To my knowledge this wording "infinite dimensional" is historical: let's take for our two independent variables x and t. In the late 19th century mathematicians mainly investigated PDEs where all derivatives with respect to one variable x, say,  (and also other dependence on x)  could be collected into a single operator A, say. Thus one obtains an ODE (ordinary differential equation) in the remaining variable t, albeit with terms including the operator A. Obviously, one needs to investigate the function space on which A operates. This function space usually is infinite dimensional and if we use a basis of this function space (which works if we consider Hilbert spaces!) we may obtain an infinite dimensional matrix representation of A leading to an infinite system of coupled (often non-linear) ODEs. The Swedish  mathematician Helge von Koch worked on this approach to handle integral equations, the German physicist Werner Heisenberg based his first theory of quantum mechanic on this approach (the famous "matrix mechanics" is the infinite-dimensional representation of the Schrödinger equation, roughly speaking) and the "father" of modern astrophysics, Subrahmanyan Chandrasekar derived from his radiative-transfer integro-differential equation an infinite system of ODEs (which he tried to solve by some approximations). Hope this helps.

The question is not quite clear. The system of equations cannot possess dimension per se. LInear spaces used to possess the one.

One possibility is a functional space where one looks for a solution. The space used to be infinite dimensional. Another possibility is the space consisting of solutions of the system. If the system is linear, the solutions space is linear too, so we may think of its dimension: in this case the dimension is often finite.

Perhaps figuring out which dimension you do mean would be beneficial.

Dear Novosyolov,

Thank you for your concern at the first place.

If we have, let say a second order time dependent ODE representing the dynamics of any system, e.g, pendulum, spring mass system or motor etc. We say that the two sate variables span a 2-D space, this is true for linear as well as nonlinear systems. Now, for PDEs there is a generic argument that the system is infinite dimensional, and this is not understandable for me.

Ali, again, the system per se does not possess dimension. The state space of the system in your examples has dimension 2, this is true for both ODE and PDE. The functional space where we look for solutions may be finite dimensional or infinite dimensional. First define what do you mean using the term "dimension of PDE system"

I guess what he means was that the solution space is infinite dimensional.

You see that you have two FREE variables. Usually, the solution is determined by its

boundary values. For example, usually, you have the function u=u(x,y). The solution

in general is suppose to be uniquely determined by the boundary values of u. Since

it is actually a function on the boundary it is infinite dimensional----

the functions on the boundary is infinite dimensional. This is different from the

ODE case since the boundary of the ODE case is just points.

To my knowledge this wording "infinite dimensional" is historical: let's take for our two independent variables x and t. In the late 19th century mathematicians mainly investigated PDEs where all derivatives with respect to one variable x, say,  (and also other dependence on x)  could be collected into a single operator A, say. Thus one obtains an ODE (ordinary differential equation) in the remaining variable t, albeit with terms including the operator A. Obviously, one needs to investigate the function space on which A operates. This function space usually is infinite dimensional and if we use a basis of this function space (which works if we consider Hilbert spaces!) we may obtain an infinite dimensional matrix representation of A leading to an infinite system of coupled (often non-linear) ODEs. The Swedish  mathematician Helge von Koch worked on this approach to handle integral equations, the German physicist Werner Heisenberg based his first theory of quantum mechanic on this approach (the famous "matrix mechanics" is the infinite-dimensional representation of the Schrödinger equation, roughly speaking) and the "father" of modern astrophysics, Subrahmanyan Chandrasekar derived from his radiative-transfer integro-differential equation an infinite system of ODEs (which he tried to solve by some approximations). Hope this helps.

To the best of my knowledge this terminology stems from the fact that certain PDEs can be viewed as infinite-dimensional dynamical systems.  

For instance, it is well known that an evolution PDE in two independent variables t and x and one dependent variable u,

u_t = F(u,u_1,...,u_n),

where u_j=(u_{j-1})_x, and u_0=u, can be written as a pair of compatible infinite-dimensional dynamical systems  for u_j, j=0,1,2,..., as follows:

(u_j)_t=D^j (F), (u_j)_x=u_{j+1}, j=0,1,2,...,

where D=\sum_{j=0}^{\infty} u_{j+1}\partial/\partial u_j.

See e.g. the paper linked below and references therein for the details on the above example.

http://iopscience.iop.org/0036-0279/43/5/R05/

many answers pointed correctly that calling a PDE infinite dimensional refers to its solution space which is a function space and a function space could be infinite dimensional. I will try to provide an example of why a function could be infinite dimensional by an intuitive example. The dimensionality of any space is defined by the cardinality of the set of basis functions that spans that space. Consider the Fourier basis and the function space of periodic functions with basic frequency f. There are infinitely many sine and cosine functions with frequencies that are the multiplier of the basic frequency f. If a periodic function with basic frequency f is written as an infinite sum of these basis functions, this periodic function would be infinite dimensional in that Fourier space. If that particular function is a solution for a PDE, one may call this PDE infinite dimensional in reference to its solution space being infinite dimensional.

Agree with Hazem above. For example, this is how Lorenz constructed his 3x3 famous system exhibiting chaotic behavior: Start with Navier - Stokes and energy conservation (clearly a PDE's  system and highly non-linear) with specific boundary conditions, expand the unknown functions to multiple spatial Fourier and keep the coefficients time dependent. Then you get an infinite by infinite ODE. Hence the name. 

Btw, in this instance Lorenz made a severe truncation throwing away all but three terms, thus reducing this system to 3x3.

Recall as an example the superposition property when we solve a PDE by using Fourier Analysis. Example: the infinite nodes of a vibrating chord.

You cannot describe solution of PDE with a finite set of numbers.

The answers above are all alluding to the same concept, which is that dimension refers to the size of the solution space.  If the system is linear, then the dimension is the cardinality of the largest linearly independent set of solutions, and given the "Fourier" approach to solving a linear PDE subject to boundary and initial conditions, the infinite number of terms in a Fourier series implies that a linear PDE, at least, is infinite dimensional.  In fact, the dimensionality of a linear system is equal to the number of initial and/or boundary values necessary to determine all arbitrary constants and thus lead to a unique solution.  

If the system is nonlinear, similar ideas hold -- dimensionality is related to number of independent arbitrary constants (integrals)  in the "general solution."  However, in the nonlinear case, the number of derivatives (the order of the system) need not be the dimensionality of the system .  A simple example of this is to consider a differential equation -- ODE or PDE -- in which the sum of the absolute squares of the dependent variable and any number of its derivatives is equal to 0.  There is only one solution -- the zero solution -- regardless of how many ordinary and partial derivatves are inccluded. 

Thus, in talking rigorously about dimensionality, the idea that a PDE is necessarily infinite dimensional almost certainly refers to a linear PDE, in which case the infinitude of Fourier modes (e.g. Demetrius) implies the infinite dimensionality (Hazem, et al).  

Initial conditions are functions and the solution describes an evolution operator on a function space. Function spaces are, in general infinite dimensional.

When solving ODEs, the state spaces or the solution spaces one deals with are the Euclidean spaces  R^n . When solving PDEs,  the state spaces are continuous function spaces, L^p spaces, and so on.  All of them are infinite-dimensional because each basis is consisted of infinite members.  For instance,  {1, sin(\pi x), cos(\pi x), sin(2\pi x),cos(2\pi x), sin(3\pi x), cos(3\pi x), ...} is a basis of L^2(0,1) , it is infinite-dimensional.

I am not studing this topic deply,But one think I want to say from my personal experience and may be I am not correct is that If we consider  to solve an ODE of order n and finds its solution then it is clear that the  solution may be expressed as  a linear combination of n independent solutions, with n undetermined constants- a vector space of dimension n.

On the same point of view if we look at the solution of the  solution to a partial differential equation of order n then the solution can be expressed  as a linear combination of n independent solutions but with n undetermined functions.and it is clear that  the functions themselves constitute an infinite dimensional vector space.

The question is not clear.   There are many singlton PDE.  However, if you consider the solution of a PDE is function space then function space is infinite dimension in general.

Many thanks to all of you who replied to my query.

I think it would be better if I asked about the infinite dimensionality of  the solution space of a PDE.

From above discussion I have further two questions

1. Why is the function space in general is infinite dimensional? (Hazem has already answered that query but I need a detailed and more elaborative answer)

2. @ Jef Knisley, Sir why is it so that the tag of infinite dimensionality is associated with linear PDEs only?

Best Regards

Why is the function space in general is infinite dimensional?

The unsver can be fount in standard unuversity tetbooks on PDE

Let us say that you have two free variables x and t. Consider the initial value condition

at t=0 and x could be arbitrary at least locally. That is, you already know u(x, 0). Then you have an equation of t for each (fixed) x, if this is true just for an example. You could get a solution for each x.

Therefore, generically speaking, the solution is depended on the initial condition u(x, 0),

which is a function depended on x.

The solution space usually is only a kind of infinite dimensional variety (or manifold if it is kind of smooth). It is only a kind of linear space if the PDE is linear (this answer your second question).

Best,

DG