General texts on systems of partial differential equations?

In summary, the conversation is discussing recommendations for textbooks on properties and solutions of systems of partial differential equations, with a focus on vector and tensor valued PDEs such as Maxwell's equations, Navier-Stokes equations, and equations governing elasticity and deformation of solids. The conversation mentions a book by Richard Haberman, "Applied Partial Differential Equations with Fourier Series and Boundary Value Problems," as a potential option, but the speaker is looking for a more modern text written for today's graduate and postdoctoral mathematicians. They specifically mention wanting a book that assumes prior knowledge of single, scalar valued PDEs and delves into vector valued systems of PDEs.
  • #1
The Bill
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What are some good general textbooks on the properties and solution of systems of partial differential equations? I'm most interested in the general theory of vector and tensor valued PDEs like Maxwell's, Navier-Stokes, and the bulk equations governing elasticity and deformation of solids, etc.

I'd like to read some books that handle all these in the framework of a general theory to help tie together what one learns from specialist books on E&M, fluid dynamics, etc.
 
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  • #3
That looks like an interesting book, but I'm looking for a more modern text, written for an audience of today's graduate and post doctoral mathematicians.
 
  • #4
The Bill said:
That looks like an interesting book, but I'm looking for a more modern text, written for an audience of today's graduate and post doctoral mathematicians.
By Richard Haberman, in 2012: "Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, 5/e". Prerequisites: multivariable calculus, linear algebra I, complex variable !, first course in ODEs for the sciences. For one of the 22 math courses offered at distance learning by Athabasca University (Alberta), the first( and still sole) public Canadian university to be recognized by the Government of USA.
 
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  • #5
Both of those look like they spend a lot of pages on single, scalar valued PDEs like the heat equation and the scalar wave equation. I've already learned a lot about those. I want a book that assumes you've already taken a course or two dealing with heat equations, Laplace's equation, etc, and dives right into vector valued systems of PDEs.
 

1. What is a system of partial differential equations?

A system of partial differential equations is a set of equations that involve multiple unknown functions and their partial derivatives. These equations are used to describe various physical phenomena, such as heat transfer, fluid dynamics, and electromagnetic fields.

2. Why are systems of partial differential equations important in science?

Systems of partial differential equations are important in science because they provide a mathematical framework for modeling complex systems and predicting their behavior. They are used in a wide range of fields, including physics, engineering, and biology.

3. What are some techniques for solving systems of partial differential equations?

There are several techniques for solving systems of partial differential equations, including separation of variables, method of characteristics, finite difference methods, and numerical methods such as finite element and finite volume methods. The choice of method depends on the specific problem and the desired level of accuracy.

4. Can systems of partial differential equations be solved analytically?

In many cases, systems of partial differential equations cannot be solved analytically, meaning that an exact solution cannot be found. Instead, numerical methods must be used to approximate the solution. However, in some special cases, such as linear systems with simple boundary conditions, an analytical solution may be possible.

5. How are systems of partial differential equations used in real-world applications?

Systems of partial differential equations are used in a wide range of real-world applications, from predicting weather patterns to designing aircraft. They are also used in medical imaging, financial modeling, and many other fields. By accurately describing complex systems, they allow scientists and engineers to make predictions and improve our understanding of the world around us.

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