Mathematical Physics

T. A. Germogenova

On the Solution of the Transport Equation for Strongly Anisotropic Scattering

(Presented by Academician M. V. Keldysh, December 3, 1956)

For the solution of the boundary-value problem

\[ \cos\theta\,\frac{\partial \psi}{\partial \tau}+\psi(\theta,\varphi,\tau) = \frac{1}{4\pi}\int_{0}^{2\pi} d\varphi' \int_{0}^{\pi} \sin\theta'\,d\theta'\, \psi(\theta',\varphi',\tau)\,P(\cos\chi), \tag{1^1} \]

\[ \cos\chi=\cos\theta\cos\theta' + \sin\theta\sin\theta'\cos(\varphi-\varphi'), \]

\[ \psi(\theta,\varphi,0)=f_1(\theta,\varphi); \qquad \psi(\theta,\varphi,h)=f_2(\theta,\varphi), \tag{1^2} \]

which arises in the study of the scattering of radiation by a plane layer of a substance of finite optical thickness, the method of spherical harmonics or interpolation methods are usually applied \((^{1-3})\). These methods give good results in the case of a slowly varying function \(P(\cos\chi)\). Strongly anisotropic scattering (scattering of light in turbid media, scattering of neutron radiation with energies of several mega-electron-volts in heavy substances, scattering of charged particles) corresponds to functions \(P(\cos\chi)\) that vary sharply, with a high maximum in the region of small angles \(\chi\). The methods of the “small-angle approximation” \((^{4,5})\) developed recently, when applied to these problems, make it possible to find the solution with a sufficient degree of accuracy only in the region of small angles \(\theta\), whereas in a number of questions the character of the solution at large angles \(\theta\) is of interest \((^6)\).

The method proposed is a generalization of methods of interpolation type. A qualitative investigation of the transport equation or of the corresponding integral equation makes it possible to establish the form of the solution and, if it turns out to be a strongly varying function of the angles \(\theta\) and \(\varphi\), to separate the assumed singularities in the form of a known factor, so that the new unknown function is a sufficiently smooth function, representable with a high degree of accuracy by a polynomial of not very high order in \(\theta\) and \(\varphi\), with coefficients depending on \(\tau\). Such a polynomial may be sought in two ways, one of which is similar to the method of spherical harmonics, the other to Chandrasekhar’s method.

Let us turn to problems in which it is possible to disregard the dependence of the solution on the azimuth \(\varphi\) and therefore to reduce equation \((1^1)\) and the boundary conditions \((1^2)\) to the form

\[ \mu\,\frac{\partial \psi}{\partial \tau}+\psi(\mu,\tau) = \frac{1}{2}\int_{-1}^{+1} p(\mu,\mu')\,\psi(\mu',\tau)\,d\mu' + f(\mu,\tau), \tag{2^1} \]

\[ \psi(\mu,0)=0,\quad 0\leq \mu\leq 1; \qquad \psi(\mu,h)=0,\quad -1\leq \mu\leq 0. \tag{2^2} \]

Here \(\mu\) denotes \(\cos\theta\), and

\[ p(\mu,\mu')=\frac{1}{2\pi}\int_{0}^{2\pi} P(\cos\chi)\,d\varphi'. \]

Let us represent the solution and the functions \(p(\mu,\mu')\) and \(f(\mu,\tau)\) as products:

\[ \psi(\mu,\tau)=w(\mu)F(\mu,\tau), \qquad p(\mu,\mu')=w(\mu)w(\mu')\gamma(\mu,\mu'), \]

\[ f(\mu,\tau)=w(\mu)g(\mu,\tau), \]

where the “weight” \(w(\mu)\) roughly describes the sharp changes of \(\psi(\mu,\tau)\) when \(\mu\) varies.

In the interpolation \(n\)-approximation we shall seek the function \(F(\mu,\tau)\) in the form of Lagrange interpolation polynomials

\[ F^{(n)}(\mu,\tau)=\sum_{i=1}^{n} L_i^{(+)} F^{(n)}(\mu_i,\tau) \qquad \text{for } \mu>0, \]

\[ F^{(n)}(\mu,\tau)=\sum_{i=-1}^{-n} L_i^{(-)} F^{(n)}(\mu_i,\tau) \qquad \text{for } \mu<0. \tag{3} \]

Here \(L_i^{(+)}\) and \(L_i^{(-)}\) denote the Lagrange factors; \(\mu_i\) are the interpolation nodes.

To determine the coefficients \(F^{(n)}(\mu_i,\tau)\), we obtain a system of linear differential equations of first order

\[ \mu_i \frac{dF^{(n)}(\mu_i,\tau)}{d\tau} +F^{(n)}(\mu_i,\tau) -\frac{1}{2}\sum_{j=-n}^{n} a_j \gamma^{(n,n)}(\mu_i,\mu_j) F^{(n)}(\mu_j,\tau) = g^{(n)}(\mu_i,\tau) \]

\[ (i=-n,\ldots,-1,+1,\ldots,+n) \tag{4¹} \]

with boundary conditions

\[ F^{(n)}(\mu_i,0)=0,\quad i>0; \qquad F^{(n)}(\mu_i,h)=0,\quad i<0. \tag{4²} \]

The quadrature formulas used for the approximate representation of the integral entering the transport equation are such that the equalities

\[ \int_{0}^{1}\pi_l(\mu)[w(\mu)]^2\,d\mu = \sum_{j=1}^{n} a_j \pi_l(\mu_j), \qquad \int_{-1}^{0}\pi_k(\mu)[w(\mu)]^2\,d\mu = \sum_{j=-1}^{-n} a_j \pi_k(\mu_j), \]

hold if \(\pi_k(\mu)\) and \(\pi_l(\mu)\) are arbitrary polynomials of degrees \(k\) and \(l\); \(k,l\le n\) for any choice of interpolation nodes, and \(k,l\le 2n-1\) if zeros of the polynomials \(p_n(\mu)\) from systems of polynomials orthogonal with weight \([w(\mu)]^2\) on the intervals \((0,1)\) and \((-1,0)\) are taken as the interpolation nodes. The functions \(\gamma(\mu,\mu')\) and \(g(\mu,\tau)\) are represented in equations (4¹) by polynomials \(\gamma^{(n,n)}(\mu,\mu')\) and \(g^{(n)}(\mu,\tau)\), which coincide with the exact functions at the nodes \(\mu_i\).

In the polynomial \(n\)-approximation we shall seek the function \(\psi(\mu,\tau)\) for \(\mu>0\) in the form of an expansion in the system of polynomials \(\{p_k^{(+)}(\mu)\}\), orthogonal with weight \([w(\mu)]^2\) on the interval \((0,1)\), restricting ourselves to \(n\) terms of the expansion of the solution \(\psi(\mu,\tau)\), as well as of the functions \(p(\mu,\mu')\), \(f(\mu,\tau)\):

\[ \psi^{(n)}(\mu,\tau)=w(\mu)\sum_{r=1}^{n}\psi_r^{(n)}(\tau)p_{r-1}^{(+)}(\mu); \]

\[ p^{(n,n)}(\mu,\mu') = w(\mu)w(\mu') \sum_{r,s=1}^{n} P_{r,s}p_{r-1}^{(+)}(\mu)p_{s-1}^{(+)}(\mu'); \tag{5} \]

\[ f^{(n)}(\mu,\tau) = w(\mu)\sum_{r=1}^{n} f_r(\tau)p_{r-1}^{(+)}(\mu). \]

For \(\mu<0\) we shall use analogous expansions in the system of polynomials \(\{p_k^{(-)}(\mu)\}\), orthogonal with weight \([w(\mu)]^2\) on the interval \((-1,0)\). The coefficients \(\psi_r^{(n)}(\tau)\) \((r=-n,\ldots,-1,+1,\ldots,n)\) are determined by the system of differential equations

\[ b_{j-1}\frac{d\psi_{j-1}^{(n)}}{d\tau} +c_j\frac{d\psi_j^{(n)}}{d\tau} +d_{j+1}\frac{d\psi_{j+1}^{(n)}}{d\tau} +\psi_j^{(n)} -\frac12\sum_{k=-n}^{n} P_{jk}\psi_k^{(n)} =f_j(\tau),\qquad j>0; \]

\[ b_{j+1}\frac{d\psi_{j+1}^{(n)}}{d\tau} +c_j\frac{d\psi_j^{(n)}}{d\tau} +d_{j+1}\frac{d\psi_{j+1}^{(n)}}{d\tau} +\psi_j^{(n)} -\frac12\sum_{k=-n}^{n} P_{jk}\psi_k^{(n)} =f_j(\tau),\qquad j<0, \tag{6^1} \]

with boundary conditions

\[ \psi_j(0)=0,\quad j>0;\qquad \psi_j(h)=0,\quad j<0. \tag{6^2} \]

In deriving the system \((6^1)\), use was made of the relation, valid for an arbitrary system of polynomials orthonormal with weight,

\[ \mu p_j(\mu)=b_{j+1}p_{j+1}(\mu)+c_{j+1}p_j(\mu)+d_{j+1}p_{j-1}(\mu) \]

(\(b_j,\ c_j,\ d_j\) are numbers, \(d_{+1},\ d_{-1}=0\)).

An investigation of the systems \((4^1)\) and \((6^1)\) makes it possible to verify the validity of the following assertion:

Theorem 1. The solutions of problems (4) and (6) coincide and, consequently, the polynomial and interpolation methods for computing the function \(\psi(\mu,\tau)\) are equivalent if: 1) as interpolation nodes in expression (3) and in system \((4^1)\) the zeros of the polynomials \(p_n^{(+)}(\mu)\) and \(p_n^{(-)}(\mu)\) from systems of polynomials orthogonal with weight \([w(\mu)]^2\) on the intervals \((0,1)\) and \((-1,0)\), respectively, are taken; 2) the approximate functions \(p^{(n,n)}(\mu,\mu')\) and \(f^{(n)}(\mu,\tau)\) are the same in both cases.

Let us introduce the source functions of the exact problem \(I(\mu,\tau)\) and of the approximate problem \(I^{(n)}(\mu,\tau)\):

\[ I(\mu,\tau)=\frac12\int_{-1}^{+1}[w(\mu')]^2\gamma(\mu,\mu')F(\mu',\tau)\,d\mu'+g(\mu,\tau); \]

\[ I^{(n)}(\mu,\tau)=\sum_{i=-n}^{n} L_i(\mu)I^{(n)}(\mu_i,\tau), \]

\[ I^{(n)}(\mu_i,\tau)=\frac12\sum_{k=-n}^{n} a_k\gamma^{(n,n)}(\mu_i,\mu_k)F^{(n)}(\mu_k,\tau)+g^{(n)}(\mu_i,\tau). \]

The study of the integral equations corresponding to the exact (2) and approximate (4) problems makes it possible to formulate the following theorem on the convergence of the approximate function \(I^{(n)}(\mu,\tau)\) and of the first iteration of the solution \(\widetilde{\psi}^{(n)}(\mu,\tau)\), obtained with the aid of the integral equation of the approximate problem, to the corresponding solutions of the exact problem as the order \(n\) of approximation is increased.

Theorem 2. If the functions \(p(\mu,\mu')\) and \(f(\mu,\tau)\) possess bounded derivatives with respect to \(\mu\) and \(\mu'\), then the polynomial and interpolation methods defined above can be used to find sequences of approximate solutions \(I^{(n)}(\mu,\tau)\) and \(\widetilde{\psi}^{(n)}(\mu,\tau)\) that converge uniformly to the solutions of the exact problem. If, moreover, for the approximation of the functions \(p(\mu,\mu')\) and \(f(\mu,\tau)\) polynomials of best approximation in \(\mu\)

and the function \(p(\mu,\mu')\) has a sufficiently large number of derivatives with respect to \(\mu'\), then the estimates

\[ \left| I-I^{(n)} \right|\sim \frac{1}{n}, \qquad \left| \psi-\widetilde{\psi}^{(n)} \right|\sim \frac{1}{n} \]

hold.

If, as the approximate functions \(p(\mu,\mu')\) and \(f(\mu,\tau)\), the corresponding segments of the Fourier series (5) are taken, analogous estimates are valid for the mean-square deviations:

\[ \int_{-1}^{+1} [w(\mu)]^{2}\left[I-I^{(n)}\right]^2\,d\mu \sim \frac{1}{n^2}, \qquad \int_{-1}^{+1} \left[\psi-\widetilde{\psi}^{(n)}\right]^2\,d\mu \sim \frac{1}{n^2}. \]

In computing the approximate solutions (3) and (5), an essential point is the determination of the roots of the characteristic equations of the corresponding differential systems. Some information about the roots is provided by a study of the coefficient matrices of these systems. In particular, the following propositions are valid.

Theorem 3. In those cases where the function \(p(\mu,\mu')\) satisfies the inequalities

\[ \text{1) } p(\mu,\mu') \geqslant 0; \qquad \text{2) } \frac{1}{2}\int_{-1}^{+1} p(\mu,\mu')\,d\mu' \leqslant 1, \tag{7} \]

and, for representing the approximate function \(p^{(n,n)}(\mu,\mu')\), the first \(n^2\) terms of the expansion of the exact function \(p(\mu,\mu')\) in some polynomial system orthogonal with weight are taken, the roots of the characteristic equations of the approximate problems are real.

Conditions (7) are satisfied, for example, for all problems on the propagation of radiation in scattering and absorbing media.

Theorem 4. When solving, by the method of spherical harmonics or by Mertens’ method, the problems of isotropic scattering and of scattering according to the law \(P(\cos \chi)=1+\omega_1\cos\chi\), the roots \(\lambda_p^{(n)}\) of the characteristic equation of an approximation of order \(n\) are bounded below in modulus, \(\left|\lambda_p^{(n)}\right|>1\), with the exception of the double zero root. As \(n\to\infty\) and for fixed \(p\), \(\left|\lambda_p^{(n)}\right|\to 1\).

In some problems it proves convenient to carry out expansions of the functions \(\psi(\mu,\tau)\), \(p(\mu,\mu')\), and \(f(\mu,\tau)\) in systems of polynomials orthogonal on different parts of the interval \((-1,+1)\). For such a “multi-interval” variant, analogous theorems hold on the equivalence of the interpolation and polynomial methods, on convergence, and on the reality of the roots.

The proposed method can also be applied in solving problems that take into account the dependence of the sought function on the azimuth, and problems connected with more complicated geometry, for example spherical geometry.

The author expresses deep gratitude to Prof. E. S. Kuznetsov for supervising the work.

REFERENCES

1. R. E. Marshak, Phys. Rev., 71, 443 (1947).
2. S. Chandrasekhar, Radiative Transfer, IL, 1953.
3. E. S. Kuznetsov, Izv. AN SSSR, ser. geofiz., No. 4 (1951).
4. H. Bethe, Phys. Rev., 89, 1256 (1953).
5. J. W. Spencer, Phys. Rev., 90, 146 (1953).
6. J. N. Cooper, J. Rainwater, Phys. Rev., 97, 492 (1955).