How to Solve This Second-Order Non-Linear ODE Involving Functions a(r) and b(r)?

In summary, the speaker is struggling with solving a PDE involving a function R(r,t) with functions a(r,t) and b(r,t) and a constant k. They are open to suggestions on how to make the problem easier and are specifically curious if removing t-dependence from a and b would affect the solution. They provide the PDE and mention they have ways of approaching the problem but would greatly appreciate a solution in terms of a, b, r, and t. They also clarify that the first term in the PDE disappears and the problem is essentially an ODE in r. They end by asking for any advice or help with solving the problem.
  • #1
ramparts
45
0
I've run across a PDE that (since I've failed to take a PDE class!) I'm finding some difficulty in solving. Does anyone have any suggestions? It's on a function R(r,t), with functions a(r,t) and b(r,t) and a constant k. If it's easier to solve with a and b not having t-dependence (just being a(r) and b(r)) I'd be curious to know - I'm not sure if that affects things.

Here's the PDE:

R = k(-e-2a Rtt + e-2b Rrr +[ar + br + 2/r] e-2b Rr)

I'd really appreciate any help! I have a couple of ways of attacking this problem (I also have R defined in terms of derivatives of a and b) but I think a solution to this PDE in terms of a, b, r, and t would be incredibly useful. Thanks!
 
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  • #2
Sorry - the first term (the derivatives in t) disappear. I don't expect any t dependence, so this is really just an ODE in r. Still, any advice would be appreciated :)
 

FAQ: How to Solve This Second-Order Non-Linear ODE Involving Functions a(r) and b(r)?

What is a second-order non-linear PDE?

A second-order non-linear PDE is a partial differential equation that involves second-order derivatives of the dependent variable and non-linear terms, meaning that the dependent variable is raised to a power other than 1. These types of equations are often used in physics and engineering to model complex systems.

How is a second-order non-linear PDE different from a linear PDE?

A linear PDE only involves first-order derivatives and linear terms, while a second-order non-linear PDE involves second-order derivatives and non-linear terms. This means that the solutions to a linear PDE can be found using superposition and other analytical methods, while non-linear PDEs often require numerical methods for solution.

What are some examples of second-order non-linear PDEs?

Some examples of second-order non-linear PDEs include the Navier-Stokes equations, which are used to model fluid flow, the Schrödinger equation, which describes the behavior of quantum particles, and the Black-Scholes equation, which is used in finance to model stock prices.

How are second-order non-linear PDEs solved?

Solving a second-order non-linear PDE often involves using numerical methods, such as finite difference or finite element methods, to approximate the solution. These methods discretize the PDE into a system of algebraic equations, which can then be solved using computer algorithms. Analytical solutions are rarely possible for non-linear PDEs.

What are some applications of second-order non-linear PDEs?

Second-order non-linear PDEs have a wide range of applications in physics, engineering, and finance. They are used to model fluid flow, heat transfer, wave propagation, and many other physical phenomena. In finance, they are used in option pricing and risk management. They also have applications in image processing, pattern recognition, and other areas of computer science.

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