Research Article | Open Access
Abdelouahab Kadem, Adem Kilicman, "Note on the Solution of Transport Equation by Tau Method and Walsh Functions", Abstract and Applied Analysis, vol. 2010, Article ID 704168, 13 pages, 2010. https://doi.org/10.1155/2010/704168
Note on the Solution of Transport Equation by Tau Method and Walsh Functions
We consider the combined Walsh function for the three-dimensional case. A method for the solution of the neutron transport equation in three-dimensional case by using the Walsh function, Chebyshev polynomials, and the Legendre polynomials are considered. We also present Tau method, and it was proved that it is a good approximate to exact solutions. This method is based on expansion of the angular flux in a truncated series of Walsh function in the angular variable. The main characteristic of this technique is that it reduces the problems to those of solving a system of algebraic equations; thus, it is greatly simplifying the problem.
The Walsh functions have many properties similar to those of the trigonometric functions. For example, they form a complete, total collection of functions with respect to the space of square Lebesgue integrable functions. However, they are simpler in structure to the trigonometric functions because they take only the values 1 and −1. They may be expressed as linear combinations of the Haar functions , so many proofs about the Haar functions carry over to the Walsh system easily. Moreover, the Walsh functions are Haar wavelet packets. For a good account of the properties of the Haar wavelets and other wavelets, see . We use the ordering of the Walsh functions due to Paley [3, 4]. Any function can be expanded as a series of Walsh functions In , Fine discovered an important property of the Walsh Fourier series: the th partial sum of the Walsh series of a function is piecewise constant, equal to the mean of , on each subinterval . For this reason, Walsh series in applications are always truncated to terms. In this case, the coefficients of the Walsh (-Fourier) series are given by where is the average value of the function in the th interval of width in the interval , and is the value of the th Walsh function in the th subinterval. The order Walsh matrix, , has elements .
Let have a Walsh series with coefficients and its integral from 0 to have a Walsh series with coefficients : . If we truncate to terms and use the obvious vector notation, then integration is performed by matrix multiplication , where and is the unit matrix, is the zero matrix (of order ), see .
2. The Three-Dimensional Spectral Solution
In the literature there several works on driving a suitable model for the transport equation in 2 and 3-dimensional case as well as in cylindrical domain, for example, see , and by using the eigenvalue error estimates for two-dimensional neutron transport, see , by applying the finite element method in an infinite cylindrical domain, see , similarly by using Chebyshev spectral- method, see , and the discrete ordinates in the infinite cylindrical domain, see .
In this paper, we consider combined Walsh function with the Sumudu transform in order to extend the transport problem for the three-dimensional case by following the similar method that was proposed in . This method is based on expansion of the angular flux in a truncated series of Walsh function in the angular variable. By replacing this development in the transport equation, this will result a first-order linear differential system. First of all we consider the three-dimensional linear, steady state, transport equation which is given by where we assume that the spatial variable varies in the cubic domain , and is the angular flux in the direction defined by and . and denote the total and the differential cross section, respectively, describes the scattering from an assumed pre-collision angular coordinates to a postcollision coordinates and is the source term. See  for further details.
Note that, in the case of one-speed neutron transport equation; taking the angular variable in a disc, this problem will corresponds to a three dimensional case with all functions being constant in the azimuthal direction of the variable. In this way the actual spatial domain may be assumed to be a cylinder with the cross-section and the axial symmetry in . Then will correspond to the projection of the points on the unit sphere (the “speed”) onto the unit disc (which coincides with ). See  for the details.
Given the functions , , and describing the incident flux, we seek for a solution of (2.1) subject to the following boundary conditions.
For the boundary terms in , for , let For the boundary terms in and for , Finally, for the boundary terms in , for ,
Theorem 2.1. Consider the integrodifferential equation (2.1) under the boundary conditions (2.2), (2.3) and (2.4), then the function satisfy the following first-order linear differential equation system for the spatial component with the boundary conditions where with
Proof. Expanding the angular flux in a truncated series of Chebyshev polynomials and leads to We insert given by (2.9) into the boundary condition in (2.3), for . Multiplying the resulting expressions by and integrating over , we get the components for , Similarly, we substitute from (2.9) into the boundary conditions for , multiply the resulting expression by , and integrating over , to define the components . For , , where To determine the components , , and we substitute , from (2.3) into (2.1) and the boundary conditions for . Multiplying the resulting expressions by , and integrating over and we obtain one-dimensional transport problems, namely, with the boundary conditions where for , and . Finally, with Now, starting from the solution of the problem given by (2.13)–(2.17) for , we then solve the problems for the other components, in the decreasing order in and . Recall that . Hence, solving one-dimensional problems, the angular flux is now completely determined through (2.9).
Remark 2.2. If we deal with different type of boundary conditions, then we consider the first components and for and will satisfy one-dimensional transport problems subject to the same of boundary conditions of the original problem in the variable .
Now, we solve the first-order linear differential equation system with isotropic scattering, that is, constant. Assuming isotropic scattering, the equation (2.13) is written as for , , and .
Then, we have the following theorem that is subject to the boundary conditions (2.14).
Proof. For this problem we expand the angular flux in terms of the Walsh function in the angular variable with its domain extended into the interval . To end this, the Walsh function are extended in an even and odd fashion as follows, see : for . The important feature of this procedure relies on the fact that a function is defined in the interval it can be expanded in terms of these extended functions in the manner: where the coefficients and are determined as So, in order to use the Walsh function for the solution of the problem (3.1), the angular flux is approximated by the truncated expansion: Inserting this expansion into the linear transport (3.1), it turns out Multiplying (3.7) by , and integrating over the interval , results: Similarly, multiplying (3.7) by , and integrating yields: The integrals appearing in (3.8) and (3.9) are known and are given  as or where the notation denotes the sum of the binary digits and 
4. Operational Tau Method and Converting Transport Equation
The operational approach to the Tau method proposed by  describes converting of a given integral, integrodifferential equation or system of these equations to a system of linear algebraic equations based on three simple matrices: We recall the following properties .
Lemma 4.1. Let be a polynomial as then we have(i), ,(ii), ,(iii), where and .
In order to convert (5.1) to a system of linear algebraic equations we define the linear differential operator of order with polynomial coefficients defined by we will write for where is the degree of and , . We notice that is the independent variable and will be defined in a finite interval.
4.1. Matrix Representation for the Different Parts
Let be a polynomial basis given by , where is nonsingular lower triangular matrix and degree , for . Also for any matrix , .
Matrix Representation for the Integral Form
Let us assume that then we can write where with , , and for .
5. Error Estimation
Consider now the discrete ordinates approximation of the equation (2.13) for , In this part, we evaluate an error estimator for the approximate solution of (2.13), we suppose that equations (2.13) and (5.1) have the same boundary conditions. Let us call this error function of the approximate solution to where is the exact solution of (2.13). Hence, satisfies the following problem: We can evaluate the perturbation term by substituting the computed solution into the equation after doing some algebraic manipulations, the error functions satisfies the problem
In general, obtaining solutions of some integrodifferential equations are usually difficult. In our recent works we have used Walsh functions, Chebyshev polynomials and Lengendre polynomials in order to reduces these kind of equations. However our present work suggests that the Tau method can be a good approximation to the exact solutions. The application of the Tau method by using the orthogonal polynomials will be considered as a future work.
The authors thank the referee(s) for the helpful and significant comments that bring the attention of authors to the references [7–11] which update the bibliography. The authors gratefully also acknowledge that this research was partially supported by the University Putra Malaysia under the ScienceFund Grant no. 06-01-04-SF1050 and Research University Grant Scheme no. 05-01-09-0720RU.
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