Adaptive power series solution for second order ordinary differential equations with initial conditions

Solution of differential equations is essential to analyze problems in many academic fields. Time varying solution schemes for differential equations are required in nonstationary environments. Numeric solutions are essential for nonlinear differential equations where explicit solutions do not exist, especially, for second and higher orders. This paper proposes efficient adaptive numeric solutions for second order ordinary differential equations (ODE) with initial conditions. The proposed technique implements neural networks with transfer functions that follow a power series. The proposed technique does not use sigmoid or tanch non-linear transfer functions commonly employed in conventional neural networks at the output. Instead, linear transfer functions are adopted which leads to explicit power series formulae for the ODE solution. This provides continuous solutions and enables interpolation and extrapolation. The efficient and accurate solutions provided by the proposed technique are illustrated through simulated examples. It is shown that the performance of the proposed technique outperforms existing conventional methods.