• Title/Summary/Keyword: higher-order boundary value problems

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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.

A NOTE ON THE EXISTENCE OF SOLUTIONS OF HIGHER-ORDER DISCRETE NONLINEAR STURM-LIOUVILLE TYPE BOUNDARY VALUE PROBLEMS

  • Liu, Yuji
    • Journal of applied mathematics & informatics
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    • v.27 no.1_2
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    • pp.205-215
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    • 2009
  • Sufficient conditions for the existence of at least one solution of the boundary value problems for higher order nonlinear difference equations $\{{{{{\Delta^n}x(i-1)=f(i,x(i),{\Delta}x(i),{\cdots},\Delta^{n-2}x(i)),i{\in}[1,T+1],\atop%20{\Delta^m}x(0)=0,m{\in}[0,n-3],}\atop%20\Delta^{n-2}x(0)=\phi(\Delta^{n-1}(0)),}\atop%20\Delta^{n-1}x(T+1)=-\psi(\Delta^{n-2}x(T+1))}\$. are established.

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ON PERIODIC BOUNDARY VALUE PROBLEMS OF HIGHER ORDER NONLINEAR FUNCTIONAL DIFFERENCE EQUATIONS WITH p-LAPLACIAN

  • Liu, Yuji;Liu, Xingyuan
    • Communications of the Korean Mathematical Society
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    • v.24 no.1
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    • pp.29-40
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    • 2009
  • Motivated by [Linear Algebra and its Appl. 420(2007), 218-227] and [Linear Algebra and its Appl. 425(2007), 171-183], we, in this paper, study the solvability of periodic boundary value problems of higher order nonlinear functional difference equations with p-Laplacian. Sufficient conditions for the existence of at least one solution of this problem are established.

EXISTENCE OF MULTIPLE POSITIVE SOLUTIONS FOR THE SYSTEMS OF HIGHER ORDER BOUNDARY VALUE PROBLEMS ON TIME SCALES

  • Rao, A. Kameswara
    • Journal of applied mathematics & informatics
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    • v.33 no.1_2
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    • pp.1-12
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    • 2015
  • This paper is concerned with boundary value problems for systems of n-th order dynamic equations on time scales. Under the suitable conditions, the existence and multiplicity of positive solutions are established by using abstract fixed-point theorems.

HIGHER ORDER NONLOCAL NONLINEAR BOUNDARY VALUE PROBLEMS FOR FRACTIONAL DIFFERENTIAL EQUATIONS

  • Khan, Rahmat Ali
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.2
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    • pp.329-338
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    • 2014
  • In this paper, we study the method of upper and lower solutions and develop the generalized quasilinearization technique for the existence and approximation of solutions to some three-point nonlocal boundary value problems associated with higher order fractional differential equations of the type $$^c{\mathcal{D}}^q_{0+}u(t)+f(t,u(t))=0,\;t{\in}(0,1)$$ $$u^{\prime}(0)={\gamma}u^{\prime}({\eta}),\;u^{\prime\prime}(0)=0,\;u^{\prime\prime\prime}(0)=0,{\ldots},u^{(n-1)}(0)=0,\;u(1)={\delta}u({\eta})$$, where, n-1 < q < n, $n({\geq}3){\in}\mathbb{N}$, 0 < ${\eta},{\gamma},{\delta}$ < 1 and $^c\mathcal{D}^q_{0+}$ is the Caputo fractional derivative of order q. The nonlinear function f is assumed to be continuous.

ON THE BOUNDARY VALUE PROBLEMS FOR LOADED DIFFERENTIAL EQUATIONS

  • Dzhenaliev, Muvasharkhan T.
    • Journal of the Korean Mathematical Society
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    • v.37 no.6
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    • pp.1031-1042
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    • 2000
  • The equations prescribed in Ω⊂R(sup)n are called loaded, if they contain some operations of the traces of desired solution on manifolds (of dimension which is strongly less than n) from closure Ω. These equations result from approximations of nonlinear equations by linear ones, in the problems of optimal control when the control when the control actions depends on a part of independent variables, in investigations of the inverse problems and so on. In present work we study the nonlocal boundary value problems for first-order loaded differential operator equations. Criterion of unique solvability is established. We illustrate the obtained results by examples.

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THREE-POINT BOUNDARY VALUE PROBLEMS FOR HIGHER ORDER NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS

  • Khan, Rahmat Ali
    • Journal of applied mathematics & informatics
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    • v.31 no.1_2
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    • pp.221-228
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    • 2013
  • The method of upper and lower solutions and the generalized quasilinearization technique is developed for the existence and approximation of solutions to boundary value problems for higher order fractional differential equations of the type $^c\mathcal{D}^qu(t)+f(t,u(t))=0$, $t{\in}(0,1),q{\in}(n-1,n],n{\geq}2$ $u^{\prime}(0)=0,u^{\prime\prime}(0)=0,{\ldots},u^{n-1}(0)=0,u(1)={\xi}u({\eta})$, where ${\xi},{\eta}{\in}(0,1)$, the nonlinear function f is assumed to be continuous and $^c\mathcal{D}^q$ is the fractional derivative in the sense of Caputo. Existence of solution is established via the upper and lower solutions method and approximation of solutions uses the generalized quasilinearization technique.

SOLUTION OF TENTH AND NINTH-ORDER BOUNDARY VALUE PROBLEMS BY HOMOTOPY PERTURBATION METHOD

  • Mohyud-Din, Syed Tauseef;Yildirim, Ahmet
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.14 no.1
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    • pp.17-27
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    • 2010
  • In this paper, we apply homotopy perturbation method (HPM) for solving ninth and tenth-order boundary value problems. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The proposed iterative scheme finds the solution without any discretization, linearization or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method. The fact that the proposed homotopy perturbation method solves nonlinear problems without using Adomian's polynomials can be considered as a clear advantage of this technique over the decomposition method.

SOLUTIONS OF STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS FOR HIGHER-ORDER DIFFERENTIAL EQUATIONS

  • Liu, Yuji
    • Journal of applied mathematics & informatics
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    • v.24 no.1_2
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    • pp.231-243
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    • 2007
  • The existence of solutions of a class of two-point boundary value problems for higher order differential equations is studied. Sufficient conditions for the existence of at least one solution are established. It is of interest that the nonlinearity f in the equation depends on all lower derivatives, and the growth conditions imposed on f are allowed to be super-linear (the degrees of phases variables are allowed to be greater than 1 if it is a polynomial). The results are different from known ones since we don't apply the Green's functions of the corresponding problem and the method to obtain a priori bound of solutions are different enough from known ones. Examples that can not be solved by known results are given to illustrate our theorems.