Corrigendum to: “A kinetic approach to comparison properties for degenerate parabolic–hyperbolic equations with boundary conditions” [J. Differential Equations 230 (2006) 682–701]

Thenumerical solution of ordinary differential equations (ODEs)with boundary conditions is studied here. Functions are approximated by polynomials in a Chebychev basis. Sections then cover spectral discretization, sampling, interpolation, differentiation, integration, and the basic ODE. Following Trefethen et al., differential operators are approximated as rectangular matrices. Boundary conditions add additional rows that turn them into square matrices. These can then be diagonalized using standard linear algebra methods. After studying various simple model problems, this method is applied to the Orr–Sommerfeld equation, deriving results originally due to Orszag. The difficulties of pushing spectral methods to higher dimensions are outlined.

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The inverse problem of recovering the coefficients of a differential equations on a graph

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2020-0070 ◽

2020 ◽

Vol 28 (5) ◽

pp. 727-738

Author(s):

Victor Sadovnichii ◽

Yaudat Talgatovich Sultanaev ◽

Azamat Akhtyamov

Keyword(s):

Inverse Problem ◽

Inverse Problems ◽

Differential Equations ◽

Boundary Value Problem ◽

Boundary Conditions ◽

Finite Number ◽

Characteristic Equation ◽

Arbitrary Number ◽

New Class ◽

Finite Set

AbstractWe consider a new class of inverse problems on the recovery of the coefficients of differential equations from a finite set of eigenvalues of a boundary value problem with unseparated boundary conditions. A finite number of eigenvalues is possible only for problems in which the roots of the characteristic equation are multiple. The article describes solutions to such a problem for equations of the second, third, and fourth orders on a graph with three, four, and five edges. The inverse problem with an arbitrary number of edges is solved similarly.

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An Optimal Control Problem by a Hybrid System of Hyperbolic and Ordinary Differential Equations

Games ◽

10.3390/g12010023 ◽

2021 ◽

Vol 12 (1) ◽

pp. 23

Author(s):

Alexander Arguchintsev ◽

Vasilisa Poplevko

Keyword(s):

Optimal Control ◽

Differential Equations ◽

Optimal Control Problem ◽

Control Problem ◽

Ordinary Differential Equations ◽

Hyperbolic Equations ◽

Control Methods ◽

The Right ◽

The Cost ◽

Optimal Control Methods

This paper deals with an optimal control problem for a linear system of first-order hyperbolic equations with a function on the right-hand side determined from controlled bilinear ordinary differential equations. These ordinary differential equations are linear with respect to state functions with controlled coefficients. Such problems arise in the simulation of some processes of chemical technology and population dynamics. Normally, general optimal control methods are used for these problems because of bilinear ordinary differential equations. In this paper, the problem is reduced to an optimal control problem for a system of ordinary differential equations. The reduction is based on non-classic exact increment formulas for the cost-functional. This treatment allows to use a number of efficient optimal control methods for the problem. An example illustrates the approach.

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On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions

Open Mathematics ◽

10.1515/math-2021-0069 ◽

2021 ◽

Vol 19 (1) ◽

pp. 760-772

Author(s):

Ahmed Alsaedi ◽

Bashir Ahmad ◽

Badrah Alghamdi ◽

Sotiris K. Ntouyas

Keyword(s):

Nonlinear System ◽

Differential Equations ◽

Boundary Conditions ◽

Fractional Differential Equations ◽

Fixed Point Theory ◽

Point Theory ◽

Numerical Examples ◽

Multipoint Boundary ◽

Multipoint Boundary Conditions ◽

Fractional Differential

Abstract We study a nonlinear system of Riemann-Liouville fractional differential equations equipped with nonseparated semi-coupled integro-multipoint boundary conditions. We make use of the tools of the fixed-point theory to obtain the desired results, which are well-supported with numerical examples.

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Ordinary Differential Equations with Nonlinear Boundary Conditions

Georgian Mathematical Journal ◽

10.1515/gmj.2002.287 ◽

2002 ◽

Vol 9 (2) ◽

pp. 287-294

Author(s):

Tadeusz Jankowski

Keyword(s):

Differential Equations ◽

Boundary Conditions ◽

Ordinary Differential Equations ◽

Nonlinear Boundary ◽

Nonlinear Boundary Conditions ◽

Existence Results ◽

Monotone Iterative Technique ◽

Lower And Upper Solutions ◽

Iterative Technique ◽

Monotone Iterative

Abstract The method of lower and upper solutions combined with the monotone iterative technique is used for ordinary differential equations with nonlinear boundary conditions. Some existence results are formulated for such problems.

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Distributions of processes governed by hyperbolic equations when also boundary conditions are assumed

Probability Theory and Mathematical Statistics ◽

10.1515/9783112319321-035 ◽

1994 ◽

pp. 605-612

Keyword(s):

Boundary Conditions ◽

Hyperbolic Equations

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Solving a linear fractional equation with nonlocal boundary conditions based on multiscale orthonormal bases method in the reproducing kernel space

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2019-0291 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Wei Jiang ◽

Zhong Chen ◽

Ning Hu ◽

Yali Chen

Keyword(s):

Differential Equations ◽

Boundary Conditions ◽

Fractional Differential Equations ◽

Reproducing Kernel ◽

Nonlocal Boundary ◽

Nonlocal Boundary Conditions ◽

Kernel Space ◽

Reproducing Kernel Space ◽

Orthonormal Bases ◽

Fractional Differential

AbstractIn recent years, the study of fractional differential equations has become a hot spot. It is more difficult to solve fractional differential equations with nonlocal boundary conditions. In this article, we propose a multiscale orthonormal bases collocation method for linear fractional-order nonlocal boundary value problems. In algorithm construction, the solution is expanded by the multiscale orthonormal bases of a reproducing kernel space. The nonlocal boundary conditions are transformed into operator equations, which are involved in finding the collocation coefficients as constrain conditions. In theory, the convergent order and stability analysis of the proposed method are presented rigorously. Finally, numerical examples show the stability, accuracy and effectiveness of the method.

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