Some Numerical and Analytical Solutions to an Enzyme-Substrate Reaction-Diffusion Problem

Stochastic differential equation models play an increasingly prominent role in a wide variety of application areas, particularly those characterized by complexity and random behavior. Examples of such areas include biology, epidemiology, mechanics, economics, and finance. This issue's Education section contains a paper, "An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations," by Desmond Higham, that makes this important material accessible to a wide range of readers. Higham's paper is written from the perspective that the best way to gain insight quickly into the topics of stochastic differential equations (SDEs) is to see examples of numerical methods in use. The author feels that it is possible to explain and understand how simple numerical methods in this area work, without requiring the reader to be familiar with SDE theory. Indeed, the paper takes the approach that simulations of the dynamic behavior of SDEs and experimentation with numerical examples can help develop an intuitive understanding of SDEs that lays the groundwork for subsequent theoretical treatments. Thus Higham's paper expects only that the reader's background includes Euler's method for deterministic ordinary differential equations and an intuitive understanding of random variables. It provides a beautifully written treatment that is aimed at upper-division undergraduates and beginning graduate students. The topics covered in Higham's paper include investigation of stochastic integration, linear stability, and strong and weak convergence. The paper is built around ten complete downloadable MATLAB programs that are used as examples and contain ample comment lines. The programs are linked to relevant discussion points in the body of the paper. This paper is suitable as a module in a numerical methods course, such as a course on numerical methods for ordinary differential equations. It could also find application in courses in the areas where SDEs arise, such as finance, epidemiology, or general mathematical modeling. And it can serve excellently as the starting point for a self-study project in the area of SDEs for advanced undergraduates or beginning graduate students. Finally, many professionals in the SIAM community will find this article useful as an introduction to an area of numerical computation that has grown in prominence since many of us were students!

This article presents a general mathematical model for describing the laminar transport phenomena during homogeneous/heterogeneous appearances in the neighborhood of external and internal surfaces of small solid specimens. The model employs an original concept to solve the governing partial differential equations in order to predict the momentum, heat and/or mass transfer at the observed solid surface as well as diffusion at and/or into solid. Therefore, an approach established on the simplified form of fundamental partial differential equations is described. On the basis of this equation the integral method procedure, reduced to balance setting of Flux Gradients (FG-codes) of the transfer phenomena that occur, is derived. A steady-state flow and an un-steady state non flow situation are evaluated. Also, determination of activation energies of observed phenomena are discussed. Examples of verification are listed.

In this paper, we present a comparative study between the modified variational iteration method (MVIM) and a hybrid of Fourier transform and variational iteration method (FTVIM). The study outlines the efficiency and convergence of the two methods. The analysis is illustrated by investigating four singular partial differential equations with variable coefficients. The solution of singular partial differential equations usually needs a coordinate transformation in order to discard the singularity of the partial differential equation. Most often this transformation is not applicable and even does not exist. Therefore in this case the solution for the singular partial differential equation does not exist. In the present study the results of simulation for the singular partial differential equations with variable coefficients using the Fourier transform variational iteration method are compared with the results of simulation using the modified variational iteration method. The comparison shows that the effectiveness and accuracy of Fourier transform variational iteration method is more than that of the modified variational iteration method for the simulation of singular partial differential equations.

Various algorithms have been proposed for developing block methods where the most adopted approach is the numerical integration and collocation approaches. However, there is another conventional approach known as the Taylor series approach, although it was utilised at inception for the development of linear multistep methods for first order differential equations. Thus, this article explores the adoption of this approach through the modification of the aforementioned conventional Taylor series approach. A new methodology is then presented for developing block methods, which is a more accurate method for solving second order ordinary differential equations, coined as the Modified Taylor Series (MTS) Approach. A further step is taken by presenting a generalised form of the MTS Approach that produces any k-step block method for solving second order ordinary differential equations. The computational complexity of this approach after being generalised to develop k-step block method for second order ordinary differential equations is calculated and the result shows that the generalised algorithm involves less computational burden, and hence is suitable for adoption when developing block methods for solving second order ordinary differential equations. Specifically, an alternate and easy-to-adopt approach to developing k-step block methods for solving second order ODEs with fewer computations has been introduced in this article with the developed block methods being suitable for solving second order differential equations directly.

A study of a solar air collector with a mixed “porous” and “non-porous” composite absorber

In this paper, we derive new method for solving particular solution of linear second order ordinary differential equations whenever one solution of their associated homogeneous differential equations is given. Also, we construct second solution of the associated homogeneous differential equation from this new method. Moreover, we have general solution method of linear second order ordinary differential equations without applying the two famous methods, undetermined coefficients method and variation of parameters method, for solving their particular solution

Explicit Euler method, implicit Euler method, trapezoidal rule and midpoint rule are wildly-used and well-known numerical methods for ordinary differential equations. Methods with two parameters are constructed in this paper, whose error analysis is also given by Taylor expansion. The new constructed methods include and extend the 4 well-known methods, and the error analysis with parameters brings us convenience for the error analysis of numerical methods. At last, we give three original methods, whose orders can be gotten directly by the error analysis with parameters.

We come back to the issue of bosonization of fermions in two spacetime dimension and give a new costruction in the steady state case where left and right moving particles can coexist at two different temperatures. A crucial role in our construction is played by translation invariant infrared states and the corresponding field operators which are naturally linked to the infrared behaviour of the correlation functions. We present two applications: a simple new derivation in the free relativistic case of a formula by Bernard and Doyon and a full operator solution of the massless Thirring model in the steady state case where the left and right movers have two distinct temperatures.

A numerical analysis of unsteady inflow wind for site specific vertical axis wind turbine: A case study for Marsabit and Garissa in Kenya

The consistency of the classical Richardson extrapolation (CRE), a simple and robust computational device, is analysed for the case where the underlying method is an explicit one-step numerical method for ordinary differential equations with order of consistency one or two. It is shown in the classical framework that the CRE increases the order of consistency by one. The convergence of the method is proved by the assumption that the time-stepping operator of the base method has the Lipschitz property in its second argument.

Continuous numerical methods for ODEs with defect control

Modelling discontinuous systems involving large frictional sliding is one of the key requirements for numerical methods in geotechnical engineering. The contact algorithms for most numerical methods in geotechnical engineering is based on the judgement of contact types and the satisfaction of contact conditions by the open–close iteration, in which penalty springs between contacting bodies are added or removed repeatedly. However, the simulations involving large frictional sliding contact are not always convergent, particularly in the cases that contain a large number of contacts. To avoid the judgement of contact types and the open–close iteration, a new contact algorithm, in which the contact force is calculated directly based on the overlapped area of bodies in contact and the contact states, is proposed and implemented in the explicit numerical manifold method (NMM). Stemming from the discretization of Kuhn–Tucker conditions for contact, the equations for calculating contact force are derived and the contributions of contact force to the global iteration equation of explicit NMM are obtained. The new contact algorithm can also be implemented in other numerical methods (FEM, DEM, DDA, etc.) as well. Finally, five numerical examples are investigated to verify the proposed method and illustrate its capability.

The article describes the features of modeling dynamic systems adapted to real production processes. An approach to solving problems using the numerical and analytical method of modeling rigid dynamic systems is described. The algorithm of this approach is based on the following procedures: recording the original system of differential equations in the form of state variables, determining the characteristic polynomial of the system, finding its roots, and determining the solution of the original system of differential equations in the analytical form under typical input effects. The analytical view of the solution allows you: to keep track of all components (both fast and slow), which is especially important when modeling hard systems; the ability to obtain a solution with the accuracy of a computer bit grid for any time, as well as complete algorithmic independence from the solution at the previous point; the lack of iteration, the ability to pre-evaluate the nature of the transition process by the type of roots. The solutions for all possible types of roots are considered (Simple roots, Multiple roots, Complex-conjugate roots). For typical input effects, analytical expressions are given and the areas of application of the proposed approach in modeling complex technical systems in specific engineering tasks are considered. The method is implemented as a software package written in the Pascal algorithmic language for a personal computer of the IBM PC type. At the user’s request, solutions are issued in the analytical, digital or graphical form.

Summary form only given. Platelets adhesion occurs at focal adhesions (FA), where cell-membrane receptors bind specifically to substrate proteins and couple to each other and to the cytoskeleton via various cellular proteins. Some of the reactions that follow the ligand-receptor binding at the FA may affect mechanical determinants of the cell-substrate attachment. The resulting molecular structure suggests that the cortex stiffens at the FA, which likely affects platelet adhesion. The present work explores that hypothesis using a numerical simulation of the 3D membrane flexible structure under steady and unsteady bond kinetic and detachment forces. The cortex is modeled as a shell anchored to the substrate by unevenly distributed adhesion forces and subjected to externally detachment forces. In the simulated models, the steady case address to a stabilized condition, when both the adhesion forces and the detachment forces are steady. In the unsteady case, however, some aspects of bond kinetics and unsteadiness in the external detachment forces are incorporated. The commercial finite-element package ADINA (Watertown, MA) is used to solve the 3D time-dependent structural equations in the model. The results show the effect of cortex stiffening at the focal adhesion sites on the membrane deformation for both the steady and unsteady cases. The consequent internal stresses are described and the effects of membrane stiffening on the bonding forces are compared for the different cases. In addition, the effect of uneven distribution of focal adhesion complexes on the cells' structural support is evaluated and the consequences on cell function and behaviors are discussed

Object Oriented Programming and Numerical Methods