Error estimate for the modified Newton method with applications to the solution of nonlinear, two-point boundary-value problems

1983 ◽  
Vol 39 (4) ◽  
pp. 489-511 ◽  
Author(s):  
M. D. Smooke
Keyword(s):  
Error Estimate ◽  
Boundary Value Problems ◽  
Newton Method ◽  
Boundary Value ◽  
Point Boundary ◽  
Modified Newton Method

scholarly journals Improved error estimate and applications of the complete quartic spline

2019 ◽  
Vol 27 (2) ◽  
pp. 71-83
Author(s):  
Alexandru Mihai Bica ◽  
Diana Curilă ◽  
Zoltan Satmari
Keyword(s):  
Error Estimate ◽  
Boundary Value Problems ◽  
Numerical Method ◽  
Numerical Integration ◽  
Error Bound ◽  
Fourth Order ◽  
Boundary Value ◽  
Continuous Functions ◽  
Point Boundary ◽  
Lipschitzian Functions

AbstractIn this paper an improved error bound is obtained for the complete quartic spline with deficiency 2, in the less smooth class of continuous functions. In the case of Lipschitzian functions, the obtained estimate improves the constant from Theorem 3, in J. Approx. Theory 58 (1989) 58-67. Some applications of the complete quartic spline in the numerical integration and in the construction of an iterative numerical method for fourth order two-point boundary value problems with pantograph type delay are presented.

scholarly journals Iterative numerical method for fractional order two-point boundary value problems

2021 ◽  
Vol 38 (1) ◽  
pp. 47-55
Author(s):  
ALEXANDRU MIHAI BICA ◽  
Keyword(s):  
Differential Equations ◽  
Error Estimate ◽  
Boundary Value Problems ◽  
Numerical Method ◽  
Fractional Order ◽  
Boundary Value ◽  
Point Boundary ◽  
Boundary Problems ◽  
Numerical Example

In this paper we develop an iterative numerical method based on Bernstein splines for solving two-point boundary problems associated to differential equations of fractional order $\alpha\in\left( 0,1\right) $. The convergence of the method is proved by providing the error estimate and it is tested on a numerical example.

Bifurcation Analysis Of Nonlinear Parameterized Two-Point Bvps With Liapunov — Schmidt Reduced Functions

2008 ◽  
Vol 8 (4) ◽  
pp. 350-359
Author(s):  
M. HERMANN ◽  
T.H. MILDE
Keyword(s):  
Boundary Value Problems ◽  
Newton Method ◽  
Bifurcation Analysis ◽  
Control Parameter ◽  
Boundary Value ◽  
External Control ◽  
Point Boundary ◽  
Bifurcation Points ◽  
Graphical Technique ◽  
Branch Switching

Abstract In this paper, we study nonlinear two-point boundary value problems (BVPs) which depend on an external control parameter. In order to determine numeri-cally the singular points (turning or bifurcation points) of such a problem with so-called extended systems and to realize branch switching, some information on the type of the singularity is required. In this paper, we propose a strategy to gain numerically this information. It is based on strongly equivalent approximations of the corresponding Liapunov — Schmidt reduced function which are generated by a simplified Newton method. The graph of the reduced function makes it possible to determine the type of singularity. The efficiency of our numerical-graphical technique is demonstrated for two BVPs.

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