• Title/Summary/Keyword: Two-point Boundary Value Problem

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A FIFTH ORDER NUMERICAL METHOD FOR SINGULAR PERTURBATION PROBLEMS

  • Chakravarthy, P. Pramod;Phaneendra, K.;Reddy, Y.N.
    • Journal of applied mathematics & informatics
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    • v.26 no.3_4
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    • pp.689-706
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    • 2008
  • In this paper, a fifth order numerical method is presented for solving singularly perturbed two point boundary value problems with a boundary layer at one end point. The two point boundary value problem is transformed into general first order ordinary differential equation system. A discrete approximation of a fifth order compact difference scheme is presented for the first order system. An asymptotically equivalent first order equation of the original singularly perturbed two point boundary value problem is obtained from the theory of singular perturbations. It is used in the fifth order compact difference scheme to get a two term recurrence relation and is solved. Several linear and non-linear singular perturbation problems have been solved and the numerical results are presented to support the theory. It is observed that the present method approximates the exact solution very well.

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POSITIVE SOLUTIONS FOR MULTI-POINT BOUNDARY VALUE PROBLEM OF FRACTIONAL FUNCTIONAL DIFFERENTIAL EQUATIONS

  • Wang, Haihua
    • Journal of applied mathematics & informatics
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    • v.30 no.1_2
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    • pp.147-160
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    • 2012
  • In this paper, we establish some sufficient conditions for the existence of positive solutions for a class of multi-point boundary value problem for fractional functional differential equations involving the Caputo fractional derivative. Our results are based on two fixed point theorems. Two examples are also provided to illustrate our main results.

THE METHOD OF QUASILINEARIZATION AND A THREE-POINT BOUNDARY VALUE PROBLEM

  • Eloe, Paul W.;Gao, Yang
    • Journal of the Korean Mathematical Society
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    • v.39 no.2
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    • pp.319-330
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    • 2002
  • The method of quasilinearization generates a monotone iteration scheme whose iterates converge quadratically to a unique solution of the problem at hand. In this paper, we apply the method to two families of three-point boundary value problems for second order ordinary differential equations: Linear boundary conditions and nonlinear boundary conditions are addressed independently. For linear boundary conditions, an appropriate Green\`s function is constructed. Fer nonlinear boundary conditions, we show that these nonlinearities can be addressed similarly to the nonlinearities in the differential equation.

ELLIPTIC BOUNDARY VALUE PROBLEM WITH TWO SINGULARITIES

  • Jung, Tacksun;Choi, Q-Heung
    • Korean Journal of Mathematics
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    • v.26 no.1
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    • pp.9-21
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    • 2018
  • We investigate existence and multiplicity of the solutions for elliptic boundary value problem with two singularities. We obtain one theorem which shows that there exists at least one nontrivial weak solution under some conditions on which the corresponding functional of the problem satisfies the Palais-Smale condition. We obtain this result by variational method and critical point theory.

Positive Solutions for Three-point Boundary Value Problem of Nonlinear Fractional q-difference Equation

  • Yang, Wengui
    • Kyungpook Mathematical Journal
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    • v.56 no.2
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    • pp.419-430
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    • 2016
  • In this paper, we investigate the existence and uniqueness of positive solutions for three-point boundary value problem of nonlinear fractional q-difference equation. Some existence and uniqueness results are obtained by applying some standard fixed point theorems. As applications, two examples are presented to illustrate the main results.

FUNCTIONAL ITERATIVE METHODS FOR SOLVING TWO-POINT BOUNDARY VALUE PROBLEMS

  • Lim, Hyo Jin;Kim, Kyoum Sun;Yun, Jae Heon
    • 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).

THE METHOD OF ASYMPTOTIC INNER BOUNDARY CONDITION FOR SINGULAR PERTURBATION PROBLEMS

  • Andargie, Awoke;Reddy, Y.N.
    • Journal of applied mathematics & informatics
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    • v.29 no.3_4
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    • pp.937-948
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    • 2011
  • The method of Asymptotic Inner Boundary Condition for Singularly Perturbed Two-Point Boundary value Problems is presented. By using a terminal point, the original second order problem is divided in to two problems namely inner region and outer region problems. The original problem is replaced by an asymptotically equivalent first order problem and using the stretching transformation, the asymptotic inner condition in implicit form at the terminal point is determined from the reduced equation of the original second order problem. The modified inner region problem, using the transformation with implicit boundary conditions is solved and produces a condition for the outer region problem. We used Chawla's fourth order method to solve both the inner and outer region problems. The proposed method is iterative on the terminal point. Some numerical examples are solved to demonstrate the applicability of the method.