B-Spline Cubic Finite Element Method for Solving Ordinary Differential Equations

This work proposes numerical solution of ordinary differential equations. The proposed method is based on applying modified cubic B-spline finite element method. The existence and uniqueness for the variational form proved. The convergence of the presented scheme is given. Numerical experiments are considered to confirm our theoretical results

Numerical Solution Of The Heat Equation By Cubic B-Spline Collocation Method

This work proposes a numerical scheme for heat parabolic problem by implementing a collocation method with a cubic B-spline for a uniform mesh. The key idea of this method is to apply forward finite difference and Crank–Nicolson methods for time and space integration, respectively. The stability of the presented scheme is proved through the Von-Neumann technique. It is shown that it is unconditionally stable. The accuracy of the suggested scheme is computed through the L_2 and L_∞-norms. Numerical experiments are also given and show that it is compatible with the exact solutions

Numerical Treatment of Allen’s Equation Using Semi Implicit Finite Difference Methods

This paper aims to propose the semi implicit finite difference method for discretizing Cahn-Allen equation. The stability and convergence analysis are proved. It is shown that the suggest scheme is stable for the usage of the Fourier-Von Neumann technique. The accuracy of the proposed method is first order in time and second order in space. A comparison between the numerical and the exact solutions is supported with two examples. Numerical results are shown that there is a good agreement between the approximate solution and exact solution

Model Reduction and Analysis for ERK Cell Signalling Pathway Using Implicit-Explicit Rung-Kutta Methods

Many complex cell signalling pathways and chemical reaction networks include many variables and parameters; this is sometimes a big issue for identifying critical model elements and describing the model dynamics. Therefore, model reduction approaches can be employed as a mathematical tool to reduce the number of elements. In this study, we use a new technique for model reduction: the Lumping of parameters for the simple linear chemical reaction network and the complex cell signalling pathway that is extracellular-signal-regulated kinase (ERK) pathways. Moreover, we propose a high-order and accurate method for solving stiff nonlinear ordinary differential equations. The curtail idea of this scheme is based on splitting the problem into stiff and non-stiff terms. More specifically, stiff discretization uses the implicit method, and nonlinear discretization uses the explicit method. This is consequently leading to a reduction in the computational cost of the scheme. The main aim of this study is to reduce the complex cell signalling pathway models by proposing an accurate numerical approximation Runge-Kutta method. This improves one's understanding of such behaviour of these systems and gives an accurate approximate solution. Based on the suggested technique, the simple model's parameters are minimized from 6 to 3, and the complex models from 11 to 8. Results show that there is a good agreement between the original models and the simplified models

Thermodynamic Analysis and Optimization of the Micro-CCHP System with a Biomass Heat Source

In this article, new multiple-production systems based on the micro-combined cooling, heating and power (CCHP) cycle with biomass heat sources are presented. In this proposed system, absorption refrigeration cycle subsystems and a water softener system have been used to increase the efficiency of the basic cycle and reduce waste. Comprehensive thermodynamic modeling was carried out on the proposed system. The validation of subsystems and the optimization of the system via the genetic algorithm method was carried out using Engineering Equation Solver (EES) software. The results show that among the components of the system, the dehumidifier has the highest exergy destruction. The effect of the parameters of evaporator temperature 1, ammonia concentration, absorber temperature, heater temperature difference, generator 1 pressure and heat source temperature on the performance of the system was determined. Based on the parametric study, as the temperature of evaporator 1 increases, the energy efficiency of the system increases. The maximum values of the energy efficiency and exergy of the whole system in the range of heat source temperatures between 740 and 750 K are equal to 74.2% and 47.7%. The energy and exergy efficiencies of the system in the basic mode are equal to 70.68% and 44.32%, respectively, and in the optimization mode with the MOOD mode, they are 87.91 and 49.3, respectively

Adaptive discontinuous Galerkin methods for elliptic interface problems

An interior-penalty discontinuous Galerkin (dG) method for an elliptic interface problem involving, possibly, curved interfaces, with flux-balancing interface conditions, e.g., modelling mass transfer of solutes through semi-permeable membranes, is considered. The method allows for extremely general curved element shapes employed to resolve the interface geometry exactly. A residual-type a posteriori error estimator for this dG method is proposed and upper and lower bounds of the error in the respective dG-energy norm are proven. The a posteriori error bounds are subsequently used to prove a basic a priori convergence result. The theory presented is complemented by a series of numerical experiments. The presented approach applies immediately to the case of curved domains with non-essential boundary conditions, too