Mathematical Methods of Physics/General theory

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As will be seen, the theory of Green's functions provides an extremely elegant procedure of solving differential equations. We wish to present here this method on a rigorous foundation.

The Dirac delta-function[edit]

The Dirac delta-function \delta (x)is not a function as it is ordinarily defined. However, we write it as if it were a function, keeping in mind the scope of the definition.

For any function f:\mathbb{R}\to\mathbb{R},we define

\int_{-\infty}^{\infty} f(x)\delta (x)dx=f(0) but for every \epsilon >0,

\int_{-\infty}^{-\epsilon}f(x)\delta (x) dx=\int_{\epsilon}^{\infty}f(x)\delta (x) dx=0

It follows that \int_{-\infty}^{\infty}\delta (x)dx=1

These conditions seem to be satisfied by a "function" \delta (x) which has value zero whenever x\neq 0, but has "infinite" value at x=0

Approximations[edit]

There are a few ways to approximate the delta function in terms of sequences ordinary functions. We give two examples

The Boxcar function[edit]

Boxcar function approximation.png

The boxcar function B_n:\mathbb{R}\to \{0,1\} such that

B_n(x) = \begin{cases} 0 & |x|>\tfrac{1}{2n} \\ 1 & |x|\leq\tfrac{1}{2n} \end{cases}

We can see that the sequence \left\langle B_n\right\rangle represents an approximation to the delta function.

The bell curve[edit]

Dirac function approximation.gif

The delta function can also be approximated by the ubiquitous Gaussian.

We write G_n(x)=\frac{n}{\sqrt{\pi}} \mathrm{e}^{-x^2n^2}

Green's function[edit]

Consider an equation of the type \mathcal{L}u(x)=F(x)...(1), where \mathcal{L} is a differential operator. The functions u,F may in general be functions of several independants, but for sake of clarity, we will write them here as if they were real valued. In most cases of interest, this equation can be written in the form

a(x)\frac{d^2u}{dx^2}+b(x)\frac{du}{dx}+c(x)=F(x) to be solved for u(x) in some closed set A, with a(x) being non-zero over A

Now, it so happens, that in problems of physics, it is much more convenient to solve the equation \mathcal{L}u(x)=f(x), when f is the delta function f(x)=\delta (x-x_0).

In this case, the solution of the operator \mathcal{L} is called the Green's function G(x,x_0). That is,

\mathcal{L}G(x,x_0)=\delta (x-x_0)

Now, by the definition of the delta-function, we have that F(x)=\int_{-\infty}^{\infty} F(x')\delta(x'-x)dx, where F(x') act as "weights" to the delta function.

Hence, we have, \mathcal{L}u(x)=\int_{-\infty}^{\infty} F(x')\mathcal{L}G(x,x')dx

Note here that\mathcal{L} is an operator that depends on x but not x'. Thus,

\mathcal{L}u(x)=\mathcal{L}\int_{-\infty}^{\infty} F(x')G(x,x')dx. We can view this as anologous to the inversion of \mathcal{L} and hence, we write

u_p(x)=\int_{-\infty}^{\infty} F(x')G(x,x')dx

The subscript p denotes that we have found a particular solution among the many possible. For example, consider any harmonic solution \mathcal{L}u_h(x)=0.

If we add u'(x)=u_h(x)+u_p(x), we see that u'(x) is still a solution of (1). Thus, we have a class of functions satisfying (1).

Boundary value problems[edit]

Problems of physics are often presented as the operator equation \mathcal{L}u(x)=F(x) to be solved for u on a closed set A, together with the boundary condition that u(x_b)=u_b(x_b) for all x_b\in\partial A (\partial A is the boundary of A).

u_b(x_b) is a given function satisfying \mathcal{L}u_b(x)=0 that describes the behaviour of the solution at the boundary of the region of concern.

Thus if a problem is stated as

\mathcal{L}u(x)=F(x) with

u(x_b)=u_b(x_b)

to be solved for u(x) over a closed set A,

The solution can be given as u_S(x)=u_p(x)+u_b(x)=\int_{-\infty}^{\infty} F(x')G(x,x')dx+u_b(x)

Green's functions from eigenfunctions[edit]

Consider the eigenvalues \lambda_n and the corresponding eigenfunctions \phi_n of the differential operator \mathcal{L}, that is \mathcal{L}\phi_n=\lambda_n\phi_n

Without loss of generality, we assume that these eigenfunctions are orthogonal. Further, we assume that they form a basis.

Thus, we can write u(x)=\sum_{i=1}^n\alpha_i\phi_i and F(x)=\sum_{i=1}^n\beta_i\phi_i.

Now \mathcal{L}u(x)=\mathcal{L}\sum_{i=1}^n\alpha_i\phi_i=\sum_{i=1}^n\alpha_i\mathcal{L}\phi_i=\sum_{i=1}^n\alpha_i\lambda_i\phi_i=\sum_{i=1}^n\beta_i\phi_i=F(x) and hence, \alpha_i=\frac{\beta_i}{\lambda_i}

by definition of orthogonality, \beta_n=\int_{-\infty}^{\infty}F(x)\phi_n(x)dx=(F(x)\cdot\phi_n(x))

Now, u(x)=\sum_{i=1}^n\alpha_i\phi_i(x)=\sum_{i=1}^n\frac{\beta_i}{\lambda_i}\phi_i(x)=\sum_{i=1}^n\frac{(F(x')\cdot\phi_i(x'))}{\lambda_i}\phi_i

and hence, we can write the Green's function as G(x,x')=\sum_{i=1}^{n}\frac{\phi_i(x)\phi_i(x')}{\lambda_i}