Approximate Analytical Solution of Nonlinear Reaction’s Diffusion Equation at Conducting Polymer Ultramicroelectrodes

A unified analytical solution of the steady-state atmospheric diffusion equation

We consider the preconditioned Krylov subspace method for linear systems arising from the finite volume discretization method of steady-state variable-coefficient conservative space-fractional diffusion equations. We propose to use a scaled-circulant preconditioner to deal with such Toeplitz-like discretization matrices. We show that the difference between the scaled-circulant preconditioner and the coefficient matrix is equal to the sum of a small-norm matrix and a low-rank matrix. Numerical tests are conducted to show the effectiveness of the proposed method for one- and two-dimensional steady-state space-fractional diffusion equations and demonstrate that the preconditioned Krylov subspace method converges very quickly.

In this work, we have theoretically analyzed the non-steady-state current response witnessed at conducting polymer-modified ultramicroelectrodes when the electrode is operating under batch amperometric conditions. Furthermore, the effect of substrate diffusion in the solution adjacent to the polymer film on both the concentration profiles and current response was examined. Analytical expressions pertaining to substrate concentration, mediator concentration, and current response pertaining to all values of diffusion parameters are presented. In addition, we have employed "reduction of order method" to evaluate the approximate analytical solutions of reaction diffusion equations.

An alternative general solution of the steady-state heat diffusion equation

A potential and stream function analysis of two-dimensional steady-state convective diffusion equations involving Laplace fields

Measurements of effective sizes and diffusivities of nano-colloids and micro-particles

Often in reactor dynamics, higher eigenfunctions of the multigroup diffusion equation must be determined. An algorithm to calculate higher eigenfunctions (modes) of the [lambda]-eigenvalue problem corresponding to the steady-state two-group neutron diffusion equation is presented. The method is based on a special type of subspace iteration for large sparse nonsymmetric eigenvalue problems. Having been tested using an International Atomic Energy Agency benchmark problem and also applied to a VVER-1000 pressurized water reactor assembly, the algorithm was found to work very effectively and reliably. In its application, the algorithm presented is not restricted to the [lambda]-eigenvalue problem only but is also generally applicable to large sparse nonsymmetric eigenvalue problems even with multiple and complex eigenvalues.

The mathematical model proposed by Chapman and Antano (Electrochimica Acta, 56 (2010), 128–132) for the catalytic electrochemical–chemical (EC’) processes in an irreversible second-order homogeneous reaction in a microelectrode is discussed. The mass-transfer boundary layer neighbouring an electrode can contribute to the electrode’s measured AC impedance. This model can be used to analyse membrane-transport studies and other instances of ionic transport in semiconductors and other materials. Two efficient and easily accessible analytical techniques, AGM and DTM, were used to solve the steady-state non-linear diffusion equation’s infinite layers. Herein, we present the generalized approximate analytical solution for the solute, product, and reactant concentrations and current for the small experimental values of kinetic and diffusion parameters. Using the Matlab/Scilab program, we also derive the numerical solution to this problem. The comparison of the analytical and numerical/computational results reveals a satisfactory level of agreement.

Evaporation from a nonspherical aerosol particle situated in an absorbing gas

Magnetohydrodynamic modeling for an OVD reactor setup

In this paper, the homotopy perturbation method (HPM) combined with Padé approximant is applied to generate the approximate analytical solution of Korteweg de-Vries (KdV) equation. The results show that the combined method gives highly accurate approximate analytical solution. It is also found that this method can achieve more suitable results in comparison to HPM alone. The approximate solution is also compared with the exact solution and it is found to be in good agreement.

PARCS code multi-group neutron diffusion constants generation using Monte Carlo method

Problems involving the dynamics of heating or cooling of physical systems submitted to certain boundary conditions are of great importance in a wide range of situations. Since simple situations involving cooling of food where the temperature of each part of the system needs to be controlled and, in general, does not involve the presence of heat sources, even more complex situations like nuclear reactors, where there is a heat source which is dependent on position and the dependence of the reactor reactivity on the temperature requires a good knowledge of the dynamics of this quantity, both for safety and efficiency. Problems of the steady-state heat diffusion equation for a series of physical systems are easy to solve and can be found in a vast literature on the subject. To find the time evolution of temperature T (x, t) of a system submitted to a heat source, however, is not a simple task and this type of problem is usually treated in an approximate way. In this work, for one-dimensional and homogeneous system, equilibrium T(x) and transient T (x, t) solutions are presented for cases of the existence of sources permeating throughout the medium. Each system considered is supposed to have constant temperature in the edges. Calculations are presented for two cases: one is the case of a source non-dependent on position and time and the other is the case of a source with sinusoidal spatial dependence, a situation close to what would occur in a one-dimensional nuclear reactor. The extension to a three-dimensional case, homogeneous sphere with constant temperature on its surface will also be presented. The case involving the contact of two subsystems of different materials will also be analyzed.

Proton radiography is an important diagnostic method for laser plasma experiments and is particularly important in the analysis of magnetized plasmas. The theory of radiographic image analysis has heretofore only permitted somewhat limited analysis of the radiographs of such plasmas. We furnish here a theory that remedies this deficiency. We show that to linear order in magnetic field gradients, proton radiographs are projection images of the MHD current along the proton trajectories. We demonstrate that in the linear regime (i.e., the small image contrast regime), the full structure of the projected perpendicular magnetic field can be reconstructed by solving a steady-state inhomogeneous 2-dimensional diffusion equation sourced by the radiograph fluence contrast data. We explore the validity of the scheme with increasing image contrast, as well as limitations of the inversion method due to the Poisson noise, discretization errors, radiograph edge effects, and obstruction by laser target structures. We also provide a separate analysis that is suited to the inference of isotropic-homogeneous magnetic turbulence spectra. Finally, we discuss extension of these results to the nonlinear regime (i.e., the order unity image contrast regime).

In this paper we describe, analyse and implement a parallel iterative method for the solution of the steady-state drift diffusion equations governing the behaviour of a semiconductor device in two space dimensions. The unknowns in our model are the electrostatic potential and the electron and hole quasi-Fermi potentials. Our discretisation consists of a finite element method with mass lumping for the electrostatic potential equation and a hybrid finite element with local current conservation properties for the continuity equations. A version of Gummel's decoupling algorithm which only requires the solution of positive definite symmetric linear systems is used to solve the resulting nonlinear equations. We show that this method has an overall rate of convergence which only degrades logarithmically as the mesh is refined. Indeed the (inner) nonlinear solves of the electrostatic potential equation converge quadratically, with a mesh independent asymptotic constant. We also describe an implementation on a MasPar MP-1 data parallel machine, where the required linear systems are solved by the preconditioned conjugate gradient method. Domain decomposition methods are used to parallelise the required matrix-vector multiplications and to build preconditioners for these very poorly-conditioned systems. Our preconditioned linear solves also have a rate of convergence which degrades logarithmically as the grid is refined relative to subdomain size, and their performance is resilient to the severe layers which arise in the coefficients of the underlying elliptic operators. Parallel experiments are given.