Solution of initial and boundary value problems by the variational iteration method

The Variational Iteration Method (VIM) is an iterative method that obtains the approximate solution of differential equations. In this paper, it is proven that whenever the initial approximation satisfies the initial conditions, vim obtains the solution of Initial Value Problems (IVPs) with a single iteration. By using this fact, we propose a new algorithm for Boundary Value Problems (BVPs): linear and nonlinear ones. Main advantage of the present method is that it does not use Green's function, however, it has the same effect that it produces the exact solution to linear problems within a single, but simpler, integral. In order to show the effectiveness of the method we give some examples including linear and nonlinear BVPs

Hybrid master equation for jump-diffusion approximation of biomolecular reaction networks

Cellular reactions have multi-scale nature in the sense that the abundance of molecular species and the magnitude of reaction rates can vary in a wide range. This diversity leads to hybrid models that combine deterministic and stochastic modeling approaches. To reveal this multi-scale nature, we proposed jump-diffusion approximation in a previous study. The key idea behind the model was to partition reactions into fast and slow groups, and then to combine Markov chain updating scheme for the slow set with diffusion (Langevin) approach updating scheme for the fast set. Then, the state vector of the model was defined as the summation of the random time change model and the solution of the Langevin equation. In this study, we have proved that the joint probability density function of the jump-diffusion approximation over the reaction counting process satisfies the hybrid master equation, which is the summation of the chemical master equation and the Fokker-Planck equation. To solve the hybrid master equation, we propose an algorithm using the moments of reaction counters of fast reactions given the reaction counters of slow reactions. Then, we solve a constrained optimization problem for each conditional probability density at the time point of interest utilizing the maximum entropy approach. Based on the multiplication rule for joint probability density functions, we construct the solution of the hybrid master equation. To show the efficiency of the method, we implement it to a canonical model of gene regulation

Solution of initial and boundary value problems by the variational iteration method

The Variational Iteration Method (VIM) is an iterative method that obtains the approximate solution of differential equations. In this paper, it is proven that whenever the initial approximation satisfies the initial conditions, vim obtains the solution of Initial Value Problems (IVPs) with a single iteration. By using this fact, we propose a new algorithm for Boundary Value Problems (BVPs): linear and nonlinear ones. Main advantage of the present method is that it does not use Green's function, however, it has the same effect that it produces the exact solution to linear problems within a single, but simpler, integral. In order to show the effectiveness of the method we give some examples including linear and nonlinear BVPs

Efficient Simulation of Multiscale Reaction

Cellular reaction systems are often multiscale in nature due to wide variation in the species abundance and reaction rates. Traditional deterministic or stochastic modeling of such systems, which do not exploit this multiscale behavior, will be computationally expensive for simulation or inference purposes. This necessitates developing simplified hybrid models combining both stochastic and deterministic approaches that can substantially speed up simulation of such reaction networks. The paper proposes a layered partitioning approach which not only split the reaction set into the usual \u27fast\u27 and the \u27slow\u27 groups, but further subdivides the fast group into subgroups of super fast and moderately fast reactions. While the occurrences of reactions from the super fast and moderately fast groups are approximated by ordinary differential equations (ODEs) and Itô diffusions respectively, the discrete counting process formulation is maintained for the reactions from the slow group. The paper develops a mathematical framework for objectively identifying these three groups and performs a rigorous error analysis for the approximation proposed. One important highlight of the paper is the utilization of the error analysis in constructing an efficient dynamic algorithm that can fully automate the partitioning process and make necessary adjustments, if needed, over the course of time

Error bound and simulation algorithm for piecewise deterministic approximations of stochastic reaction systems

In cellular reaction systems, events often happen at different time and abundance scales. It is possible to simulate such multi-scale processes with exact stochastic simulation algorithms, but the computational cost of these algorithms is prohibitive due to the presence of high propensity reactions. This observation motivates the development of hybrid models and simulation algorithms that combine deterministic and stochastic representation of biochemical systems. Based on the random time change model we propose a hybrid model that partitions the reaction system into fast and slow reactions and represents fast reactions through ordinary differential equations (ODEs) while the Markov jump representation is retained for slow ones. Importantly, the partitioning is based on an error analysis which is the main contribution of the paper. The proposed error bound is then used to construct a dynamic partitioning algorithm. Simulation results are provided for two elementary reaction systems