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For a given linear space, the collocation points must be placed judiciously to achieve optimal accuracy.

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The error is again controlled by adjusting the mesh spacing using local error estimates involving approximate solutions of varying orders of accuracy; see Ascher et al. Choosing a spline basis for collocation or more or less equivalently using certain types of Runge-Kutta formulas on the mesh leads to a nonlinear system which must be solved iteratively.

Ömür Uğur, PhD - Boundary Value Problems for Higher Order Linear Impulsive Differential Equations

At each iteration we must solve a structured linear system of equations. When the boundary conditions are separated, the system is almost block diagonal.

Similarly structured systems arise from finite difference approximations and also from multiple shooting techniques. Because of the great practical importance of this type of linear algebra problem, significant effort has been devoted to developing stable algorithms which minimize storage and maximize efficiency; see Amodio et al. The case of nonseparated boundary conditions leads to a similarly structured system whose solution poses potentially greater stability difficulties.

Another type of BVP that arises in the analytical solution of certain linear partial differential equations is the Sturm—Liouville eigenproblem. ODE eigenvalue problems can be solved using a general-purpose code shooting code that treats an eigenvalue as an unknown parameter. However, with such a code one can only hope to compute an eigenvalue close to a guess.

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Specialized codes are much more efficient and allow you to be sure of computing a specific eigenvalue; see Pryce for a survey. Numerical methods for Sturm—Liouville eigenproblems that have been implemented in software include finite difference and finite element discretizations which each lead to generalized algebraic eigenproblems where approximations to a number of the lower eigenvalues are available simultaneously.

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Boundary value problem, second-order homogeneous differential equation, distinct real roots

Post-publication activity Curator: Ian Gladwell Contributors:. Tobias Denninger. Skip Thompson. Sponsored by: Dr. Izhikevich , Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia Reviewed by : Dr. Namespaces Page Discussion. Views Read View source View history.

Contents 1 Introduction 2 Existence and uniqueness 3 Shooting or marching methods 4 Infinite intervals 5 Numerical methods 6 Sturm—Liouville eigenproblems 7 References 8 Recommended reading 9 External links 10 See also. Sponsored by: Eugene M.

Boundary Value Problems for Higher Order Linear Impulsive Differential Equations

References [1]. Coddington and N. Theory of Ordinary Differential Equations. McGraw-Hill New York, Ordinary Differential Equations in the Complex Domain. John Wiley, New York, Edwin L. Maxima by Example: Ch. Fornberg, B. A Practical Guide to Pseudospectral Methods. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, New York, Anton and C. Elementary Linear Algebra: Applications Version. WxMaxima Related Papers. Differential Equations. By grace cort.

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By ajer research. Download pdf. Gupta, Existence and uniqueness theorem for the bending of an elastics beam equation at resonance, J. Anal Appl. Ma, B.

Boundary Value Problems for Systems of Differential, Difference and Fractional Equations

Thompson, Nodal solutions for a nonlinear fourth-order eigenvalue problem, Acta Math. Yao, Local existence of multiple positive solutions to a singular cantiliver beam equation, J. Benaicha, Positive periodic solutions for a class of fourth-order nonlinear differential equations, Numerical Analysis and Applications. Benaicha, H. Djourdem, M. Benattia, Positive solutions for fourth-order two-point boundary value problem with a parameter, Romanian J.

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Benaicha, N. Bouteraa, Existence and iteration of monotone positive solution for a fourth-order nonlinear boundary value problem, Fundam. Hu, L. Wang, Multiple positive solutions of boundary value problems for systems of nonlinear second order differential equation, J. Liu, L. Liu, Y. Wu, Positive solutions for singular systems of three-point boundary value problems, Computers Math. Zhou, Y. Xu, Positive solutions of three boundary value problems for systems of nonlinear second order ordinary differential equation, J.

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