(x) \frac {\partial^2 f}{\partial x \partial y} + b(x)f=c(x); a(x), b(x), c(x)## are functions, and you need to figure out which functions f satisfy the equation. One you have likely seen is the Laplacian of a function ## U=U(x,y,...): U_{xx}+ U_{yy}+...=0## for a function U of many variables, or the Cauchy-Riemann equations , :https//en.wikipedia.org/wiki/Laplace_operator and https://en.wikipedia.org/wiki/Cauchy–Riemann_equations respectively
 
  • #15
Thanks :-)
 
  • #16
rosekidcute said:
First you have a function f(x)
Then you can take the derivative of that function from first principles
Then you get f'(x) or dy/dx

That dy/dx , the slope of the curve (derivative) at a given point is a number .

I have been unnecessarily thinking about the cause behind why differential equations have dy/dx or f'(x) in them .I guess there is no need to think about it that way .
Differential equations are equations that involve the derivatives of various orders of some unknown function f. These derivatives represent the rates of change of some quantity with respect to some other quantity. You don't start off knowing the function, so it makes no sense to talk about finding its derivative.

For example, a mass on a spring, with no damping force, can be represented by the differential equation ##m \frac{d^2 }{dt^2}\left(x(t)\right) + kx(t) = 0## or ##mx''(t) + kx(t) = 0##. Solving this equation involves finding the position x(t). See https://en.wikipedia.org/wiki/Harmonic_oscillator#Spring.2Fmass_system
 
  • Like
Likes awholenumber
  • #17
Thanks :-)
 
  • #18
Mark44 said:
Differential equations are equations that involve the derivatives of various orders of some unknown function f. These derivatives represent the rates of change of some quantity with respect to some other quantity. You don't start off knowing the function, so it makes no sense to talk about finding its derivative.

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.

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?
 
  • #19
Look at what you quoted in post #9.

I don't know why you are asking questions about partial differential equations if you can't look at an equation and tell which variable is dependent and which is independent.
 
  • #20
Ok , The independent variable is x. The dependent variable is f(x)

In this ,

eq0003_P.gif


The function 1 , 4 , 5 are functions of several variables right ?
 
  • #21
rosekidcute said:
Ok , The independent variable is x. The dependent variable is f(x)

In this ,

View attachment 205645

The function 1 , 4 , 5 are functions of several variables right ?
Correct, and only in this case, with many variables, does it make sense to talk about partial derivatives.
 
  • Like
Likes awholenumber
  • #22
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.

In a sense, these are equations where the " input data" is given in terms of partial derivatives: you are given a numerical relation involving an unknown function and some of its partial derivatives and your goal is to identify the function(s) satisfying said relations .

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)
the "wave equation" and
the "heat equation "

are such types of equations right ?
 
  • #23
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.
This quote is missing context. In general , derivatives aren't about how point mass quantites change.

In a sense, these are equations where the " input data" is given in terms of partial derivatives: you are given a numerical relation involving an unknown function and some of its partial derivatives and your goal is to identify the function(s) satisfying said relations .

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)
rosiekidcute said:
the "wave equation" and
the "heat equation "

are such types of equations right ?
Do a web search for "wave equation" and "heat equation".
 
  • #24
What is a differential equation ?

It is an equation involving an unknown function (solution) and its derivatives

differential_equation.png


This much is OK , right ?
 
  • #25
rosekidcute said:
What is a differential equation ?

It is an equation involving an unknown function (solution) and its derivatives

View attachment 205656

This much is OK , right ?
If you're interested in differential equations, get a textbook or look at an online course. Any textbook on differential equations will have a definition of the term, as will online sites such as Wikipedia.
 
  • #26
Thanks , Found a good material online .

http://ncert.nic.in/textbook/textbook.htm?lemh2=3-6
 
  • #27
The answer to your question is true. For example, U(x,t) is a function that depends on x and t. X and t are independent. Do you think is possible for a dependent function to have derivatives as functions?
 

What are partial differential equations?

Partial differential equations (PDEs) are mathematical equations that involve multiple independent variables and their partial derivatives. They are used to describe physical phenomena that vary in space and time, such as heat transfer, fluid flow, and electromagnetic fields.

Why are partial differential equations important?

PDEs are important because they are used to model and analyze many real-world problems in fields such as physics, engineering, and economics. They provide a powerful tool for predicting the behavior of complex systems and designing solutions to practical problems.

How do partial differential equations differ from ordinary differential equations?

The main difference between PDEs and ordinary differential equations (ODEs) is the number of independent variables. ODEs involve only one independent variable, while PDEs involve multiple independent variables. This makes PDEs more complex and challenging to solve, but also allows for a more accurate representation of real-world phenomena.

What are some common techniques for solving partial differential equations?

There are several techniques for solving PDEs, including separation of variables, method of characteristics, and numerical methods. Each technique has its own advantages and is suited for different types of PDEs. Choosing the appropriate technique depends on the specific problem being solved.

What are some examples of applications of partial differential equations?

PDEs have numerous applications in various fields. For example, in physics, they are used to model heat transfer, fluid dynamics, and wave propagation. In engineering, they are used to design structures and optimize processes. In economics, they are used to model financial markets and analyze business strategies.

Similar threads

  • Differential Equations
Replies
5
Views
921
Replies
1
Views
1K
  • Differential Equations
Replies
14
Views
3K
  • Differential Equations
Replies
18
Views
4K
  • Differential Equations
Replies
1
Views
1K
  • Differential Equations
Replies
5
Views
2K
  • Differential Equations
Replies
2
Views
2K
  • Differential Equations
Replies
3
Views
2K
  • Differential Equations
2
Replies
52
Views
515
Replies
9
Views
2K
Back
Top