Geometric Mesh Three-Point Discretization for Fourth-Order Nonlinear Singular Differential Equations in Polar System

Numerical method based on three geometric stencils has been proposed for the numerical solution of nonlinear singular fourth-order ordinary differential equations. The method can be easily extended to the sixth-order differential equations. Convergence analysis proves the third-order convergence of the proposed scheme. The resulting difference equations lead to block tridiagonal matrices and can be easily solved using block Gauss-Seidel algorithm. The computational results are provided to justify the usefulness and reliability of the proposed method.

Or equivalently subject to the natural boundary conditions: Fourth-order differential equations occur in various areas of mathematics such as viscoelastic and inelastic flows, beam theory, Lifshitz point in phase transition physics (e.g., nematic liquid crystal, crystals, and ferroelectric crystals) [1], the rolls in a Rayleigh-Benard convection cell (two parallel plates of different temperature with a liquid in between) [2], spontaneous pattern formation in second-order materials (e.g., polymeric fibres) [3], the waves on a suspension bridge [4,5], geological folding of rock layers [6], buckling of a strut on a nonlinear elastic foundation [7], traveling water waves in a shallow channel [8], pulse propagation in optical fibers [9], system of two reaction diffusion equation [10], and so forth.
The existence and uniqueness of the solution for the fourth and higher-order boundary value problems have been discussed in [11][12][13][14]. In the recent past, the numerical solution of fourth-order differential equations has been developed using multiderivative, finite element method, Ritz method, spline collocation, and finite difference method [15][16][17][18]. The determination of eigen values of self adjoint fourthorder differential equations was developed in [19] using finite difference scheme. The motivation of variable mesh technique for differential equations arises from the theory of electrochemical reaction-convection-diffusion problems in one-dimensional space geometry [20]. The geometric mesh method for self-adjoint singular perturbation problems using 2 Advances in Numerical Analysis finite difference approximations was discussed in [21]. The use of geometric mesh in the context of boundary value problems was studied extensively in [22][23][24]. In this paper, we derive a geometric mesh finite difference method for the solution of fourth-and sixth-order differential boundary value problems with order of accuracy being three. The simplicity of the proposed method lies in its three-point discretization without any use of fictitious nodes. The scheme is compact and applicable to both singular and nonsingular problems. The resulting difference equations are solved using block Gauss-Seidel algorithm for linear case, and corresponding Newton's method has been applied to nonlinear problems.
The paper is outlined in the following manner: in Section 2, the derivation of the method is discussed in detail. In Section 3, we define the procedure for numerical solution to singular problems in such a way that the method retains the order and accuracy even in the vicinity of singularity. In Section 4, algorithmic details are provided for the numerical solution of sixth-order differential equations. The convergence property has been discussed briefly in Section 4. The numerical illustrations based on geometric mesh as well as uniform mesh were provided in Section 5. The paper is concluded in the last section with future development and remarks.
Applying the difference schemes (5) and (15) to (19) and (20), respectively, we obtain a system of coupled difference equations for = 1 (1) where 11 = , 11 = 1, Advances in Numerical Analysis Note that the scheme (21) fails when the solution is to be determined at = 1. We overcome this difficulty by modifying the scheme in such a way that the solutions retain order and accuracy even in the vicinity of singularity = 0. We consider the following approximations: Using the similar approximations of ±1 , ±1 and ±1 and neglecting (ℎ 5 ) terms, we can rewrite (21) wherẽ 21 = ((12 ( + 1) 2 ℎ 2 + 6 ( + 1) Advances in Numerical Analysis The modified scheme (24) is free from the terms 1/( ± 1), hence easily solved for = 1(1) . The difference equation (24) along with the boundary conditions (4) gives a 2 × 2 linear system of equations for the unknowns , , = 1(1) . The resulting block tridiagonal system can be easily solved using block Gauss-Seidel algorithm.

Extension to Sixth-Order Differential Equations
The proposed method can be easily extended to the sixthorder differential equations: subject to the necessary boundary conditions: or equivalently, subject to the natural boundary conditions: We outline the similar algorithm for (28) as follows: where the values of are same as obtained in Section 2.

Convergence Analysis
In this section, we derive the difference scheme of singular problem and investigate the convergence property of the proposed scheme. Consider the model problem For the convergence, the coefficients , , and associated with (5) and (15) must be negative (see, [22]), from which we obtain the condition | − √ 5/2| < 1/2. Now applying the methods (5) and (15) to (33) and using the similar technique discussed in Section 3 for singular coefficients ( ) and ( ), we obtain the following system of difference equations: where Incorporating the boundary values 0 = 0 , 0 = 1 , +1 = 0 , and +1 = 1 , the system of difference equations (34) in the matrix-vector form can be written as (38) Also, we obtain Thus for sufficiently small ℎ or equivalently as ℎ → 0, we obtain the relations ‖P ‖ ∞ = , = 2(1) and ‖R ‖ ∞ = 1, = 1(1) − 1. Hence, the graph G(M) of the matrix M is strongly connected, and thus the matrix M is irreducible (see, [28]).

Advances in Numerical Analysis
For = 1, For = 3(2)2 − 3, , For = 2 − 1, With the help of (48), we obtain the following bounds: From (38) and (52), we obtain the following error estimates: This proves the third-order convergence of the proposed method. We generalize the above results in the following theorem.

Computational Illustrations
To illustrate the geometric mesh finite difference method, we have solved both linear and nonlinear problems. The boundary conditions may be obtained from the analytical solution as a test procedure. The numerical accuracy of results are tested using maximum absolute errors and root mean square errors with the error tolerance being ≤10 −15 . For the simplicity in computation, we choose = = constant, for = 1(1) and define the geometric mesh as follows ( [24]): The subsequent mesh spacing is determined by ℎ +1 = ℎ , = 1(1) . If the boundary value problems exhibit layer behaviour near the left boundary (see, [21]), the solution value can be captured by choosing > 1. If the layer occurs at the right boundary, we choose < 1. If the layer occurs in the interior region, then mesh in the first half of the interval may be arranged by choosing > 1 and second half of the interval by choosing < 1.
All the numerical computations are performed using long double length arithmetic in under Linux operating system with 2 GB operational memory. Example 1. Consider the fourth-order linear problem (see, [31]) in the polar form: The analytical solution is ( ) = . The errors estimates for various values of are reported in Tables 1 and 2 for uniform mesh ( = 1) and geometric mesh ( ̸ = 1), respectively.

Example 2.
Consider the boundary value problems that arise from time-dependent Navier-Stokes equation (see, [32]) for axis symmetric flow of an incompressible fluid contained between infinite disks The analytical solution is ( ) = (1 − 2 ) . The errors estimates are reported in Table 3 for various values of and = 10 3 .
Advances in Numerical Analysis 9 The analytical solution is ( ) = sinh( ). The errors estimates for various values of are reported in Tables 4 and 5 for uniform mesh ( = 1) and geometric mesh ( ̸ = 1), respectively.
The analytical solution is ( ) = sinh( ). The errors estimates are reported in Table 6 for various values of and = 10 3 .

Conclusion
The numerical results confirm that the proposed geometric mesh finite difference scheme converges and applicable to both singular and nonsingular differential equations. The numerical accuracy obtained using geometric mesh shows superiority over corresponding uniform mesh. The optimum mesh ratio parameter within the specified convergent region may be obtained by simulations. We have employed block Gauss-Seidel method to solve the block matrix systems. The method can be extended to general even-order nonlinear differential equations. Application to the proposed scheme to nonlinear singular elliptic problems is an open problem.