Solving Non-Smooth Delay Differential Equations with Multiquadrics

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ISSN: 978-3-642-25100-9 (online)
ISBN: 978-3-642-25099-6 (print)
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Springer Verlag Berlin-Heidelberg
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We put forward a discretization scheme for the numerical solution of neutral differential equations (NDEs). The solution to the NDE in an interval I = [a, b] is approximated by a multiquadric (MQ) interpolant, whose coefficients are found by collocation on a set of N nodes in I. This approach, also known as Kansa’s method, enjoys an exponential rate of convergence and great flexibility regarding the location of the nodes, as long as the solution to the differential equation is smooth. However, the critical difficulty posed by NDEs is precisely that they propagate, forward in time and without damping, low-order discontinuities of the history function. Here, we exploit the sensitivity of the MQ interpolant to discontinuities in order to detect them in computing time. This allows for a partition of I into smooth subintervals, which are then sequentially solved by Kansa’s method.
Proceeding of: 16th Conference of the European Consortium for Mathematics in Industry (ECMI 2010), Wuppertal, Germany, Wuppertal, from July 26 to July 30, 2010.
Delay differential equation, Jump discontinuity, History function, Gibbs phenomenon, Neutral differential equation
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Günther M., Bartel A., Brunk M., Schöps S., Striebel M. (eds). (2012) Progress in Industrial Mathematics at ECMI 2010. (Mathematics in Industry. TECMI; 17). Springer-Verlag Berlin Heidelberg. Pp. 603-609.