• Title, Summary, Keyword: Boundary-Value Problems

### REPRODUCING KERNEL METHOD FOR SOLVING TENTH-ORDER BOUNDARY VALUE PROBLEMS

• Geng, Fazhan;Cui, Minggen
• Journal of applied mathematics & informatics
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• v.28 no.3_4
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• pp.813-821
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• 2010
• In this paper, the tenth-order linear boundary value problems are solved using reproducing kernel method. The algorithm developed approximates the solutions, and their higher-order derivatives, of differential equations and it avoids the complexity provided by other numerical approaches. First a new reproducing kernel space is constructed to solve this class of tenth-order linear boundary value problems; then the approximate solutions of such problems are given in the form of series using the present method. Three examples compared with those considered by Siddiqi, Twizell and Akram [S.S. Siddiqi, E.H. Twizell, Spline solutions of linear tenth order boundary value problems, Int. J. Comput. Math. 68 (1998) 345-362; S.S.Siddiqi, G.Akram, Solutions of tenth-order boundary value problems using eleventh degree spline, Applied Mathematics and Computation 185 (1)(2007) 115-127] show that the method developed in this paper is more efficient.

### ANALYSIS OF SOME NONLOCAL BOUNDARY VALUE PROBLEMS ASSOCIATED WITH FEEDBACK CONTROL

• Lee, Hyung-Chun
• Bulletin of the Korean Mathematical Society
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• v.35 no.2
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• pp.325-338
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• 1998
• Some nonlocal boundary value problems which arise from a feedback control problem are considered. We give a precise statement of the mathematical problems and then prove the existence and uniqueness of the solutions. We consider the Dirichlet type boundary value problem and the Neumann type boundary value problem with nonlinear boundary conditions. We also provide a regularity results for the solutions.

### NON-POLYNOMIAL QUARTIC SPLINE METHOD FOR SOLVING TWELFTH ORDER BOUNDARY VALUE PROBLEMS

• Proceedings of the Jangjeon Mathematical Society
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• v.21 no.4
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• pp.645-660
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• 2018
• In this paper, a non-polynomial quartic spline method is presented to obtain the approximate solution of twelfth-order boundary value problems with two point boundary conditions. For the employment of the method, the given problem is decomposed into a system of sixth order boundary value problems. Convergence analysis of the method for second and fourth order has been discussed. Numerical examples are given to demonstrate the accuracy and efficiency of the developed method. Also, the results obtained by this method have been compared with the other existing methods.

### POSITIVE SOLUTIONS OF SINGULAR FOURTH-ORDER TWO POINT BOUNDARY VALUE PROBLEMS

• Li, Jiemei
• 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.

### EXISTENCE OF POSITIVE SOLUTIONS TO NONLOCAL BOUNDARY VALUE PROBLEMS WITH BOUNDARY PARAMETER

• Lee, Eun Kyoung;Park, Yena
• East Asian mathematical journal
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• v.32 no.5
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• pp.621-633
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• 2016
• We establish the existence of positive solutions to nonlocal boundary value problems with integral boundary condition and non-negative real boundary parameter by mainly using the Schauder-Fixed point theorem.

### 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.

### Prior Estimates and Solutions of Boundary Value Problems for Higher-Order Nonlinear Finite Difference Equations

• Liu, Yuji
• Kyungpook Mathematical Journal
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• v.47 no.4
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• pp.569-578
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• 2007
• Sufficient conditions for the existence of at least one solution of boundary value problems for higher order nonlinear difference equations are established. We allow $f$ to be at most linear, superlinear or sublinear in obtained results.

### MULTIPLE POSITIVE SOLUTIONS OF INTEGRAL BOUNDARY VALUE PROBLEMS FOR FRACTIONAL DIFFERENTIAL EQUATIONS

• Liu, Xiping;Jin, Jingfu;Jia, Mei
• Journal of applied mathematics & informatics
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• v.30 no.1_2
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• pp.305-320
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• 2012
• In this paper, we study a class of integral boundary value problems for fractional differential equations. By using some fixed point theorems, the results of existence of at least three positive solutions for the boundary value problems are obtained.

### AN EXISTENCE RESULT OF POSITIVE SOLUTIONS FOR SINGULAR SUPERLINEAR BOUNDARY VALUE PROBLEMS AND ITS APPLICATIONS

• Lee, Yong-Hoon
• 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.

### A FIFTH ORDER NUMERICAL METHOD FOR SINGULARLY PERTURBED DIFFERENTIAL-DIFFERENCE EQUATIONS WITH NEGATIVE SHIFT

• Chakravarthy, P. Pramod;Phaneendra, K.;Reddy, Y.N.
• Journal of applied mathematics & informatics
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• v.27 no.1_2
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• pp.441-452
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• 2009
• In this paper, a fifth order numerical method is presented for solving singularly perturbed differential-difference equations with negative shift. In recent papers the term negative shift has been using for delay. Similar boundary value problems are associated with expected first exit time problem of the membrane, potential in models for neuron and in variational problems in control theory. In the numerical treatment for such type of boundary value problems, first we use Taylor approximation to tackle terms containing small shifts which converts it to a boundary value problem for singularly perturbed differential equation. 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 and is solved using the boundary conditions. Several numerical examples are solved and compared with exact solution. It is observed that present method approximates the exact solution very well.