Solving electromagnetic wave equations directly in antenna theory

[rppearso]

Is it possible to solve these partial differential equations directly, relating to Antenna Theory;

$$∇^2 E - μ_0 ε_0 \frac{∂^2E}{∂t^2} = -μ_0 \frac{∂J}{∂t}.$$ AND $$∇^2 B - μ_0 ε_0 \frac{∂^2B}{∂t^2} = -μ_0 ∇ x J.$$

I don't like the idea of having to make up fields that don't exist in order to make the math work. The x is a cross product not a variable or multiplication.

 

[Simon Bridge]

Here, let me help with that:$$\nabla^2 \vec B - \mu_0 \epsilon_0 \frac{\partial^2\vec B}{\partial t^2} = -\mu_0 \nabla \times \vec J.$$... better? (Hit "quote" to see how I did that.)

There may be some geometries where the equations can be solved directly, I've not heard of any for antennas. Follows that you have to use a Trick. Welcome to realmaths.

 

[rppearso]

What is the best textbook on PDE's that cover greens function in detail?