# Partial Differential Equations?

• Success
In summary, for physics and electrical engineering majors, partial differential equations is a pre-requisite for studying topics like harmonic functions, elasticity, hydrodynamics, EM and electrodynamics. Other topics that are usually learned around the same time are ordinary differential equations, linear algebra, vector/tensor calculus, calculus of variations, complex variables, integral transforms, probability/statistics, and numerical analysis. There is no specific order to learn these topics, but some key starting points include linear algebra, differential equations (ordinary and partial), and complex variables. Depending on the course, complex analysis can be either pure math or applied math. After PDE, it is recommended to take a course in complex analysis or applied functional analysis.
Success
What math subject comes after partial differential equations for physics and electrical engineering majors?

For undergrad, that's more than you need.

In physics and many other graduate engineering courses, PDEs are a pre-requisite to studying things like harmonic functions, elasticity, hydrodynamics, EM and electrodynamics, etc. Also, you will need a healthy dose of vector calculus and complex analysis.

So next is complex analysis? Is that pure math or applied math?

There is not a certain order to learn topics. The usual things to learn around the same time as elementary partial differential equations (and covered in the same books) are ordinary differential equations, linear algebra, vector/tensor calculus, calculus of variations, complex variables, integral transforms, probability/statistics, and numerical analysis. One always seems to have more mathematics to learn than time to learn it.

I can't speak to engineering too much, but from a physics point of view any comprehensive math course is a useful math course. Even things like number theory find themselves popping up in research.

That being said, some of the key starting points are linear algebra, differential equations (ordinary and partial), and complex variables.

Basically piggybacking off of lurflurf "One always seems to have more mathematics to learn than time to learn it."

Success said:
So next is complex analysis? Is that pure math or applied math?

It can go either way. Typically, complex analysis is usually introduced to engineering undergrads as part of vector calculus.

Is complex variables a hard course?

Depends on how complex is taught. If you see it in a sort of "Math Methods" kind of class, it could be a lot of fun. Doing problems like contour integration and such.

If you take a math majors complex analysis, it is going to be a stranger version of real analysis which is all proof based.

Now some find doing proofs hard, some find it challenging but fun.

So the course is called complex variables or complex analysis?

Just check your course roster for physics/math. Some schools offer both a pure complex analysis class and an applied complex analysis class whereas others might only offer one or the other. You would have to choose based on your needs/requirements and interests, amongst other things. For example here are the course descriptions for the introductory pure and applied complex analysis classes at my university:

"MATH 4180 - Complex Analysis
...
Students interested in the applications of complex analysis should consider MATH 4220 rather than MATH 4180; however, undergraduates who plan to attend graduate school in mathematics should take MATH 4180.
...
Theoretical and rigorous introduction to complex variable theory. Topics include complex numbers, differential and integral calculus for functions of a complex variable including Cauchy's theorem and the calculus of residues, elements of conformal mapping."

"MATH 4220 - Applied Complex Analysis
...
...
Covers complex variables, Fourier transforms, Laplace transforms and applications to partial differential equations. Additional topics may include an introduction to generalized functions."

1 person
Should have probably taken complex analysis before PDE's. Also a numerical analysis class which focuses on differential equations would be good.

A good course to do after PDE is one that involves applied functional analysis, in which you can apply the theory of Banach and Hilbert spaces to problems involving ODE, PDE and distributions.

Alright, thanks everyone.

## 1. What are partial differential equations (PDEs)?

Partial differential equations (PDEs) are mathematical equations that involve multiple independent variables, and their derivatives, to describe the relationships between different quantities. They are commonly used in physics, engineering, and other scientific fields to model and solve problems involving continuous systems.

## 2. What is the difference between partial differential equations and ordinary differential equations?

The main difference between partial differential equations and ordinary differential equations is the number of independent variables involved. In partial differential equations, there are multiple independent variables, while in ordinary differential equations, there is only one independent variable. This means that PDEs are used to describe systems that vary in space and time, while ODEs are used for systems that only vary in time.

## 3. What are some real-world applications of partial differential equations?

Partial differential equations have a wide range of applications in the real world, including modeling heat transfer, fluid dynamics, electromagnetism, and quantum mechanics. They are also used in finance to model stock prices and in biology to model population growth.

## 4. How are partial differential equations solved?

There are various methods for solving partial differential equations, including analytical methods, numerical methods, and computer simulations. Analytical methods involve finding exact solutions using mathematical techniques, while numerical methods use algorithms to approximate solutions. Computer simulations use software to solve PDEs and visualize the results.

## 5. What are some common challenges in solving partial differential equations?

Some common challenges in solving partial differential equations include dealing with complex boundary conditions, finding appropriate numerical methods for accurate solutions, and handling high-dimensional problems. PDEs can also be difficult to solve analytically, requiring advanced mathematical techniques and specialized knowledge in the specific field of application.

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