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Numerical solution of neutral functional differential equations by Adams methods in divided difference form. (English) Zbl 1090.65084

An algorithm of variable step and variable order for the numerical solution of neutral functional differential equations (NFDE) is described. The form of NFDE are: \[ \begin{aligned} y'(t) &= f\bigl(t,y(\bullet),y'(\bullet)\bigr), \;t\in[a,b]\\ y(t)& =g(t),\;t\in[\tau,\alpha],\;y'(t)=g'(t),\;t\in[\tau,\alpha] \end{aligned} \] with \(\tau\leq\alpha<b\) and \(g\in C_n^1[\tau,\alpha]\) is a given initial function. The algorithm for the solution of the above system is based on the variable step formulation of the Adams methods represented in divided difference form in which the Adams-Bashforth and Adams-Moulton methods are implemented in predictor-corrector mode. The restarting of the integration at each discontinuity point (derivative discontinuities) relies on the step size and order changing strategy based on the estimates of the local discretization errors. The algorithm reduces the computational cost and increases the reliability and efficiency by obtaining asymptotically correct estimates of the local discretization errors. The method is tested in three test examples.

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
65L12 Finite difference and finite volume methods for ordinary differential equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations
34K40 Neutral functional-differential equations
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