Partial Differential Equation Puns

I have Partial Differential Equation for maths this sem and I have been struggling due to missing out on classes here and there. Usually Khan Academy works when I have to learn something but I coudnt find anything about this topic on there and googling only gives some articles on differential eq. Heres what I have to learn :-

Partial differential equations

, Formation of partial differential equations –elimination of arbitrary constants-elimination of arbitrary functions, Solutions of a partial differential equations, Equations solvable by direct integration, Linear equations of the first order- Lagrange’s linear equation, Non-linear equations of the first order -Charpit’s method, Solution of equation by method of separation of variables.

Applications

One dimensional wave equation- vibrations of a stretched string, derivation, solution of the wave equation using method of separation of variables, D’Alembert’s solution of the wave equation, One dimensional heat equation, derivation, solution of the heat equation

Lately I found an interest in solving PDEs numerically. I've done some simulations on rectangular grids(heat equation, wave equation with location dependent velocity etc), and I'd like to do something more sophisticated. My end goal is to perform numerical simulations on curved surfaces, but simple things first.

Flat domain

Given I've got a flat domain discretized into a bunch of triangles with a value associated at each point. I would like to calculate the spatial derivatives at point P given the values at P's immediate neighbors. I figured I could calculate a linear function that interpolates values at vertices belonging to each neighboring triangle, calculate the derivatives based on each interpolant and then do a weighted average. However I'm not sure whether the weight should be with respect to the surface area S, or the edge length l?

Abstracting the ambient space

Is it possible/sensible to express the gradient/laplace operator not in terms of derivatives with respect to the axes of ambient space, but in terms of some sort local barycentric coordinates spanned on the neighboring vertices? Would the differential equation change in any way? E.g would the equation u_{t} = ∇^(2)u remain the same, or would some additional factors show up?

Curved domains

What considerations would I need to take if the domain weren't flat? For example if the domain would approximate the surface of a sphere or a torus? I feel like there's a need to account for the curvature of the surface in the differential operators. This is why I thought about expressing the gradient not with respect to ambient space but with respect to the local space defined by the simplicial complex(should it be a good idea).

All of this is for fun/learning, so one can assume everything is sufficiently nice, smooth and not an edge case.

Need someone who can solve some partial differential equations 50$ per question solved. Needs to be 90+% correct.

Due 24 October

If you are good we can work together a lot

As the title. You start with a variety of worlds, each one is a different "level", and requires different skills. This keeps complexity low at the start.

Maybe your tutorial level is putting the finishing touches on a planet that's 95% of the way done.

Need more oxygen, so you introduce a new type of algae. Quick little touches.

Then you move onto bigger and substantially more complicated planets as you unlock new bio-technology and work with harder and harder challenges.

Foreword:

The purpose of this post is to share an interesting paper I came across recently, and hopefully to generate interest (especially among analysts) in stochastic analysis.

Arxiv link to the paper.

Introduction:

The Feynmann Kac connection is a celebrated family of theorems in analysis that relates partial differential equations and stochastic analysis.

They state that you can write the solution to a wide class of second order partial differential equations by running an Ito diffusion (that is, the solution to a linear SDE) for some time and taking an expected value.

The most classical and well known case is that of the heat equation - you can write the solution u(x, t) to the heat equation with initial distribution f by running a Brownian motion X starting at x up to time t and taking the expected value of f(X_t), weighted by an exponential decay factor. More precisely, we have

u(x, t) = E[e^-t f(X_t)| X_0 = x].

The paper:

Recently I encountered a cool paper that does something similar but for second order ODE’s instead. Like the classical Feynman Kac theorems, it expresses the solution to an wide class of second order ODE in terms of a diffusion.

What is most striking about the paper however is that instead of linking the ODE to the expected values of the diffusion, what is used instead is a different measure of “centrality” - the so called mode of the diffusion.

Informally, the mode of a diffusion is the “most likely path” the diffusion will take, similar to how the mode of a discrete random variable is the most likely value.

Formally, it is defined as the solution to a certain minimisation problem involving the Onsanger Machlup functional, which describes the infinitesimal time evolution of the probability density function of a diffusion. It is the stochastic analog of the Lagrangian in classical mechanics, in a way that is made precise in the paper.

The connection is in fact established for arbitrarily large systems of ODE, which has large implications for numerical computation. Namely, the classical deterministic algorithms for solving ODE (eg Runge-Kutta) scale very badly for large systems of equations. On the oth

I took it last fall and because it was remote (and my professor wasn’t the best) I did not learn anything! I have no idea how to solve most PDEs and I barely passed that class.

Fortunately, my college allows you to repeat classes. Should I retake the PDEs class?

I am using the method of characteristics to solve the next PDE

(u+e^x)u_x+(u+e^y)u_y=u^2-e^{x+y}

I have that

\frac{dx}{u+e^x}=\frac{dy}{u+e^y}=\frac{du}{u+e^{x+y}}

But, I'm struggling with obtain an ODE that can be solved.

I've been reading into Separation of Variables and Fourier Series for functions that are orthogonal/multiplied together:

•υ(x,t)≡X(x)T(t) such that they can be summed together as a Linear Combination of Fourier Series set on 2L [-L,L] periodic intervals.

Anyone else find this fascinating to read about!? It amazes me that almost any form of modeling on a system in real life can be broken down into Sinusoidal waves and combinations of them!

Has anyone here taken 6.339? What's your opinion about this class? I'm thinking of taking it this fall

I am taking Computational Science and Engineering minor and looking for PDF of Numerical Methods for Partial Equations. It looks like this book was written just for this course so I am having hard time finding it.

Greetings everyone,

I am a university student interested in geometric analysis and several complex variables. These subjects both require the prerequisites of a study in differential topology and partial differential equations.

I am looking for a dedicated study partner(s) to join me in a study through either of the two subjects (or both, if you wish). I am currently reading through the following two books:

Introduction to Smooth Manifolds by Jack Lee, and

Partial Differential Equations by Jurgen Jost.

For differential topology, I would like to read up to at least the 18th chapter on the de Rham theorem, with a treatment of the 22nd chapter on symplectic manifolds. For PDEs, I would like to read through most of the book (~400 pages).

In terms of organizing, we have options in terms of using a private stackexchange (Stackexchange teams), Discord, Overleaf, or other means that support LaTeX. If someone can run a server or knows how to, we can also use MediaWiki for a private wikipedia in our study.

Prerequisites:

If you would like to study with me, for differential topology it is best that you have learned point-set topology that includes a treatment of precompactness, paracompactness, and at least a surface-level survey of the fundamental group and covering spaces. Analysis is also recommended.

For partial differential equations, real analysis goes without saying. A course in complex variables would also be helpful. There is some measure theory in the book, but that is not until later and we can always review.

Please DM me if you are interested, and we can then speak further about how we will go about managing our study. Thank you for reading :)

Hey everybody, there’s an upper division math class at my school that’s an mix of complex analysis, partial differential equations, and Fourier series on top of vector field theory (Gauss, Stokes, and Green).

It’s not a required course, but do y’all think I should take it? Would I need any of these concepts to be a mechanical engineer?

I'm feeling decent on all the stuff from exam 1 handling eigenvalue solutions and systems of DEs, I did very well on the laplace transforms from exam 2, but for the life of me I have no fucking idea how to solve a PDE. From what my prof said there are going to be 5 questions from each of the 3 sections on the exam, so I know I'll deal with not just 1 or 2 but several of them.

My assignment is to solve 1-D homogenous heat equation and compare the numerical and analytical solutions. The numerical solution didn't pose a problem, but I'm struggling with the analytical. I've outlined all the methods in attached picture.

At first I tried to solve using Fourier's transformation, but the resulting integral is quite complex and I wasn't able to solve it by hand. Therefore I used the 'integral' command in Matlab. The solution is mostly correct, but there are two issues:

1. Evaluating the integral is rather slow, but that's to be expected.

2. Random 'spikes' appear in the solution. This can be seen in the attached pictures. My guess is that it has to do with internal workings of the integral command, which I'm not familiar with.

As a second option I tried to find the solution in the form of Fourier series. This gave me integrals that were simple enough to solve by hand, so I could avoid the numerical integration. But, unfortunately, the solution is wildly different from the expected one. I really don't know what could be wrong here, the coefficients look correct and I'm not doing any numerical integration in Matlab.

The third option was to, instead of using the classical Fourier transform, use FFT which is suited for computational use. We haven't really learned this method, so I based my attempts on this video. This method didn't return any meaningful results, the inverse transform always returns complex numbers, which is obv. not correct.

Here is my Matlab code.

And here are the pictures.

Any help is greatly appreciated.

Final notes: I showed the solution only for a=1, but the same issues arise for a=10 and 100

The time discretization is given by the requirements for the stability of numerical solution. I know I could lengthen the time step, but if I solve it on the same mesh then I can compare the solutions easily.

The expected solution is a disippating sine wave, starting from the initial condition of two superimposed sine waves, where the higher frequency wave dies out rapidly.

Edit: Forgot to mention, I also tried the 'pdepe' command, which gave correct results. But as I haven't solved the equation, only used a prepared solver, I can't present this as a solution to the teacher.

An Introduction to Partial Differential Equations with MATLAB, 2nd edition by Matthew P. Coleman, Chapman & Hall/CRC, 2013.

ISBN: 1439898464

Please help

Hi everyone,

Sharing some work from the graph ML team at Twitter showing how a new class of GNNs can be constructed by discretising diffusion PDEs. Was on arxiv today and it will be at ICML21.

Thinking of GNNs as partial differential equations leads to a new broad class of GNNs that are able to address in a principled way some of the prominent issues of current Graph ML models such as depth, oversmoothing, bottlenecks, and graph rewiring.

Many popular GNNs can be formalised as discretised diffusion PDEs with explicit single-step Euler scheme with a time step of 1, where an iteration corresponds to a convolutional or attentional layer of the graph neural network, and running the diffusion for multiple iterations amounts to applying a GNN layer multiple times. In the Neural PDEs formalism, the diffusion time parameter acts as a continuous analogy of the layers—an interpretation allowing us to exploit more efficient and stable numerical schemes that use adaptive steps in time.

Blog post: https://bit.ly/3gUOEL8
Paper: https://arxiv.org/abs/2106.10934
Code: https://github.com/twitter-research/graph-neural-pde

https://preview.redd.it/68zohy4ebs671.png?width=647&format=png&auto=webp&s=439f7b1428bd5fd9b5751ecbe4975140084eb677

I took it last fall and because it was remote (and my professor wasn’t the best) I did not learn anything! I have no idea how to solve most PDEs and I barely passed that class.

Fortunately, my college allows you to repeat classes. Should I retake the PDEs class?

Greetings everyone,

I am a university student interested in geometric analysis and several complex variables. These subjects both require the prerequisites of a study in differential topology and partial differential equations.

I am looking for a dedicated study partner(s) to join me in a study through either of the two subjects (or both, if you wish). I am currently reading through the following two books:

Introduction to Smooth Manifolds by Jack Lee, and

Partial Differential Equations by Jurgen Jost.

For differential topology, I would like to read up to at least the 18th chapter on the de Rham theorem, with a treatment of the 22nd chapter on symplectic manifolds. For PDEs, I would like to read through most of the book (~400 pages).

Prerequisites:

If you would like to study with me, for differential topology it is best that you have learned point-set topology that includes a treatment of precompactness, paracompactness, and at least a surface-level survey of the fundamental group and covering spaces. Analysis is also recommended.

For partial differential equations, real analysis goes without saying. A course in complex variables would also be helpful. There is some measure theory in the book, but that is not until later and we can always review.

Please DM me if you are interested, and we can then speak further about how we will go about managing our study. Thank you for reading :)

Greetings everyone,

I am a university student interested in geometric analysis and several complex variables. These subjects both require the prerequisites of a study in differential topology and partial differential equations.

I am looking for a dedicated study partner(s) to join me in a study through either of the two subjects (or both, if you wish). I am currently reading through the following two books:

Introduction to Smooth Manifolds by Jack Lee, and

Partial Differential Equations by Jurgen Jost.

For differential topology, I would like to read up to at least the 18th chapter on the de Rham theorem, with a treatment of the 22nd chapter on symplectic manifolds. For PDEs, I would like to read through most of the book (~400 pages).

In terms of organizing, we have options in terms of using a private stackexchange (Stackexchange teams), Discord, Overleaf, or other means that support LaTeX. If someone can run a server or knows how to, we can also use MediaWiki for a private wikipedia in our study.

Prerequisites:

If you would like to study with me, for differential topology it is best that you have learned point-set topology that includes a treatment of precompactness, paracompactness, and at least a surface-level survey of the fundamental group and covering spaces. Analysis is also recommended.

For partial differential equations, real analysis goes without saying. A course in complex variables would also be helpful. There is some measure theory in the book, but that is not until later and we can always review.

Please DM me if you are interested, and we can then speak further about how we will go about managing our study. Thank you for reading :)