Ordinary Differential Equations with Inverted Functions

Authors: Thomas Kampke

Abstract:

Equations with differentials relating to the inverse of an unknown function rather than to the unknown function itself are solved exactly for some special cases and numerically for the general case. Invertibility combined with differentiability over connected domains forces solutions always to be monotone. Numerical function inversion is key to all solution algorithms which either are of a forward type or a fixed point type considering whole approximate solution functions in each iteration. The given considerations are restricted to ordinary differential equations with inverted functions (ODEIs) of first order. Forward type computations, if applicable, admit consistency of order one and, under an additional accuracy condition, convergence of order one.

Keywords: Euler method, fixed points, golden section, multi-step procedures, Runge Kutta methods.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1332076

References:

[1] Birkhoff, G., Rota, G.-C., "Ordinary differential equations", 2nd ed., Wiley, New York, 1969.
[2] Gilsinn, D.E., Potra, F.A., "Integral operators and delay differential equations", Journal of Integral Equations and Applications 18, 2006, p. 297-336.
[3] Huylebrouck, D., "Golden Gray", Forma 19, 2004, p. 111-120.
[4] K¨ampke, T., "On duality transformations between Lorenz curves and distribution functions", FAW/n, Ulm, 2011, www.faw-neu-ulm.de/artikel.
[5] Koepf, W., "Taylor polynomials of implicit functions, of inverse functions, and of solutions of ordinary differential equations", Complex Variables 25, 1994, p. 23-33.
[6] PlanetMath, "Quasiperiodic functions", planetmath.org/encyclopedia/ QuasiperiodicFunction.html, accessed June 30, 2011.
[7] Scilab, "The free platform for numerical computation", www.scilab.org.
[8] Stoer, J., Bulirsch, R., "Numerische Mathematik 2", 5th ed., Springer, Berlin, 2005.
[9] Smart, D.R., "Fixed point theorems", 2nd printing, Cambridge University Press, Cambridge, 1980.
[10] Schnell, S., Mendoza, C., "A formula for integrating inverse functions", The Mathematical Gazette 84, 2000, p. 103-104.
[11] Zill, D.G., Wright, W.S, "Advanced engineering mathematics", 4th ed., Jones & Bartlett Learning, Burlington, 2010.