Team Quanta gladly presents all short questions of Methods of Mathematical Physics – II’s Chapter#5: Green’s Function.
Q.1 what is the definition of Green’s Function?
Answer: Consider linear differential operator $\alpha$ such that
$$dy\left(x\right)=f\left(x\right)\ \ \ldots\ldots\ldots\ldots\ldots(1)$$
Suppose x is continuous in interval [a,b]. There exists a function ) which has characteristics that when alpha operator on ) we get Dirac Delta function.
$\alpha G\left(x,x^\prime\right)=\delta(x,x^\prime)$ ………(2)
From eq(1) $\alpha y\left(x\right)=f\left(x\right)$
$$y\left(x\right)=\alpha^{-1}f(x)$$
Where is equal to differential operator. $\alpha^{-1}$ is integration operator and $G\left(x,x^\prime\right)$ is Green’s function.
$$y\left(x\right)=\int_{\ }^{\ }{f(x^\prime})G\left(x,x^\prime\right)dx$$
Q.2 Explain the properties of Green’s function?
Answer: It has following properties such as;
- The interval $a\le x\le b$ divided by parameter t. we label
$$G\left(x\right)=G_1\left(x\right)\ \ \ for\ \ \ a\le x\le t$$
$$G\left(x\right)=G_2\left(x\right)\ \ \ for\ \ \ t\le x\le b$$
- $G_1\left(x\right)and\ G_2\left(x\right)$ Satisfy homogeneous equation.
$${\mathcal{L}G}_1\left(x\right)=0\ \ \ \ \ for\ \ \ a\le x\le t$$
$${\mathcal{L}G}_2\left(x\right)=0\ \ \ \ \ for\ \ \ t\le x\le b$$
- At x=0, satisfy boundary condition impose on y(x) at x=b, $G_2\left(x\right)$ satisfy boundary condition impose on y(x)
$$y\left(a\right)=0\ \ \ ,\ \ y^\prime\left(a\right)=0$$
$$\alpha y\left(a\right)\pm\beta y^\prime\left(a\right)\ and\ similar\ at\ x=b$$
$$y\left(b\right)=0\ \ \ ,\ \ y^\prime\left(b\right)=0$$
$$\alpha y\left(b\right)+\beta y^\prime\left(b\right)=0$$
- G is continuous
$$\lim_{x\rightarrow t^-}{G_1\left(x\right)=\lim_{x\rightarrow t^+}{G_2(x)}}$$
- G’ is discontinuous
$$\frac{d}{dx}G_2\left(x\right)\left|\begin{matrix}\ \t\\end{matrix}-\frac{d}{dx}G_1\left(x\right)\right|\begin{matrix}\ \t\\end{matrix}=-\frac{1}{p(t)}$$
Q.3 what is the function of green?
Answer: In mathematics, a Green’s function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
Q.4 what is green function in interval equation?
Answer: The Green’s function integral equation method (GFIEM) is a method for solving linear differential equations by expressing the solution in terms of an integral equation, where the integral involves an overlap integral between the solution itself and a Green’s function.
Q.5 what is Green function in mathematical physics?
Answer: The Green function is the Kernel of the integral operator inverse to the differential equation and the homogeneous boundary conditions.
Q.6 what is Green function in physics?
Answer: A Green’s function is a solution to an inhomogeneous differential equation with a ‘’driving team’’ that is a delta function. It provides a convenient method for solving more complicated inhomogeneous differential equation.
Q.7 what is the general formula of Green’s function?
Answer: The formula is;
$$G\left(x,t\right)=\left{\begin{matrix}C_1u\left(x\right)\ \ \ \ \ ,\ \ \ \ a\le x\le t\C_2v\left(x\right)\ \ \ \ ,\ \ \ t\le x\le b\\end{matrix}\right}$$
$$G\left(x,t\right)=\left{\begin{matrix}-v(t)\frac{u(x)}{A}&a\le x\le t\-u(t)\frac{v(x)}{A}&t\le x\le b\\end{matrix}\right}$$
Q.8 why is Green function important?
Answer: The primary use of Green’s function in mathematics is to solve non-homogeneous boundary value problems. In modern theoretical physics, Green’s function is also usually used as propagations in Feynman diagrams; the term Green’s function is often further used for any correlation function.
Q.9 Is Green function continuous?
Answer: The Green function of L is the function G(x,e) that satisfies the following conditions;
- G(x,e) is continuous and has continuous derivatives with respect to x up to order n.
- For all values of x and e in the interval [a,b].
Q.10 why do we use Green’s function in solving boundary value problems?
Answer: The Green’s function satisfies several properties, which we will explore further in the next section. For example, the Green’s function satisfies the boundary conditions at x=a and x=b.
Q.11 what is the Green function of a given differential equation?
Answer: In mathematics, a Green’s function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
Q.12 what is green formula?
Answer: The Greens formulas are obtained by integration by parts of intervals of the divergence of a vector field that is continuous in D=D+T and that is continuously differentiable.
Q.13 Write the formula of Green’s function in terms of Eigen value?
Answer: The Green’s function in terms of Eigen value is;
$$G\left(x,x^\prime\right)=\frac{\sum_{n}{u_n\left(x\right)u_n^\ast\left(x\right)f(x^\prime)}}{\lambda_n-\lambda}$$
Q.14 why Eigen expansion is important in Green function?
Answer: The Eigen function expansion of Green’s function in makes the symmetry property;
$$G\left(r_1,r_2\right)=G(r_2,r_1)$$
Explicit and is often useful when comparing with solutions obtained by other means.
Q.15 what is the Green’s function depend on?
Answer: Green’s function will depends on the B.C.
$$y\left(r_1\right)=\int{G\left(r_1,r_2\right)f\left(r_2\right)dr_2}$$
- $G\left(r_1,r_2\right)$ is solution of
- It gives integral solution of DE.
- $G\left(r_1,r_2\right)$ gives effect of unit point source charge at r1 in producing potential at r2 ;
$$\ \nabla^2G=-\delta(r_1-r_2)$$
Q.16 what is Helmholtz homogeneous equation formula?
Answer: Helmholtz homogeneous equation is;
$$\nabla^2\mathrm{\Psi}_n+k_n^2\mathrm{\Psi}_n=0$$
Q.17 what is u(x) in the Green function?
Answer: U(x) is the solution of homogeneous equation that satisfies at x=0
Q.18 what is v(x) in Green’s function?
Answer: V(x) is the solution of homogeneous equation that satisfies at x=b
Q.19 when the boundary value problem is called as non-homogeneous?
Answer: A boundary value problem is homogeneous if in addition to g(x)=0
We also have If any of these are not zero we will call the boundary value problem (BVP) non-homogeneous.
Q.20 If the function is symmetric then the Green function is?
Answer: By symmetry,
$$G\left(x,t\right)=G\left(t,x\right)$$
i.e $\left|r_1-r_2\right|=\left|r_2-r_1\right|$
Q.21 why is discontinuous?
Answer: We require that $G^\prime\left(x\right)$ is be discontinuous, specially that
$$\frac{d}{dx}G_2\left(x\right)\left|\begin{matrix}\ \t\\end{matrix}-\frac{d}{dx}G_1\left(x\right)\right|\begin{matrix}\ \t\\end{matrix}=-\frac{1}{p(t)}$$
Where p(t) comes from the self-adjoint operator. Note that the first derivative discontinuous, the second derivative does not exist.
Q.22 what is Abel’s theorem in Green function?
Answer: Abel’s theorem;
$$C_1=-\frac{V(t)}{A}$$
$$C_2=-\frac{U(t)}{A}$$
