Compact alternating group explicit method for the cubic spline solution of two point boundary value problems with significant nonlinear first derivative terms

In this paper, we report the application of two parameter coupled alternating group explicit (CAGE) iteration and Newton-CAGE iteration methods for the cubic spline solution of non-linear differential equation u" = f(r,u,u') subject to given natural boundary conditions. The error analysis for CAGE iteration method is discussed in details. We compared the results of proposed CAGE iteration method with the results of corresponding two parameter alternating group explicit (TAGE) iteration method to demonstrate computationally the efficiency of the proposed method.


Introduction
Consider the two point boundary value problem L u r ð Þ ½ ≡ À u ″ r ð Þ þ f r; u; u 0 ð Þ¼0; 0 < r < 1 ð1Þ with natural boundary conditions where A and B are constants. We assume that for 0 < r < 1 and − ∞ < u,v < ∞ These conditions assure that the boundary value problem (1)-(2) has a unique solution (see Keller [1]).
During last four decades, there has been a growing interest in the theory of splines and their applications (see [2][3][4][5][6]). Bickley [7], Albasiny and Hoskins [8,9], and Fyfe [10] have demonstrated the use of cubic spline function for obtaining the second order approximation solution for two point boundary value problems. Later, Chawla and Subramanian [11] have constructed a fourth order cubic spline method for second-order mildly nonlinear two point boundary value problems. In 1983, Jain and Aziz [12] have first developed fourth order accurate numerical method based on cubic spline approximation for the solution of more general nonlinear two point boundary value problems. In the recent past, many authors (see [13][14][15][16][17][18][19]) have suggested various numerical methods based on cubic spline approximations for the solution of linear singular two point boundary value problems.
In 1985 Evans [20] developed group explicit method for solving large linear systems arising due to the discretization of differential equations. Later, Sukon and Evans [21] have introduced two parameter alternating group explicit (TAGE) iterative methods for the solution of tri-diagonal linear system of equations. Using the technique given in [20,21], Mohanty et al. [22][23][24] have discussed the application of TAGE iterative method to fourth order accurate cubic spline approximation for the solution of non-linear singular two point boundary problems. In this paper, we discuss two parameter coupled alternating group explicit (CAGE) and Newton-CAGE iteration methods, and fourth order cubic spline finite difference approximation and their application to linear and nonlinear differential equations with singular coefficients. In the next section, we discuss cubic spline approximation and its application to singular problems. In section 3, we discuss the CAGE and Newton-CAGE iteration methods and convergence analysis. In section 4, we compare the computational results obtained by using the proposed CAGE iterative method with the corresponding TAGE iterative method. Concluding remarks are given in section 5.

Cubic spline approximation and application
To obtain a cubic spline solution of the boundary value problem (1) and (2), we choose uniform mesh spacing h > 0 along the r-direction. The interval [0, 1] is divided into a set of points with interval of h = 1/(N + 1), N being a positive integer. The cubic spline approximation to equation (1) is obtained on [0,1] which consists of the central point r k = kh and the two neighboring points r k + 1 = r k + h and r k − 1 = r k − h, k = 1(1)N, where r 0 = 0 and r N+ 1 = 1. Let U k = u(r k ) be the exact solution of u at the grid point r k and is approximated by u k .
At each internal mesh point r k , we denote: Given the values u 0 , u 1 , . . ., u N + 1 of the function u(r) at the mesh points r 0 , r 1 , . . ., r N + 1 and the values of the second derivatives of u at the end points u 0 ″ and u N + 1 ″ , there exists a unique interpolating cubic spline function S(r) with the following properties: The interpolating cubic spline polynomial may be written as: We consider the following approximations: ; ð4:3Þ Then the cubic spline method with order of accuracy four for the differential equation (1) may be written as: where T k = O(h 6 ) (See Jain and Aziz [12]) with u 0 = A and u N + 1 = B. Let us discuss the application of the difference formula (5) to the following singular problems and where v = R − 1 > 0 is a constant and D(r) = − α/r and E(r) = α/r 2 , B(r) = − αv/r and E(r) = αv/r 2 . For α = 1 and 2, the linear singular equation (6) becomes cylindrical and spherical problems, respectively, and for α = 0, 1 and 2, the non-linear singular problem (7) represents steady-state Burger's equation in Cartesian, cylindrical and spherical coordinates respectively. Now applying the difference formula (5) to the singular equations (6) and (7) and using the technique discussed by Mohanty et al. [22], we may obtain the following fourth order difference scheme for the numerical solution of the differential equation (6), where and the following fourth order difference scheme for the numerical solution of the differential equation (7), where and In order to avoid the numerical complexity, we consider η = 1.
If the differential equation is linear, we can apply the two parameter CAGE iterative method and in the non-linear case, we can use the Newton-CAGE iterative method to obtain the solution.
We shall be concerned here with the situation where G 1 and G 2 are small (2 × 2) block systems. Now we discuss the case when N is odd (with x 0 = 0, x N + 1 = 1). Let and NÂN So that the system (10) can be re-written as Then a two parameter AGE method for solving the above system may be written as where z ðsÞ is an intermediate vector.
Eliminating z ðsÞ and combining equations (12) and (13), we obtain the iterative method where The new iterative method (14) or (15) is called the two parameter CAGE iterative method and the matrix T w is called the CAGE iteration matrix.
To prove the convergence of the method, we need to prove that S(T w ) ≤ 1, where S(T w ) denotes the spectral radius of T w .
Let λ i and μ i , i = 1(1)N, be the eigen values of G 1 and G 2 , respectively.
Now we discuss the CAGE algorithm, when N is odd. For simplicity we denote: and for ( p k p k+1 − c k a k+1 ) ≠ 0, we define d k = 1/(p k p k+1 − c k a k+1 ) for k = 1(1)N − 1. By carrying out the necessary algebra in equation (14), we obtain the following CAGE parallel algorithm: Let For k = 3(2) N-2 Δ ¼ r k r kþ1 À c k a kþ1 ≠ 0; Finally, for k = N: Similarly, we can write the CAGE algorithm when N is even. Now we discuss the two parameter Newton-CAGE iterative method for the non-linear difference equation (9). We follow the approaches given by Evans [25].
Let us define Then the Jacobian of φ(u) can be written as the Nthorder tri-diagonal matrix Now with any initial vector u (0) , we define where Δu (s) is the solution of the nonlinear system For the Newton-CAGE method, we consider the case when N is odd. We split the matrix T as T = T 1 + T 2 , where and Figure 3 Graph of the exact solution and the approximate solution for N = 60, R = 100 for problem 3.
Further the matrices (T 2 + ω 2 I) − 1 (T 1 + ω 1 I) − 1 (T 2 − ω 1 I) and (T 2 + ω 2 I) − 1 (T 1 + ω 1 I) − 1 can be evaluated in a manner suitable for parallel computing. In order for this Newton-CAGE method to converge, it is sufficient that the initial vector u (0) be close to the solution.
In a similar manner, we can write the Newton-CAGE algorithm when N is even.

Numerical illustrations
To illustrate the proposed CAGE iterative methods, we have solved the following four problems whose exact solutions are known. We have also compared the proposed CAGE iterative methods with the corresponding TAGE iterative methods. The right-hand side functions and boundary conditions can be obtained by using the exact solutions. The initial vector 0 is used in all iterative methods, and iterations were stopped when |u (s + 1) − u (s) | ≤ 10 − 10 was achieved. While solving non-linear difference equations, we have considered five inner iterations only.
The exact solution is u(x) = (1 − e − β(1 − r) )/(1 − e − β ).The root mean square (RMS) errors and the number of iterations both for CAGE and TAGE methods are presented in Table 1 for various values of β. The graph of the exact solution and the approximate solution for N = 80, β = 100 is give in the Figure 1.

Problem 2
u″ þ α r u 0 À α r 2 u ¼ f r ð Þ; 0 < r < 1; The exact solution is u r ð Þ ¼ e r 4 . The RMS errors and the number of iterations for both CAGE and TAGE methods are presented in Table 2 for α = 1,2. The graph of the exact solution and the approximate solution for N = 80, α = 2 is give in the Figure 2.
The exact solution is u(r) = β[1 − tanh(βr/2ν)]. The root mean square (RMS) errors and the number of iterations both for both Newton-CAGE and Newton-TAGE methods are presented in Table 3 for β = 1/2 and various values of R = v -1 . The graph of the exact solution and the approximate solution for N = 60, R = 100 is given in the Figure 3.
The exact solution is u(r) = r 2 cosh(r). The RMS errors and the number of iterations for both Newton-CAGE and Newton-TAGE methods are presented in Table 4 for α = 1,2 and various values of Re.The graph of the exact solution and the approximate solution for N = 80, R = 50 is given in the Figure 4.

Final remarks
The TAGE method requires two sweeps to solve a problem and also, it requires a lot of algebra for computational work, whereas the CAGE method requires only one-sweep to solve the problem. Experimentally, as compared to the TAGE method the corresponding CAGE method is requires very less number of iterations and better time because it uses less intermediate variables. We have solved four benchmark problems and numerical results show the efficiency of the proposed CAGE method. The results conclude that the two parameter CAGE method is competitive to solve the one-dimensional problem and it can be extended to solve multi-dimensional problems.