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A class of non-uniform mesh three point arithmetic average discretization for $y^{\prime\prime} = f(x, y, y^{\prime}$ and the estimates of $y^{\prime}$. (English) Zbl 1104.65315
Summary: We propose two new non-uniform mesh three point arithmetic average discretization strategy of order two and three, to solve non-linear ordinary differential equation $y^{\prime\prime}= f(x, y, y^{\prime})$, $a < x < b$ and the estimates of first-order derivative $y^{\prime}$, where $y = y(x)$, subject to the essential boundary conditions $y(a) = A, y(b) = B$. Both methods are compact and directly applicable to singular problems. There is no need to discuss any special scheme for the singular problems. Error analysis of a proposed method is discussed. Numerical experiments are performed to study the convergence behaviors and efficiency of the proposed methods.

##### MSC:
 65L10 Boundary value problems for ODE (numerical methods) 34B16 Singular nonlinear boundary value problems for ODE 65L20 Stability and convergence of numerical methods for ODE 65L12 Finite difference methods for ODE (numerical methods) 65L70 Error bounds (numerical methods for ODE)
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##### References:
 [1] Keller, H. B.: Numerical methods for two points boundary value problems. (1968) · Zbl 0172.19503 [2] Bogucz, F. A.; Walker, J. D. A.: Fourth order finite difference methods for two point boundary value problems. IMA. J. Numer. anal. 4, 69-82 (1984) · Zbl 0544.65052 [3] Fox, L.; Mayers, D. F.: Numerical solution of ordinary differential equations. (1990) · Zbl 0643.34001 [4] Jain, M. K.; Iyengar, S. R. K.; Subramanyam, G. S.: Variable mesh methods for the numerical solution of two point singular perturbation problems. Comput. methods appl. Mech. eng. 42, 273-286 (1984) · Zbl 0514.65065 [5] Mohanty, R. K.; Evans, D. J.; Dey, Shivani: Three points discretization of order four and six for (du/dx) of the solution of non-linear singular two point boundary value problems. Int. J. Comput. math. 78, 123-139 (2001) · Zbl 0984.65075 [6] Evans, D. J.; Mohanty, R. K.: On the application of the SMAGE parallel algorithms on a non-uniform mesh for the solution of non-linear two-point boundary value problems with singularity. Int. J. Comput. math. 82, 341-353 (2005) · Zbl 1064.65064 [7] Mohanty, R. K.: A family of variable mesh methods for the estimates of (du/dr) and the solution of non-linear two-point boundary value problems with singularity. J. comput. Appl. math. 182, 173-187 (2005) · Zbl 1071.65113 [8] Chawla, M. M.; Shivakumar, P. N.: An efficient finite difference method for two-point boundary value problems. Neural parallel sci. Comput. 4, 387-396 (1996) · Zbl 1060.65626 [9] Stephenson, J. W.: Single cell discretization of order two and four for biharmonic problems. J. comput. Phys. 55, 65-80 (1984) · Zbl 0542.65051 [10] Hageman, L. A.; Young, D. M.: Applied iterative methods. (1981) · Zbl 0459.65014