# Solving 2D Diffusion Problem: Analytical Solution Needed

• AndersFK
In summary, the conversation discusses the search for an analytical solution to a 2D diffusion problem in the xy-plane with a known value at the origin and a constant diffusion constant. The individual has attempted to expand a solution from a similar 1D case, but has not been successful. However, they have found a solution in Carslaw & Jaeger's book that solves the problem for a constant boundary condition, and Duhamel's theorem can be used to generalize it to a time-dependent boundary condition. Any comments or suggestions are welcome.
AndersFK
I'm trying to find an analytical solution (probably containing a convolution integral) to a 2D diffusion problem in the xy-plane, when the value h(t) at the origin is known for all times t>=0. The diffusion constant is the same everywhere.

The last problem solved under the section http://en.wikipedia.org/wiki/Heat_equation#Homogeneous_heat_equation solves a similar problem for a semi-infinite 1D case. I've tried to expand their solution to the 2D case, but no luck so far.

Any comments/suggestions would be greatly appreciated.

I've found the solution. Carslaw & Jaeger (Conduction of Heat in Solids) solve the problem for $h(t)\equiv h_0$ constant. Duhamel's theorem can then be used to generalize the solution to the problem with a time-dependent boundary condition $h(t)$.

## 1. What is a 2D diffusion problem?

A 2D diffusion problem is a mathematical problem that involves the spreading of a substance or quantity in a two-dimensional space over time. This can occur in various fields such as physics, chemistry, and biology.

## 2. What is an analytical solution?

An analytical solution is a solution to a problem that can be obtained through mathematical calculations and does not require numerical methods or simulations. It provides an exact and closed-form solution to the problem.

## 3. What is the importance of solving 2D diffusion problems analytically?

Solving 2D diffusion problems analytically allows for a better understanding of the underlying mechanisms and behavior of the diffusion process. It also provides a more accurate and precise solution compared to numerical methods.

## 4. What are the steps involved in solving a 2D diffusion problem analytically?

The steps involved in solving a 2D diffusion problem analytically include setting up the diffusion equation, applying boundary conditions, solving the equation using separation of variables or other analytical methods, and interpreting the solution in the context of the problem.

## 5. What are some real-life applications of solving 2D diffusion problems analytically?

Some real-life applications of solving 2D diffusion problems analytically include modeling the spread of pollutants in a body of water, understanding the diffusion of drugs in the human body, and predicting the distribution of heat in a material. It can also be applied in fields such as meteorology, economics, and environmental science.

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