Cubic Hermite Collocation Method for Solving Boundary Value Problems with Dirichlet , Neumann , and Robin Conditions

Cubic Hermite collocation method is proposed to solve two point linear and nonlinear boundary value problems subject to Dirichlet, Neumann, andRobin conditions. Using several examples, it is shown that the scheme achieves the order of convergence as four, which is superior to various well known methods like finite difference method, finite volume method, orthogonal collocation method, and polynomial and nonpolynomial splines and B-spline method. Numerical results for both linear and nonlinear cases are presented to demonstrate the effectiveness of the scheme.

In this study, cubic Hermite collocation method (CHCM) involves cubic Hermite basis function to reduce mathematical complexity.Different linear and nonlinear differential equations are solved subject to Dirichlet, Neumann, and Robin boundary conditions using the present method.Moreover, the decoupling technique [4] used to solve elliptic problems with Neumann and Dirichlet conditions is a particular case of present technique.In this paper, linear and nonlinear boundary value problems reported in recent papers [2,3,7,9,13,14] are solved using CHCM.It is worth mentioning that CHCM is giving better results than finite difference method, finite element method, finite volume method, Bspline method, and polynomial and nonpolynomial spline approach with fourth order of convergence.
The paper comprises five sections.Section 1 deals with general introduction of the problem.Section 2 gives brief description of cubic Hermite collocation method.In Section 3, symbolic solution of (1) is presented.Seven numerical examples are discussed in Section 4 and finally overall conclusions are given in Section 5.

Proposed Technique
In the present method, the domain is divided into finite elements and then orthogonal collocation method with cubic Hermite as basis function is applied within each element.
The grid points,   , are often called the "knots" of the piecewise polynomial since they are points where polynomials are "tied together." The Hermite polynomials do not require the subsidiary condition to make first derivative continuous.This fact reduces the number of equations by ( − 1), where  is the number of elements.
The global variable  varies in the th element, where  = 1, 2, . . ., .A new variable  = ( −   )/ℎ  is introduced in th element in such a way that as  varies from   to  +1 ,  varies from 0 to 1. Orthogonal collocation is applied on local variable .
Approximation of function () in the th element is given as [6] To apply the collocation method, one must evaluate the trial function (6) and its derivatives at two internal collocation points  =   ( = 1, 2).These are given by where the Hermite polynomials and their first and second derivatives are defined as where   's are the zeros of shifted orthogonal Legendre polynomial  (0,0) 2 () with  1 = 0.2113248654 and  2 = 0.7886751346, as shown in Figure 1.
It includes all parameters of the system and the dependent variables at the boundaries.The support of each Hermite cubic basis function spans at most two subintervals; therefore, a band matrix is obtained with bandwidth two (Figure 2).Of these, two unknowns are found using boundary conditions and rest 2 are from discretized system of (9), using Mathematica.After substituting the appropriate values of 's in (7), the result can be obtained for any element.  , where ℎ =  − ,  = /384.Also the placement of the collocation points plays a critical role in obtaining the (ℎ 4 ) estimate [20,21].For Gauss Legendre roots, as collocation points, an error estimate of (ℎ 4 ) is obtained whereas for other choices of the collocation points, only second-order accuracy is obtained.
Step 5. Carry out discretization of the model using Step 4.
Step 7. The obtained system in Step 6 is solved using any software.

Numerical Examples and Discussion
In this section, seven examples demonstrate the efficiency and accuracy of the method.Following formulae are used for estimation of error in this study.
Relative error is obtained by Example 1. Solve (1) for subject to Robin's boundary conditions  (0) = 1,  (1) (1) = 0, which has exact solution [9], for planer geometry The problem is solved for different values of dimensionless parameter  by taking 10 to 40 elements.The exact and numeric results are plotted in Figure 3.The results reported by [9] for 10 elements using finite difference method are matching with exact solution up to 3 decimal places, whereas using CHCM the results are matching up to 9 decimal places.This shows the superiority of cubic Hermite collocation method over the finite difference method.Relative error between CHCM and exact values is presented in Table 1.Example 2. Solve (1) for subject to Dirichlet boundary conditions which has exact solution [3,7]  () =  (1 −  −1 ) .
The exact and numeric results are plotted in Figure 4.The CHCM results are matching up to 11 decimal places with the exact ones in Table 2. On comparing present results with the results of [3,7], shown in Table 3, a big difference of errors between CHCM with finite difference method, finite element method, finite volume method, and B-spline method is observed.This indicates the supremacy of the present method.
subject to Dirichlet boundary conditions which has exact solution [2], for  2 = 10 and The exact and numeric results are plotted in Figure 5.The results are matching up to 9 decimal places as shown in Table 4. From Table 5, order of convergence is found to be 4.
An excellent matching is found between the exact and CHCM results in Figure 7 for  = 50.In Table 9, the order of convergence is again found to be 4 for Examples 6 and 7.

Conclusion
In this paper, cubic Hermite collocation method is tested for seven problems.The numerical results obtained are quite satisfactory and comparable with the existing solution available in the literature.The superiority over the finite difference method, finite element method, finite volume method, Bspline method, and polynomial and nonpolynomial spline approach shows the strength of this method.The convergence of the CHCM technique is of order 4.

Figure 1 :
Figure 1: Subdivision of mesh points on the global domain.The four coefficients, in each  element, are estimated by using four collocation points  0 ,  1 ,  2 , and  3 .

Figure 2 :
Figure 2: Pattern of nonzero elements of banded matrix arising from CHCM for  = 6, where ×'s represent nonzero elements and 0's are represented by dots.

Figure 3 :
Figure 3: Temperature profiles in a rectangular fin for dimensionless heat transfer coefficient .

Figure 4 :
Figure 4: Comparison of CHCM with exact result for Example 2.

Figure 7 :
Figure 7: Comparison of CHCM with exact result for Example 7.

Table 1 :
Relative error for different values of parameter  for Example 1.

Table 2 :
Relative error between CHCM and exact values for Example 2.

Table 3 :
Max norm of errors for five methods with respect to exact solution.

Table 4 :
Relative error between CHCM and exact values for Example 3.

Table 5 :
Rate of convergence of CHCM for Example 3.

Table 6 :
Maximum absolute errors obtained by different methods in Example 4.

Table 7 :
Maximum absolute errors obtained by different methods in Example 5.

Table 8 :
Rate of convergence of CHCM for Examples 4 and 5.

Table 9 :
Rate of convergence of CHCM for Examples 6 and 7.