• Journal of applied mathematics & informatics
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• v.27 no.5_6
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• pp.1361-1370
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• 2009

In this paper, we consider singular fourth-order two point boundary value problems $u^{(4)}$ (t) = f(t, u), 0 < t < 1, u(0) = u(l) = u'(0) = u'(l) = 0, where $f:(0,1){\times}(0,+{\infty}){\rightarrow}[0,+{\infty})$ may be singular at t = 0, 1 and u = 0. By using the upper and lower solution method, we obtained the existence of positive solutions to the above boundary value problems. An example is also given to illustrate the obtained theorems.

• Journal of the Korean Mathematical Society
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• v.37 no.5
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• pp.823-833
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• 2000

In this paper, we develop the method of quasilinearization comvined with the methos of upper and lower solutions for singular second order boundary value problems in weighted spaces. The sequences constructed converge uniformly and monotonically to the unique of the second singular order boundary value problem. Further we prove the rate of convergence is quadratic.

• 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.

• Journal of the Korean Mathematical Society
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• v.34 no.1
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• pp.247-255
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• 1997

In this paper, we are concerned with the existence of positive solutions for the boundary value problems of the form; $$(1) u"(t) + q(t)g(u(t)) = 0, 0 < t < 1 $$ $$ u(0) = 0 = u(1), $$ where q is singular at 0 and /or 1.or 1.

• Journal of the Korean Mathematical Society
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• v.37 no.5
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• pp.822-822
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• 2000

• Journal of applied mathematics & informatics
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• v.27 no.3_4
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• pp.895-902
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• 2009

In this paper, the singular three-point boundary value problem $$\{{{u"(t)\;+\;f(t,\;u)\;=\;0,\;t\;{\in}\;(0,\;1),}\atop{u(0)\;=\;0,\;u(1)\;=\;{\alpha}u(\eta),}}\$$ is studied, where 0 < $\eta$ < 1, $\alpha$ > 0, f(t,u) may be singular at u = 0. By mixed monotone method, the existence and uniqueness are established for the above singular three-point boundary value problems. The theorems obtained are very general and complement previous know results.

• Journal of applied mathematics & informatics
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• v.23 no.1_2
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• pp.141-152
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• 2007

In this paper a numerical method is presented to solve singularly perturbed two points boundary value problems for second order ordinary differential equations consisting a discontinuous source term. First, in this method, an asymptotic expansion approximation of the solution of the boundary value problem is constructed using the basic ideas of a well known perturbation method WKB. Then some initial value problems and terminal value problems are constructed such that their solutions are the terms of this asymptotic expansion. These initial value problems are happened to be singularly perturbed problems and therefore fitted mesh method (Shishkin mesh) are used to solve these problems. Necessary error estimates are derived and examples provided to illustrate the method.

• Journal of applied mathematics & informatics
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• v.28 no.1_2
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• pp.411-423
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• 2010

In this paper, a numerical method for singular perturbation problems arising in chemical reactor theory for general singularly perturbed two point boundary value problems with boundary layer at one end(left or right) of the underlying interval is presented. The original second order differential equation is replaced by an approximate first order differential equation with a small deviating argument. By using the trapezoidal formula we obtain a three term recurrence relation, which is solved using Thomas Algorithm. To demonstrate the applicability of the method, we have solved four linear (two left and two right end boundary layer) and one nonlinear problems. From the results, it is observed that the present method approximates the exact or the asymptotic expansion solution very well.

• Journal of applied mathematics & informatics
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• v.34 no.3_4
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• pp.193-206
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• 2016

In this paper, we study boundary value problems for fractional integrodifferential equations involving Caputo derivative in Banach spaces. A generalized singular type Gronwall inequality is given to obtain an important priori bounds. Some sufficient conditions for the existence solutions are established by virtue of fractional calculus and fixed point method under some mild conditions.

• Journal of applied mathematics & informatics
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• v.5 no.1
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• pp.111-124
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• 1998

Existence results of multiple solutions are obtained un-der suitable conditions for impulsive integrodifferential boundary value problems on the half-line which may be singular at the boundary.