• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.24 no.3
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• pp.243-291
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• 2020

In this paper, multi-block generalized backward differentiation methods for numerical solutions of ordinary differential and differential algebraic equations are introduced. This class of linear multi-block methods is implemented as multi-block boundary value methods (MB₂ VMs). The root distribution of the stability polynomial of the new class of methods are determined using the Wiener-Hopf factorization of a matrix polynomial for the purpose of their correct implementation. Numerical tests, showing the potential of such methods for output of multi-block of solutions of the ordinary differential equations in the new approach are also reported herein. The methods which output multi-block of solutions of the ordinary differential equations on application, are unlike the conventional linear multistep methods which output a solution at a point or the conventional boundary value methods and multi-block methods which output only a block of solutions per step. The MB₂ VMs introduced herein is a novel approach at developing very large scale integration methods (VLSIM) in the numerical solution of differential equations.

• Journal of applied mathematics & informatics
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• v.31 no.5_6
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• pp.733-745
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• 2013

In this paper, we first propose a new technique of the functional iterative methods VIM (Variational iteration method) and NHPM (New homotopy perturbation method) for solving two-point boundary value problems, and then we compare their numerical results with those of the finite difference method (FDM).

• Journal of the Korean Mathematical Society
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• v.45 no.1
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• pp.79-95
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• 2008

The local existence of solutions for the initial-boundary value problem of a generalized Boussinesq equation on the half line is considered. The approach consists of replacing he Fourier transform in the initial value problem by the Laplace transform and making use of modern methods for the study of nonlinear dispersive wave equation

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.23 no.3
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• pp.179-202
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• 2019

This paper presents second derivative generalized extended backward differentiation formulas (SDGEBDFs) based on the second derivative linear multi-step formulas of Cash [1]. This class of second derivative linear multistep formulas is implemented as boundary value methods on stiff problems. The order, error constant and the linear stability properties of the new methods are discussed.

• Journal of applied mathematics & informatics
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• v.5 no.3
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• pp.759-770
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• 1998

In this paper we use uniform cubic spline polynomials to derive some new consistency relations. These relations are then used to develop a numerical method for computing smooth approxi-mations to the solution and its first second as well as third derivatives for a second order boundary value problem. The proesent method out-performs other collocations finite-difference and splines methods of the same order. numerical illustratiosn are provided to demonstrate the practical use of our method.

• Journal of applied mathematics & informatics
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• v.8 no.1
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• pp.129-136
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• 2001

We introduce and discuss a new numerical method for solving system of second order boundary value problems, where the solution is required to satisfy some extra continuity conditions on the subintervals in addition to the usual boundary conditions. We show that the present method gives approximations which are better than that produced by other collocation, finite difference and spline methods. Numerical example is presented to illustrate the applicability of the new method. AMS Mathematics Subject Classification : 65L12, 49J40.

• Korean Journal of Mathematics
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• v.23 no.4
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• pp.665-732
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• 2015

Existence results for multiple positive solutions of two classes of boundary value problems for bilateral difference systems are established by using a fixed point theorem under convenient assumptions. It is the purpose of this paper to show that the approach to get positive solutions of boundary value problems of finite difference equations by using multi-fixed-point theorems can be extended to treat the bilateral difference systems with one-dimensional Laplacians. As an application, the sufficient conditions are established for finding multiple positive homoclinic solutions of a bilateral difference system. The methods used in this paper may be useful for numerical simulation. An example is presented to illustrate the main theorems. Further studies are proposed at the end of the paper.

• Journal of the Korean Mathematical Society
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• v.44 no.2
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• pp.249-260
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• 2007

By variational methods, we prove the existence of positive solutions of a class of indefinite weight semilinear elliptic boundary value problems on critical Sobolev exponent.

• Journal of applied mathematics & informatics
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• v.5 no.3
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• pp.749-758
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• 1998

We investigate some numerical methods for computing approximate solutions of a system of second order boundary value problems associated with obstacle unilateral and contact problems. We show that cubic spline method gives approximations which are better than that computed by higer order spline and finite difference techniques.

• Journal of the Korean Mathematical Society
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• v.59 no.5
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• pp.891-909
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• 2022

The main objective of this paper is to develop a concrete inverse formula of the system induced by the fourth-order finite difference method for two-point boundary value problems with Robin boundary conditions. This inverse formula facilitates to make a fast algorithm for solving the problems. Our numerical results show the efficiency and accuracy of the proposed method, which is implemented by the Thomas algorithm.