A differential equation, which involves differentiating a function with more than one parameter. For example:

(d2/dt2 - v2 d2/dx2) y(x,t) = 0

where t is time, x is position, v is velocity and y is displacement, is a partial differential equation (abbreviated PDE), which can be used to describe a vibrating guitar string.

A differential equation is a relation between a mathematical function and its derivates. If the function depend on more than one variable, the differential equation is said to be partial. These are a lot harder to solve analytically and numerically than the ordinary differential equations. The theories for PDEs has been developed mainly for the field of theoretical physics, where they play a very important part, especially in quantum mechanics. Below I list the most important PDEs in physics and how you would solve them.

(An excuse goes out to all Netscape users who may not be able to see the equations below.)

One of the most important PDEs in physics is Poisson's partial differential equation, which is what mechanical problems usually reduces to:

     n   ∂2u 
Δu =  ----- = f       (1)
    j=1  ∂x2j

Where u is the sought function, f is a known function, and x is a coordinate in space. In physics, n usually equals 3, for our 3 dimensional world. When f=0 the equation is called Laplace's equation. The Newton potential gives solutions to this as:

        1         f(y)
u(x) = ---- ∫∫∫------- dy1dy2dy3
        4π       |x-y|       

In physical applications of this, you usually search for solutions within a domain Ω where you have a Dirichlet boundary condition u = φ on the boundary ∂Ω. There are many methods to find this solution analytically, and usually they are focused on finding upper and lower values of the function . This PDE can also be solved for other boundary conditions such as Neumann boundary condition. Equation (1) is and example of a elliptical differential equation. The term "elliptical" refers to the characteristic polynomial of the equation, which is a way to classify PDEs. 

Another common PDE is the heat equation, or diffusion equation:

∂u
--- - Δu = f         (2)
∂t            

where u is a function both in the room (x1, x2, x3... xn) and time (t). For n=3 this equation describes the distribution of heat in an homogenous material or some other process of diffusion in the physical world. If we have the initial value of u, u(t=0) = u0, the solutions for t > 0 can be derived from


u(t,x) = (4πt)-π/2e-|x-y|2/4t·u0(y)dy 

For solutions within a domain Ω you can also apply boundary conditions which makes the PDE solvable. Equation (2) is a hypo elliptical differential equation. In quantum mechanics, the Schrödinger equation is an important PDE, and it is fairly similar to the heat equation. The difference is that it has a imaginary factor:

 ∂u
i--- - Δu + Vu = 0         (3)
 ∂t  

This equation is not easily solved, but great progress has been made during the last decades. In physics, where V is the same potential function as in the N-body problem, and u = u(x,y,z), this is the equation for the quantum mechanical, non-relativistic N-body problem. For each φ where |φ(x)|2dx <  there is a solution u to (3) which is equal to φ for t = 0 and  |u(t,x)|2dx is independent of t. 

The wave equation, which describes the movement of light and electromagnetic waves, is also a PDE:

2u
----- - Δu = f             (4)
∂t2

Maxwell's equations reduce to the above. For u = u(x,y,z) a solution is given by

             f(t-|y|,x-y)
u(t.x)= ∫∫∫------------ dy1dy2dy3
               4π|y|

The wave equation (4) is an example of a hyperbolical differential equation. Cauchy's problem, to find a solution for u=φ and ∂u/∂t=Ψ, can easily be derived from this last equation. Other problems lead to more complex calculations. For instance, the Dirichlet problem leads to a complicated study of singularities, which is related to the study of refraction and diffraction in optics

All of the above are examples of linear partial differential equations. However, many of the fundamental equations in physics are non-linear, such as Navier-Stokes equations in fluid dynamics. These are a bit harder to solve analytically and are usually the subject to numerical methods. Usually one has to study the linear solution to the non-linear problem, and then based upon this adjust the solution in the surrounding you are interested in.

Reference: ne.se

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