Direct Integrators for the General Third-Order Ordinary Differential Equations with an Application to the Korteweg–de Vries Equation
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We construct a new class of implicit continuous linear multistep methods (LMMs) which are used as boundary value methods for the numerical integration of the general third order initial and boundary value problems in ordinary differential equations, including the Korteweg–de Vries equation. The boundary value methods obtained from these continuous LMMs are weighted the same and are used to simultaneously generate approximate solutions to the exact solutions in the entire interval of integration. We established the convergence analysis of the methods and several numerical examples are given to show the performance of the methods.
KeywordsBoundary value methods Third order problems Convergence Linear multistep methods Korteweg–de Vries equation
Mathematics Subject Classification65L05 65L06 65L10 65L12
The authors are very grateful to the referee whose valuable suggestions greatly improved the manuscript.
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