scholarly journals On Sturm–Liouville boundary value problems for second-order nonlinear functional finite difference equations

2008 ◽  
Vol 216 (2) ◽  
pp. 523-533 ◽  
Author(s):  
Yuji Liu
Keyword(s):  
Finite Difference ◽  
Boundary Value Problems ◽  
Difference Equations ◽  
Boundary Value ◽  
Second Order ◽  
Nonlinear Functional ◽  
Finite Difference Equations ◽  
Sturm Liouville

Application of discrete differential operators of periodic functions to solve 1D boundary-value problems

2020 ◽  
Vol 39 (4) ◽  
pp. 885-897 ◽  
Author(s):  
Tadeusz Sobczyk ◽  
Marcin Jaraczewski
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Difference Equations ◽  
Differential Operators ◽  
Nonlinear Differential Equations ◽  
Boundary Value ◽  
Periodic Functions ◽  
Content Type ◽  
Finite Difference Equations

Purpose Discrete differential operators (DDOs) of periodic functions have been examined to solve boundary-value problems. This paper aims to identify the difficulties of using those operators to solve ordinary nonlinear differential equations. Design/methodology/approach The DDOs have been applied to create the finite-difference equations and two approaches have been proposed to reduce the Gibbs effects, which arises in solutions at discontinuities on the boundaries, by adding the buffers at boundaries and applying the method of images. Findings An alternative method has been proposed to create finite-difference equations and an effective method to solve the boundary-value problems. Research limitations/implications The proposed approach can be classified as an extension of the finite-difference method based on the new formulas approximating the derivatives. This can be extended to the 2D or 3D cases with more flexible meshes. Practical implications Based on this publication, a unified methodology for directly solving nonlinear partial differential equations can be established. Originality/value New finite-difference expressions for the first- and second-order derivatives have been applied.

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