In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to how x is thought of as an unknown number, to be solved for, in an algebraic equation like x2 − 3x + 2 = 0. However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations. Among the many open questions are the existence and smoothness of solutions to the Navier–Stokes equations, named as one of the Millennium Prize Problems in 2000.
Partial differential equations are ubiquitous in mathematically-oriented scientific fields, such as physics and engineering. For instance, they are foundational in the modern scientific understanding of sound, heat, diffusion, electrostatics, electrodynamics, fluid dynamics, elasticity, general relativity, and quantum mechanics. They also arise from many purely mathematical considerations, such as differential geometry and the calculus of variations; among other notable applications, they are the fundamental tool in the proof of the Poincaré conjecture from geometric topology.
Partly due to this variety of sources, there is a wide spectrum of different types of partial differential equations, and methods have been developed for dealing with many of the individual equations which arise. As such, it is usually acknowledged that there is no "general theory" of partial differential equations, with specialist knowledge being somewhat divided between several essentially distinct subfields.Ordinary differential equations form a subclass of partial differential equations, corresponding to functions of a single variable. Stochastic partial differential equations and nonlocal equations are, as of 2020, particularly widely studied extensions of the "PDE" notion. More classical topics, on which there is still much active research, include elliptic and parabolic partial differential equations, fluid mechanics, Boltzmann equations, and dispersive partial differential equations.
TL;DR Summary: I am taking a math methods course this semester of which a large part are PDEs. I don't understand the context behind the order in which we are solving PDEs. I am interested in learning how other people were taught PDEs and any book recommendations you might have.
I am taking a...
I want to solve the following system of PDEs:
\frac{\partial\nu}{\partial t}=\frac{\partial u}{\partial h}
\frac{\partial u}{\partial t}=\frac{\partial}{\partial h}\left(f(\nu)\frac{\partial u}{\partial h}\right)
I know the usual Fourier analysis that are applied to the stencil for single...
HI
HI! While trying to solve problem in Hydrodynamic stability I have got a system of Two Partial Diffential equations :
Can anyone help me to solve this analytically? Is there any general method to solve system of PDEs?
Some may say that ##\frac{ \partial g }{ \partial t }## is correct because it is a term in a partial differential equation, but since ##g## is a one variable function with ##t## only, I think ##\frac{ dg }{ dt }## is correct according to the original usage of the derivative and partial...
A PDE is solved by finite elements. The PDE then becomes a discrete system solved by Newton iteration. Every iteration step comes with a residual error. When the solution is completed, how far is the residual error of the PDE from the residual error of the finite element discrete system?
Use a numerical method to solve a PDE f[u(x),u'(x),...]=0, where f is an operator, e.g. u'(x)+u(x)=0, and obtain a numerical solution v(x). Define f[v(x),v'(x),...] as the residual of the original PDE. Is this residual of the PDE widely used as the convergence criteria of the numerical solution...
hello
I own mathematica 10.02
it is virtually impossible to solve PDE's ,even with NDSolve,if the initial conditions contain a derivative
I write
Derivative[1,0] [0,x] == f[x]
I mean
the first t derivative of u[t,x] for x at t=0 is f[x]
I own a book based on Mathematica 10.3
Even if a...
Came across this today, a fourth order PDE - the Kuramoto–Sivashinsky equation, apparently used to model flames
https://en.wikipedia.org/wiki/Kuramoto%E2%80%93Sivashinsky_equation
Any other examples of high order PDEs with actual applications?
Hello everyone,
I'm currently going through Strauss "introduction to differential equations" and i can't get around a certain proof that he
gives on chapter 11.5 page(327 (2nd edition)).Specifically, the proof refers to a certain version of Fredholm's alternative theorem.
Assume that we are...
Summary:: I'm looking for some resources to study PDEs.
Hello everyone,
I'm a sophomore majoring in Physics and this semester I am taking a course on Mathematical Methods focusing on PDEs and I'm really struggling in the course. Can someone suggest some resources to self-study PDEs? The...
I've been trying to get change of variables in PDEs down (I don't particularly like my textbook or professor's approach to it), and I want to ask here if I am getting this right. Let ##\vec{x}=(x_1,x_2,...,x_n)^T## and ##\partial_\vec{x}=(\partial_{x_1},\partial_{x_2},...,\partial_{x_n})^T##. I...
(1) ok.
(2) We start with ##\sigma(ξ) = a_{11} ξ_1^2 +2a_{12}ξ_1ξ_2 +a_{22}ξ_2^2ξ##
and we replace every ##ξ_iξ_j## with ##\partial_i\partial_ju##,
giving ##a_{11}\partial_x^2+2a_{12}\partial_x\partial_yu+1_{22}\partial_2^2##
(3) The given equation is the following.
##\sigma(ξ) = ξ^t A ξ ##...
Hey, I realized there are some parts (okay, a lot of parts) of Physics that I can't learn more about until I actually get a bit of practice solving PDEs. I'll cover it 'properly' next year but for now I'm just interested to learn about the most common solution techniques, types of boundary...
This paper getting some press, with promises that NNs can crack Navier-Stokes solutions more efficiently than traditional numerical methods.
https://www.technologyreview.com/2020/10/30/1011435/ai-fourier-neural-network-cracks-navier-stokes-and-partial-differential-equations/...
In solving some PDEs such as the heat/diffusion equation or the wave equation, when the equation itself, as well as its associated boundary conditions, are independent of some variable (for example the azimuthal angle), we often use the trick to assume that the solution (and eigenfunctions) are...
FTCS scheme
My Matlab code:
%Problem 3
%Solve diffusion problem using Richardson scheme or DuFort-Frankel scheme
clear all;
scheme = menu('Choose method of solving diffusion equation:', 'Richardson', 'DuFort-Frankel');
tau = input('Enter time step: ');
N = input('Enter the number of grid...
Hi! I am looking into a mechanical problem which reduces to the set of PDE's below. I would be very happy if you could help me with it.
I have the following set of second order PDE's that I want to solve. I want to solve for the generic solutions of the functions u(x,y) and v(x,y). A, B and C...
With a 1000 time speed-up too, this could be a game-changer.
From: https://www.technologyreview.com/2020/10/30/1011435/ai-fourier-neural-network-cracks-navier-stokes-and-partial-differential-equations/
They did it by solving in "...Fourier space (rather) than to wrangle with PDEs in Euclidean...
Hi,
I have been learning about Laplace's equation recently, and have been wondering: how would we approach the problem if the region was a parallelogram (or some other shape that isn't a standard rectangle or circle)? Is this something that could feasibly be solved by hand, or would it require...
Hi,
I understand the underlying concept of changing variables in PDEs (so that we can reduce it to a simpler form), however, I am just not completely clear on the mathematics of it so I have a quick question about it.
For example, if we have the transmission line equation \frac{\partial...
I want to start off here saying I took the problem has finding a potential function, and not a general solution, so I worked to only find one function that works.
I already confirmed that this function can be written as a curl of a vector function and the gradient of a scalar function.
Since...
Dear Everyone,
I am having trouble with how to start with one part of the question:
"In this exercise, we derived the PDE that models the vibrations of a hanging chain of length $L$. For convenience, the x-axis placed vertically with the positive direction pointing upward, and the fixed end...
Hello,
I have a problem with a solution of PDEs. I understand it is impossible to find my problem but some hint how to look at such problem would be very useful. I have to say it is my first encounter with a numerical solution of PDEs, so be patient with my description.
I have a code (in...
I am unsure how to choose the boundary conditions for a system of PDEs or for a single PDE for that matter.
The situation I am stuck with involves a system of 4 PDEs describing plasma in a cylinder. The dependent variables being used are Vr, Vt, Vz, ni, and the independent variables are Rr...
Homework Statement
"Show that a solution of the homogeneous PDE ##au_x+bu_y+cu=0## cannot be zero at one, and only one point in the plane."
My interpretation of this is that ##u(x,y)## is zero everywhere on the plane except on that point ##(x_0,y_0)##.
Homework Equations
##w=bx-ay##
##z=y##...
Homework Statement
Show that k(x,0)=δ(x).
Where k(x,t) is the heat kernel and δ(x) is the Dirac Delta at x=0.
Homework Equations
k(x,t) = (1/Sqrt[4*π*D*t])*Exp[-x^2/(4*D*t)]
The Attempt at a Solution
I am just clueless from the beginning. I am guessing this is got to do with convolution...
This past semester, I just took an introductory course on G.R., which translates to a lot of differential geometry and then concluding with Schwarzschild's solution. We really didn't do any cosmology. However, one of the themes that kept creeping up again and again is that in 4-dimensions...
Homework Statement
(a) Light waves satisfy the wave equation ##u_{tt}-c^2u_{xx}## where ##c## is the speed of light.
Consider change of coordinates $$x'=x-Vt$$ $$t'=t$$
where V is a constant. Use the chain rule to show that ##u_x=u_{x'}## and ##u_{tt}=-Vu_{x'}+u_{t'}##
Find ##u_{xx},u_{tt},##...
I used to like math when I was in high school. Calculus (integration and derivatives) seem intuitive to me and made me understand math so much better.
Now I'm currently in university majoring in civil engineering taking Calc III and I feel overwhelmed by everything taught in class. No matter...
Hi, my background is in mathematics, and theoretical physics.
I am new to the realm of solving PDEs using Finite element methods, does anyone know of any good introductory level textbooks for course notes?
I had a poke around online and couldn't find anything overly useful.
Also I am...
What are useful practical applications of numerical conformal mapping that are most limited by map computation speed or boundary complexity? I'm betting some of the applications will be be physics PDEs, so I chose this DE subforum to ask.
As part of an engineering project I've implemented...
Hey! I've been trying to tackle this problem but I'm a little lost at the moment and any references or suggestions would be greatly appreciated. Essentially the problem boils down to solving (at least) 3 coupled partial differential equations with (at least) 2 independent variables. Now the...
Suppose we are solving a diffusion equation.
##\frac{\partial}{\partial t} T = k\frac{\partial^2}{\partial x^2} T##
On the domain ##0 < x < L##
Subject to the conditions
##T(x,0) = f(x) ## and ##T = 0 ## at the end points.
My question is:
Suppose we solve this with some integration scheme...
One of my friends needs to numerically solve this two dimensional boundary value problem but has now idea where to begin. Could anybody help?
## [(K H )(f g_x-gf_x)]_x+[(K H )(f g_y-gf_y)]_y=0 #### K H G^2 (f^2+g^2)+\frac 1 2 [KH (f^2+g^2)_x]_x+\frac 1 2 [K H (f^2+g^2)_y]_y-K...
Homework Statement
Graph snapshots of the solution in the x-u plane for various times t if
\begin{align*}
f(x) =
\begin{cases}
& 3, \text{if } -4 \leq x \leq 0 \\
& 2, \text{if } 4 \leq x \leq 8 \\
& 0, \text{otherwise}
\end{cases}
\end{align*}
Homework Equations
Assuming that c=1 and g(x)...
Suppose a PDE for a function of that depends on position, ##\mathbf{x}## and time, ##t##, for example the wave equation $$\nabla^{2}u(\mathbf{x},t)=\frac{1}{v^{2}}\frac{\partial^{2}}{\partial t^{2}}u(\mathbf{x},t)$$ If I wanted to solve such an equation via a Fourier transform, can I Fourier...
Homework Statement
Imagine a chain with ##n## links across a river. Now imagine, the chain is in a straight horizontal line at time ##t=0##. The problem wants me to calculate the movement of the chain links (center of mass) due to the gravity field. There are other forces in the system but this...
Greetings all,
Quick question. I know that all 4 Maxwell's equations are said to be first-order, coupled PDEs, where each equation has an unknown field. I see that with Faraday's and Ampere's law, because, E and H appear in each of those equations.
But Gauss' laws, I'm not seeing that...
Hi everyone. In electrical engineering, when you study control theory, you're taught that electrical circuits can be used to simulate the behaviour of complex systems. What I don't understand is, what are the limitation of this sistem, and why it can't be obviouslly used in a general way to...
Does anyone know of any books or online resources that do a good job discussing systems of linear 1st order PDEs with several (more than 2) independent variables? I am not a mathematician, but can handle graduate level classical physics with the associated applied math. Analytical and...
When looking at Elliptic PDEs that describe a physical system, do these typically not involve a time term?
I have yet to see an elliptic PDE involving a time term, which seem to be associated with parabolic/hyperbolic PDEs rather than elliptic.
Can anyone confirm?
So here I have Laplace's equation with non-homogeneous, mixed boundary conditions in both x and y.
1. Homework Statement
Solve Laplace's equation \begin{equation}\label{eq:Laplace}\nabla^2\phi(x,y)=0\end{equation} for the following boundary conditions:
\phi(0, y)=2;
\phi(1, y)=0;
\phi(x...
I just wanted to check something. The equation
∂2φ / ∂x2 + ∂2φ / ∂y2 = sin(xy)
Was given as an example of a separable equation. I can't separate it, and I found online that to use separation of variables the equation should be linear, which this isn't? Is there a way of separating this?
Hi. I'm a bit confused on determining whether a certain PDE is linear or non-linear.
For example, for the wave equation, we have: u_{xx} + u_{yy} = 0, where a subscript denotes a partial derivative.
So, my textbook says to write:
$L = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial...
I want to solve a system of 2 coupled pde (in MATLAB) of the format:
c1*(df/dt)+c2*(df/dz)+c3*(f)+c4*(g)=0
(dg/dt)=c5*f+c6*g
with Initial conditions as
f(0,t)=1, g(z,0)=0 and f(z,0)=0
0<f,g,z,t<1
I tried using the MATLAB function pdepe to do this but got errors and if I go for numerical...
Hello,
I have a problem in the search for symmetries in pde.
I would use Mathematica(c), does anyone know how to set up the code to obtain generators and then symmetries?
Thanks for all.
What are some good books (or other resources) on numerical methods of solving PDEs in 3 space and 1 time variable?
I am interested both in finite element and finite volume methods. I could be interested in other methods but I don't know about them. I am interested in being able to take...
Homework Statement
Hi - looking at 'discretizing elliptical PDEs'.
I understand the normal lattice approach, but this approach uses the variational principle. I have a couple of questions please. The text says:
$$ \: Given\: E=\int_{0}^{1} \,dx\int_{0}^{1} \,dy\left[\frac{1}{2}\left(\nabla...
I am using the text by Farlow to study elementary methods of solving PDEs, and there is a point in his illustration of separation of variables where I am not seeing something. I am clear on everything that comes after and before this point, but after having returned to a certain step a few times...