Numerical behavior of singular two-point boundary value problems in a comparative way

Article history: Received: 29 June 2017 Accepted: 15 November 2017 Available Online: 23 November 2017 This article concentrates on discovering numerical behavior of the singular twopoint boundary value problems through various numerical techniques. This is carried out in a comparative way by mainly using differential quadrature and finite element methods. Also a discussion has been done by means of advantages and disadvantages of the numerical methods of interest.To properly understand the behavior of the physical processes represented by the model equation, the calculated solutions have been discussed in detail.


Differential quadrature method (DQM)
The DQM was presented by Bellman at the beginning of the 1970s for solving differential equations [18].In the DQM, derivatives of a function with respect to a coordinate direction are expressed as linear weighted sums of all the functional values at all grid points along that direction.In this study we used the polynomial-based differential quadrature (PDQ) but a Fourier expansion-based differential quadrature can also be used depending on the physical structure of the problem [19,22].

Finite difference method (FDM)
The finite difference approaches for derivatives are one of the simplest and oldest methods for solving differential equations in the early 18th century.To solve differential equations numerically we can replace the derivatives in the equation with finite difference approximations on a discretized domain.A number of algebraic equations transformed from the differential equation can be solved by using a suitable method [26].In this study, we used the second-order finite difference (FD2) approximation and the fourthorder finite difference (FD4) approximation for solving the model equations.The details can be found, for instance, in reference [11].

Finite element method (FEM)
The FEM is a numerical method that appeared at the beginning of the 1950s to solve various problems of science [27][28].This method is based on the principle of mesh discretization of a continuous domain into a set of discrete subdomains, usually called elements.The process is to construct an integral of the inner product of the residual and the weight functions and set the integral to zero.In this study, we used the Galerkin FEM for solving the model problem.The process steps of the method can also be found in the literature [15,27].In summary, as pointed out in the above references, the FDM can be considered to be simpler and easier to implement than the FEM.However, the FEM can be seen to be relatively more effective on nonlineartiy and irregular domains.It is possible to find the results with sufficient accuracy by dividing the solution region into many elements in the FEM.If solution is achieved by separating the element into too many subregions, the required computational capacity and time will increase.However, the DQM requires less number of grids comparison to its rival methods.The FDM is easy to use and produce computer codes but is relatively less accurate.In order to observe those advantages and disadvantages of the methods properly, here, we used comparatively the three methods in solving the singular two-point BVPs.

Numerical illustrations
To demonstrate the efficiency and accuracy of the DQM, the FDM and the FEM, we have solved the following two problems(the first is a linear and the second is a non-linear) whose exact solutions are known.The performances of the approches are measured by the absolute and relative errors.
We used here the MATLAB code we produced for each method.The relative and absolute errors are presented, for N=7 in Table 1 and for N=50 in Tables show the absolute and relative errors for the DQM, FD2, FD4 and FEM results.The error measurements stemmed from the DQM is less than the others, as long as less number of grids is used.When the number of grids increases, the most effective results obtained from the FD4 among the methods of interest.
The relative and absolute errors are presented, for N=7 in Table 3 and for N=15 in Table 4 for uniform grids, respectively.The relative errors are plotted, for N=7 in Figure 3, for N=11 in Figure 4, respectively, with  0 = 0.5,  0 = 1.
From the produced results both qualitatively and quantitatively, the DQM has been seen to be the most accurate one among the methods for the problems of interest.

Conclusion
This study has focused on the singular two-point BVPs with a linear or non-linear nature through different numerical methods.It has been concluded that the DQM is the most accurate one among the corresponding methods for this problem.However, the FDM and FEM can take opportunity to catch the same accuracy for very large number of grid points.Note that for higher dimensional problems, the same discussion could be an important milestone in numerical modeling.In such a probable discussion, especially the advantages of the FEM and FDM may come out.

Table 1 .
Comparison of the relative and absolute errors in

Table 2 .
Comparison of the relative and absolute errors in

Table 3 .
Comparison of the relative and absolute errors in Problem 2 for N=7 (a) Sub-table 1.

Table 4 .
Comparison of the relative and absolute errors in Problem 2 for N=15 .(a) Sub-table 1.