Two step runge-KUTTA-nystrom method for solving second-order ordinary differential equations

In this research, methods that will be able to solve the second order initial value problem (IVP) directly are developed. These methods are in the scheme of a multi-step method which is known as the two-step method. The two-step method has an advantage as it can estimate the solution with less function evaluations compared to the one-step method. The selection of step size is also important in obtaining more accurate and efficient results. Smaller step sizes will produce a more accurate result, but it lengthens the execution time. Two-Step Runge-Kutta (TSRK) method were derived to solve first-order Ordinary Differential Equations (ODE). The order conditions of TSRK method were obtained by using Taylor series expansion. The explicit TSRK method was derived and its stability were investigated. It was then analyzed experimentally. The numerical results obtained were analyzed by making comparisons with the existing methods in terms of maximum global error, number of steps taken and function evaluations. The explicit Two-Step Runge-Kutta-Nyström (TSRKN) method was derived with reference to the technique of deriving the TSRK method. The order conditions of TSRKN method were also obtained by using Taylor series expansion. The strategies in choosing the free parameters were also discussed. The stability of the methods derived were also investigated. The explicit TSRKN method was then analyzed experimentally and comparisons of the numerical results obtained were made with the existing methods in terms of maximum global error, number of steps taken and function evaluations. Next, we discussed the derivation of an embedded pair of the TSRKN (ETSRKN) methods for solving second order ODE. Variable step size codes were developed and numerical results were compared with the existing methods in terms of maximum global error, number of steps taken and function evaluations. The ETSRKN were then used to solve second-order Fuzzy Differential Equation (FDE). We observe that ETSRKN gives better accuracy at the end point of fuzzy interval compared to other existing methods. In conclusion, the methods developed in this thesis are able to solve the system of second-order differential equation (DE) which consists of ODE and FDE directly

Optimal Rate of Direct Estimators in Systems of Ordinary Differential Equations Linear in Functions of the Parameters

Many processes in biology, chemistry, physics, medicine, and engineering are modeled by a system of differential equations. Such a system is usually characterized via unknown parameters and estimating their 'true' value is thus required. In this paper we focus on the quite common systems for which the derivatives of the states may be written as sums of products of a function of the states and a function of the parameters. For such a system linear in functions of the unknown parameters we present a necessary and sufficient condition for identifiability of the parameters. We develop an estimation approach that bypasses the heavy computational burden of numerical integration and avoids the estimation of system states derivatives, drawbacks from which many classic estimation methods suffer. We also suggest an experimental design for which smoothing can be circumvented. The optimal rate of the proposed estimators, i.e., their $\sqrt n$-consistency, is proved and simulation results illustrate their excellent finite sample performance and compare it to other estimation approaches

Fast derivatives of likelihood functionals for ODE based models using adjoint-state method

We consider time series data modeled by ordinary differential equations (ODEs), widespread models in physics, chemistry, biology and science in general. The sensitivity analysis of such dynamical systems usually requires calculation of various derivatives with respect to the model parameters. We employ the adjoint state method (ASM) for efficient computation of the first and the second derivatives of likelihood functionals constrained by ODEs with respect to the parameters of the underlying ODE model. Essentially, the gradient can be computed with a cost (measured by model evaluations) that is independent of the number of the ODE model parameters and the Hessian with a linear cost in the number of the parameters instead of the quadratic one. The sensitivity analysis becomes feasible even if the parametric space is high-dimensional. The main contributions are derivation and rigorous analysis of the ASM in the statistical context, when the discrete data are coupled with the continuous ODE model. Further, we present a highly optimized implementation of the results and its benchmarks on a number of problems. The results are directly applicable in (e.g.) maximum-likelihood estimation or Bayesian sampling of ODE based statistical models, allowing for faster, more stable estimation of parameters of the underlying ODE model.Comment: 5 figure