A new interpolation procedure for adapting Runge-Kutta methods to delay differential equations

Hout, K. J. in't

doi:10.1007/bf01994847

Cited by 86 publications

(7 citation statements)

References 9 publications

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“…Solving such a system requires the development and application of special methods with increased stability [167,177]. Usually, these are algorithms from the class of implicit Runge-Kutta [168,178,179] and BDF methods [175,176,180]. The schemes of Rosenbrock type [181][182][183][184] are also suitable.…”

Section: Methods Of Lines Numerical Integration Using Mathematicamentioning

confidence: 99%

Nonlinear Reaction–Diffusion Equations with Delay: Partial Survey, Exact Solutions, Test Problems, and Numerical Integration

Сорокин

Vyazmin

2022

Mathematics

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The paper describes essential reaction–diffusion models with delay arising in population theory, medicine, epidemiology, biology, chemistry, control theory, and the mathematical theory of artificial neural networks. A review of publications on the exact solutions and methods for their construction is carried out. Basic numerical methods for integrating nonlinear reaction–diffusion equations with delay are considered. The focus is on the method of lines. This method is based on the approximation of spatial derivatives by the corresponding finite differences, as a result of which the original delay PDE is replaced by an approximate system of delay ODEs. The resulting system is then solved by the implicit Runge–Kutta and BDF methods, built into Mathematica. Numerical solutions are compared with the exact solutions of the test problems.

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Section: Methods Of Lines Numerical Integration Using Mathematicamentioning

confidence: 99%

Nonlinear Reaction–Diffusion Equations with Delay: Partial Survey, Exact Solutions, Test Problems, and Numerical Integration

Сорокин

Vyazmin

2022

Mathematics

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“…According to [34,36], we know a piecewise interpolation operator Π h ni defined by (7) or by (8) and ( 9) which satisfies the canonical condition…”

Section: Derivation Of the Numerical Methodsmentioning

confidence: 99%

“…VFDEs are widely applied in many fields of science and technology (see [1][2][3][4][5] and their references), for which there have been remarkable theoretical and numerical analysis research results. As for VDDEs, please refer to [6][7][8][9][10][11][12][13][14][15][16][17][18][19], as for VIDEs, please refer to [20][21][22][23], and as for VDIDEs, please refer to [24][25][26][27][28][29]. In recent decades, Li [30][31][32][33][34][35] has carried on systematic research for stiff general VFDEs and the numerical methods for them.…”

Section: Introductionmentioning

confidence: 99%

Classical Theory of Linear Multistep Methods for Volterra Functional Differential Equations

2021

Discrete Dynamics in Nature and Society

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Based on the linear multistep methods for ordinary differential equations (ODEs) and the canonical interpolation theory that was presented by Shoufu Li who is exactly the second author of this paper, we propose the linear multistep methods for general Volterra functional differential equations (VFDEs) and build the classical stability, consistency, and convergence theories of the methods. The methods and theories presented in this paper are applicable to nonneutral, nonstiff, and nonlinear initial value problems in ODEs, Volterra delay differential equations (VDDEs), Volterra integro-differential equations (VIDEs), Volterra delay integro-differential equations (VDIDEs), etc. At last, some numerical experiments verify the correctness of our theories.

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“…This delay term needs to be evaluated using interpolation. Several techniques involving the approximation of the delay term have been discussed by various authors such as [7][8][9]. In this paper, we use Lagrange and Hermite interpolations to approximate the delay term.…”

Section: Methods Implementationmentioning

confidence: 99%

Solving delay differential equations by predictor-corrector method using lagrange and hermite interpolations

Ishak

Ahmad

2011

2011 IEEE Colloquium on Humanities, Science and Engineering

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Delay differential equations (DDEs) appear naturally in modeling many real life phenomena. DDEs differ from ordinary differential equations since the derivative of the unknown function contains the expression of the unknown function at earlier and present states as well. DDEs that cannot be solved analytically are solved numerically. In this work, we solve DDEs using predictor-corrector multistep method where the corrector is iterated until convergence. The predictor uses the Adams-Bashforth four-step explicit method and the corrector uses Adams-Moulton three-step implicit method. Two types of interpolation polynomials which are Lagrange and Hermite interpolations are used to approximate the delay solutions. The accuracy of the adapted Adams-Bashforth-Moulton methods using these two polynomials is compared.

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A new interpolation procedure for adapting Runge-Kutta methods to delay differential equations

Cited by 86 publications

References 9 publications

Nonlinear Reaction–Diffusion Equations with Delay: Partial Survey, Exact Solutions, Test Problems, and Numerical Integration

Nonlinear Reaction–Diffusion Equations with Delay: Partial Survey, Exact Solutions, Test Problems, and Numerical Integration

Classical Theory of Linear Multistep Methods for Volterra Functional Differential Equations

Solving delay differential equations by predictor-corrector method using lagrange and hermite interpolations

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