Symmetry and Solution of Neutron Transport Equations in Nonhomogeneous Media

and Applied Analysis 3


Introduction
The group-theoretical analysis is known to be used for the construction of exact solutions of a number of linear and nonlinear equations of mathematical physics [1,2].One of the most efficient methods for the obtaining of explicit solutions is the method of group reduction [1][2][3][4][5][6][7][8][9][10][11].Finite transformations of the invariance group of differential equations can also be applied to generate new solutions (both exact and approximate).Moreover, based on the notion of the equivalence group of transformations [2], we broaden possibilities of applications of the group-theoretical methods by extending the class of admissible operators.These operators are not the symmetry operators in the rigorous classical sense.In the present paper we show that the group analysis can be applied to the construction of solutions for equations of mathematical physics with varying coefficients characterizing properties of the media.Using the equivalence group we show the connections between nonhomogeneity of the medium and the group transformations.The nonhomogeneity manifests itself by dependency of the coefficients in the equation on the variable .We show that from the solution for a homogeneous medium one can generate the solution for a nonhomogeneous medium and even obtain the results for the equations, in which the coefficients depend on time (nonstationary media), and for media with anisotropy.This approach enables one to obtain the class of not only the integrable, but also the nonintegrable equations in some sense.Regardless of integrability or nonintegrability of a given equation it is important whether or not the coefficients of this equation lie on one orbit of the action of the equivalence group, as we will show in Section 3. The following property takes place: if a potential lies on an orbit of a nonintegrable potential, then it is also nonintegrable, whilst if it lies on an orbit of an integrable potential, then it is also integrable.If the system of Lie equations is integrable and its solution is expressed by elementary functions, then by means of this group of transformations one can generate new nonintegrable potentials.

Application of Finite Group of Transformations to the Neutron Diffusion Problem
The group-theoretical methods are proved to be very effective for construction of exact solutions for many important equations in mathematical physics.In particular they enable one to reduce partial differential equation to ordinary differential equation by using the classical and conditional symmetry of initial equation.Integrating this reduced equation one can obtain exact solutions in explicit form, such as one-and many-soliton solutions.Moreover, the existence of Lie-Bäcklund symmetry enables one to construct conservation laws for initial equation [3][4][5][6][7][8][9][10][11].In this section we use another property of an invariance group, namely, the generation of the new solutions by action of the group of finite transformations on the known one.
While studying the problems of the theory (and interpretation) of geophysical fields, it is necessary to solve boundaryvalue problems for equations of the form where  is a parameter characterizing the nonhomogeneity of the medium and Φ is a smooth function.In mathematical formulation these problems are reduced to solve boundaryvalue problems for these equations.
Assume that (, , ) is a function parameter contained in (1) along with (, , ).Then in the extended space (, , , , ) (1) admits a sufficiently wide group of transformations of the form where () = (, ) + (, ) is an arbitrary analytic function.These groups of transformations were used in [12] for investigation of inverse problems of geophysics.In this paper we use the group of transformations to solve the direct problem, namely, the problem of stationary diffusion of neutrons in a nonhomogeneous medium for the linear and point source in two-and three-dimensional space.
As an example illustrating the efficiency of the given approach, consider the following problem of stationary diffusion equation: Here  is the diffusion length of thermal neutrons, which is piecewise continuous function depending on the distance  = √ 2 +  2 from point (, ) to the origin of coordinates in two-dimensional Euclidean space.As a geometrical model for an investigated heterogeneous medium we take three-zone cylindrical system with parameters  and  (coefficients of diffusion) given by relations Thus, the coefficient  is a piecewise constant function.This setting of the problem can be used for investigation of diffusion of particles (e.g., thermal neutrons) in a heterogeneous three-zone system that simulates a real system of "borehole layer" [13,14].Gradient variation of diffusion length  in the second zone is caused by penetration of the borehole fluid in the layer with absorption parameters different from the same parameters of fluid.
The solution of the problem in question in the first and third zones coincides with the corresponding solution for the infinite homogeneous medium with  = const.The solution in the middle zone is obtained from such solution [14] by the action of a finite group of transformations.
It is known that the sought solution must satisfy the continuity conditions for the flow of neutrons as well as for the normal component of a neutrons current across interfaces.It means that the functions Φ,  ⋅ (Φ/) must be continuous at points   ,  = 1, 2. These properties together with the condition at infinity and the condition of the neutrons balance where  is a number of neutrons, emitted by 1 cm length of linear source per 1 second, are used to determine the constants   ,   , on which the solution of the boundary-value problem for different source models of thermal neutrons depends.
The formulas for the coefficients   ,   seem to be rather complicated and therefore we give system of algebraic equations instead of them.From this (linear) system of equations one can easily calculate coefficients   ,   in each particular case.Finally the solution of this problem can be presented in the following form: (i) for a linear source of heat neutrons located on the symmetry axis of the system (iiA) for a thin cylinder layer emitting heat neutrons in case when a cylinder source is located in the inner homogeneous zone 0 ≤  <  1 ; (iiB) for a thin cylinder layer emitting heat neutrons in case when the source is located in the middle zone  1 <  <  2 ; that is, the gradient is nonhomogeneous in the radial direction; (iiC) for a thin cylinder layer emitting heat neutrons in case when a cylinder source is located in the exterior homogeneous zone  1 <  < ∞.
Here  * is a radius of a cylinder layer emitting thermal neutrons;  0 and  0 are modified Bessel functions of the first and second kinds.Solutions ( 8), ( 17), ( 26), (27), and (36) are obtained from the corresponding solutions for the homogeneous medium with  =  2 by using the transformations Further we consider the case when  is not constant but a smooth function on .By using the polar coordinates and assuming Φ = Φ(),  = (), Σ = Σ(), one can write the stationary neutron diffusion equation − div ( (, ) ⋅ ∇Φ (, )) + Σ (, ) ⋅ Φ (, ) = 0, (46) where (, ) is the coefficient of thermal neutron diffusion and Σ(, ) is the macroscopic cross-section of neutron absorption in the following form: After the substitution we obtain the equation which can be written in the form where the potential () is given by formula We have constructed the generator of one parameter Lie group of equivalence transformations for (50): where () is an arbitrary smooth function.Hence we obtain the finite group transformations: where τ() = ∫((1/())).Now we can generate new solution   for new potential   from a given solution.Let  = (),  =  1 (), where ,  1 are given smooth functions.Then we obtain If  =  2 = const, Σ = Σ 2 = const, then the solution of (47) is given by where  2 = const,  2 = const, and Thus  = √ 2  ⋅ Φ() and  =  1 = (1/4 2 ) − (Σ 2 / 2 ).By using (54), one can construct new solution   for new potential   and therefore So, the function Φ  (56) is the solution of (47) provided that () and Σ() satisfy the following equation: Thus, by virtue of the group method, we can construct the solution of neutron diffusion equation in nonhomogeneous medium where both coefficients  and Σ are not constant but variable functions on .Obviously, such a model extends to more adequately describe the properties of the nonhomogeneous medium.Note that two functions (), Σ() have to satisfy one differential equation.Hence, there is the class of nonhomogeneous media for which the function ( 56) is the solution of diffusion equation ( 47).The media are described by coefficients , Σ satisfying (57).By analogy with the previous example, making use of the conformal transformations, we can construct solutions of diffusion equations in the three-dimensional space (for a point source) for a nonhomogeneous medium where ( If we consider the parameter  as a dependent variable, then (58) is invariant with respect to the continuous Lie group of conformal transformations.We choose the following transformations, forming a discrete group with two elements: Applying transformations (60) to the solution of (58) in a homogenous medium for  =  2 , we construct the solution of ( 58), (59) for the point source in the following form: For the medium where  ̸ = const the stationary thermal neutron diffusion equation has the form By using the spherical coordinates  =  cos  sin ,  =  sin  sin ,  =  cos , one can write it in the following form: Then, after substitution: Φ() = ()/( ⋅ √()), we obtain the equation in the form where We can construct the solution of (63) in nonhomogeneous media from the solution in homogeneous medium when Σ() and () are constants, by using the group transformations (53).If  =  2 = const, Σ = Σ 2 = const, then the solution of ( 63) is given by where  2 = const,  2 = const, and Thus we have and solution for nonhomogeneous medium Then we can obtain Therefore the function Φ  (69) is the solution of (63) provided that () and Σ() satisfy the following differential equation: Thus, by using the group method, we can construct the solution of neutron diffusion equation for point source in nonhomogeneous medium.

Further Applications of the Method
In the framework of this approach one can also study nonstationary problems.It turns out that the group of transformations ( 2) is sufficiently wide to solve the Cauchy problem for (1).Using the invariance property of (1) with respect to transformations (2), we can easily verify that the following assertion is true.Thus, if the medium has a nonhomogeneity defined by the relation then the solution of the Cauchy problem for (1) can be obtained from the solution of the same problem for the homogeneous medium ( =  0 ), by transformations (2).One can obviously map a known solution of the Schrödinger equation with a given potential to the solution of the equation for a changed (in general time-dependent) potential as it has been done above for the neutrons diffusion equation.The group equivalence transformations are also used in this case.
Consider the Schrödinger equation in the -dimensional space.Symmetry properties of this equation were investigated in [15,16].For arbitrary (, ⃗ , ||), this equation admits only the group of identity transformations: If we treat  as a new independent variable, then we obtain quite a broad invariance group of (74).This approach is equivalent to the construction of the group of equivalence transformations [16,17].
where   (), (), and () are arbitrary smooth functions of  and we mean by the repeated index  the summation from 1 to .The upper dot stands for the derivative with respect to time.
Theorem 2 has been proved by using the Lie infinitesimal criterion of invariance in [16].Note that the invariance algebra (76) includes subalgebras such as the Galilean algebra and the projective algebra.
By using the Lie equations we obtain the following finite group transformations generated by operators from (76).
For operators   we get Thus any solution of the integrable (79) is transformed into the solution of (80).
For operators   one gets (86)