High Accuracy Arithmetic Average Discretization for Non-Linear Two Point Boundary Value Problems with a Source Function in Integral Form

conditions on a non-uniform mesh. The proposed variable mesh approximation is directly applicable to the integro-differential equation with singular coefficients. We need not require any special discretization to obtain the solution near the singular point. The convergence analysis of a difference scheme for the diffusion convection equation is briefly discussed. The presented variable mesh strategy is applicable when the internal grid points of the solution space are both even and odd in number as compared to the method discussed by authors in their previous work in which the internal grid points are strictly odd in number. The advantage of using this new variable mesh strategy is highlighted computationally.     1


Introduction
We consider the non-linear differential equation with a source function in integral form: The two point boundary conditions associated with (1) are given by: where 0  , 1  are finite constants.We assume that K(x, s) is a real valued function of both variables in the range 0 ≤ x, s ≤ 1.Let Then we may re-write (1) as Keller [1] has given the conditions under which the differential Equation (3) together with the boundary conditions (2) has a unique solution.We assume that these conditions are satisfied in the problem that we are considering.In addition, we assume that     6 0,1 u x C  and     4 , 0 K x s C  , 1 .Many physical problems from fluid mechanics, fluid dynamics, elasticity, magneto-hydrodynamics, plasma dynamics, oceanography, biological model, boundary layer theory, …etc are described mathematically by nonlinear integro-differential equations.Davis and Rabinowitz [2], Philips [3], Linz [4], Lakshmikantham and s Rao [5], Atkinson [6], and Agarwal and O'Regan [7] have discussed various techniques for numerical integration and methods for approximate solution of integrodifferential equations and their applications to various physical models.Most of the nonlinear differential equations cannot be solved analytically.So it is required to obtain efficient numerical methods.Jain et al. [8] have discussed variable mesh methods for the solution of two point nonlinear boundary value problems; however, their methods are not applicable to differential equations with singular coefficients.In recent years, ) have discussed a family of third order variable mesh methods for the solution of two point non-linear boundary value problems and obtained convergent solution for singular problems.More recently, Mohanty and Dhall [13] have proposed a three-point third order variable mesh method for the solution of non-linear integro-differential Equation (1), which is applicable only when the internal grid points of the solution region are odd in number.In this paper, we propose an efficient third order variable mesh method based on arithmetic average discretization for the solution of non-linear integro-differential Equation (1), which is applicable when the internal grid points of the solution region are both odd and even in number.In next section, we give mathematical details of the method.In Section 3, we discuss the application of the proposed method to an integro-differential equation with singular coefficients and study the convergence analysis.In Section 4, we give numerical results to justify the utility of the proposed new strategy.Final remarks are given in Section 5.

Mathematical Details of the Discretization
We discretize the solution region [0,1] with the nonuniform mesh such that be the variable mesh size in x-direction, where .Grid points are given by . The mesh ratio is , then it reduces to the constant mesh case.The off-step points are defined by Let the exact solution of u(x) at the grid point k x be denoted by and u be the approximate value of .

 
Let us construct a numerical method for evaluating the Using the derivation technique discussed in [13], we obtain a fourth order accurate integral formula based on Simpson's 1 3 rd rule (see Evans [14]).
  where Then on the variable mesh the value of the integral can be found by the repeated application of (4).Now we discuss the third order numerical method based on arithmetic average discretization for the differential Equation (3).
At the grid point k x , we denote   , , , Using Taylor expansion, from (3), we obtain We need the following approximations: and let From (9a) and (9b), it follows that It is then easy to see that   where "a" and "b" are parameters to be determined.
With the help of (10a) and (10b), from (11a) and (11b), we obtain and with the help of (12a) and (12b), it follows that Then at each internal grid point x k , the differential Equation (3) is discretized by where Now with the help of the approximations (9a), (9b) and ( 14), from ( 6) and ( 15), we obtain the local truncation error The proposed numerical method (15) to be of   16) must be zero and we obtain the values of parameters and the local truncation error given by ( 16) becomes  .For  = 1 or 2, the equation above represents cylindrical or spherical problem, respectively.Replacing the variable x by r and applying the formula (15) to the integro-differential equa-

Application to Singular Problems
Consider the linear second order model integro-differenttion (17), we obtain where 18) is directly applicable to singu he scheme (18), we use the follo a   (17) and do not require any fictitious points outside the solution region to compute the scheme.The scheme is also applicable when the internal grid points of the solution region are both even and odd in number as compared to the scheme discussed by Mohanty and Dhall [13] in which the internal grid points are strictly odd in number.
For convergence of t wing approximations: Similarly, we can define the approximations for Using the approximations (19) in (18), neglecting high order terms and simplifying we get the modified scheme in compact form where and In e boundary values   5 .
, , , , , e    U btracting (21) f be the discretization error (in the abse round of errors) at the grid point r k and be the error vector.
), we obtain the error equation

 
, , where G 1 , G 2 , G 3 are some positive constants.If p i, j be the (i, j) th -element of P, then Thus for suffic , the matrix (D reducible (see V oung [16]). , iently small h k arga [15] and Y + P) is i r Let S k be the sum of elements of the k th -row of (D + P), then It is straightforward to show that for sufficiently small h k , (D + P) is Monotone (see Varga [15] and Young [16]).Hence (D + P) -1 exists and (D + P) -1 ≥ 0.
From error Equation (23), we have for sufficiently small h k , it is easy to show that (26) For any matrix M, we define With the help of (28a), (28b), (29), from (26), we obtain This establishes the third order convergence of the method (20).

merical Results
In this section, we consider another ne method for the solution of non-linear in Equation (1) as )

Nu
where The approximations associated with ntegral by the trapezoidal rule (see Evans [14]).
In this section, we have solved two benchmark problems using the proposed method described by equation ( 15) and compared our results with those obtained by using the variable mesh method discussed by Mohanty and Dhall [13] only for the cases whe internal grid points are odd in number.We have also puted our ts using uniform mesh (when ) for all values e boundary conditions ma tained using the exact solution as a test procedure.The linear difference equation has been solved using a tri-diagon lver, whereas non-linear difference equations have been solved using the Newton-Raphson method (see Kelly [17] and of N. Th al so Ev aph the iterations were stopped when absolute error tolerance 1 , ans [18]).While using the Newton-R son method, ≤10 -12 was achieved.
The unit interval [0,1] in the space-direction is divided into (N + 1) points with We may write For simplicity, we consider (N + 2), we can compute the value of h 1 from (34).This is the first mesh spacing on the left of the boundary and the remaining mesh is determined by 2, , N, then from (33) we have By prescribing the total number of mesh points to be N. For variable mesh, we ch lues of , 1.2.

 
All computations were carr g double precision arithmetic.

ied out usin
Example 4.1 . The maximum absolute errors are tabulated in Table 1 for various values of N.   oposed va e but applicable to the lution space having both odd and even number of internal grid points.In addition, the proposed methods are directly applicable to singular problems and we do not require any fictitious points near the boundaries to inand appli tabulate either in increasing or in decreasing order.So it is not possible to estimate the order of convergence of the proposed method.Order of convergence can be estimated for unifo m mesh using the formula U new n ing three variable mesh points, we have discussed a   3 k O h he so ithmetic averag r rd order variable discretization, m wh mber of grid points.Although the pr riable sh method involve more algebra, m so corporate the singular point.The numerical results indicate that the proposed method is computationally nearly equal to the method discussed in [13] cable to the solution space with all internal grid points.We have d maximum absolute errors for different mesh sizes.Our mesh sizes are    (15) proposed method (15) [for uniform mesh 1

(
Model Burger's equation in polar coordinates) (36)The exact solution is given by   2 e r u r r   .The maximum absolute errors are tabulated in

errors for Example 4.2 Table 2. The maximu .
*: Results obtained by using the method discussed in[13].mabsolute