Numerical Integration of Nonlinear Klein–Gordon Type Equations with Delay by the Method of Lines
https://doi.org/10.56304/S2304487X19030131
Abstract
Qualitative features of numerical integration of initial-boundary value problems for partial differential equations with delay by the method of lines have been described. The method of lines is based on the approximation of spatial derivatives by corresponding finite differences, which allows reducing the initial equation to an approximate system of ordinary differential equations with delay. The system is then solved by the Runge–Kutta methods of the second and fourth orders and by the BDF method, which are built into Wolfram Mathematica. Test problems for nonlinear Klein–Gordon type equations with a constant delay τ whose solutions are expressed in terms of elementary functions have been formulated. The extensive comparison of numerical and exact solutions of the test problems on a significant time interval from 0 to 50 τ has been made. It has been found that the numerical method under consideration with moderate delay times ensures high accuracy of the results obtained.
Keywords
nonlinear hyperbolic equations,
Klein–Gordon type equations,
differential-difference equations,
equations with delay,
numerical integration,
method of lines
Supporting agencies: The work was carried out on the topic of the state assignment (state registration No. AAAAA-A17-117021310385-6)
About the Author
V. G. Sorokin
Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences
Russian Federation
Russian Federation
119526
Moscow
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For citations:
Sorokin V.G. Numerical Integration of Nonlinear Klein–Gordon Type Equations with Delay by the Method of Lines. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2019;8(3):232-247. (In Russ.) https://doi.org/10.56304/S2304487X19030131
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