Dimensional Diffusion Equation Research Articles

A study of one dimensional nonlinear diffusion equations by Bernstein polynomial based differential quadrature method

The purpose of this work is to study numerical solutions of nonlinear diffusion equations such as Fisher’s equation, Burgers’ equation and modified Burgers’ equation by applying Bernstein differential quadrature method (BDQM). These nonlinear diffusion equations occur in many applications of theoretical, engineering and environmental sciences. Therefore, finding numerical solutions of these equations is very important. In BDQM, Bernstein polynomials are used as base functions to find weighting coefficients of differential quadrature method. We have applied BDQM on eleven different test problems taken from the literature and the computed results confirm that BDQM is an efficient method for finding solution of nonlinear partial differential equations. It is found that BDQM produces very good results even at small number of grid points.

IMPLEMENTATION OF THE HALF-SWEEP AOR ITERATIVE ALGORITHM FOR SPACE-FRACTIONAL DIFFUSION EQUATIONS

In this paper, we consider the numerical solution of one dimensional space-fractional diffusion equation. The half-sweep AOR (HSAOR) iterative method is applied to solve linear system generated from discretization of one dimensional space-fractional diffusion equation using Caputo’s derivative operator and half-sweep implicit finite difference scheme. Furthermore, the formulation and implementation of HSAOR iterative method to solve the problem are also presented. Two examples and comparisons with FSAOR iterative method are given to show the effectiveness of the proposed method. From numerical results obtained, it has shown that the HSAOR iterative method is superior as compared with the FSAOR methods.

Galerkin finite element method for higher dimensional multi-term fractional diffusion equation on non-uniform meshes

In this paper, we study Galerkin finite element methods for a class of higher dimensional multi-term fractional diffusion equations. The finite difference approximation of Caputo derivative on non-uniform meshes is used in temporal direction and Galerkin finite element method is used in spatial direction. We prove that semi-discrete and fully discrete finite element schemes are unconditionally stable. Meanwhile, -norm convergence properties of the two schemes are proved rigorously. To confirm our theoretical analysis, we give some numerical examples in both two-dimensional (2D) and three-dimensional (3D) spaces. Finally, a moving local refinement technique in temporal direction is used to improve the accuracy of numerical solution.

PMMA壁面の上方向への燃え拡がりのLarge eddy simulation

A fully coupled fluid-solid approach for flame spread simulations has been developed using FireFOAM. Radiative heat transfer and soot treatment are fully coupled with the pyrolysis calculations. For gaseous combustion, the newly extended eddy dissipation concept (EDC) for the large eddy simulation (LES) is used. Due to low Reynolds number, the laminar combustion model based on viscous diffusion is presented and coupled with the EDC model. The soot model is based on the laminar smoke point concept recently extended to turbulent fires using the partially stirred reactor (PaSR) concept. For radiation, the finite volume discrete ordinate method (FVDOM) is used with the gaseous absorption coefficients evaluated using the polynomial coefficient of temperature. In the solid region, one dimensional diffusion equation for sensible enthalpy is solved. The effect of in-depth radiation is taken into account using the relatively simple formation according to Beer's law. The Arrhenius type pyrolysis model developed by FM Global is used along with their measurements of the PMMA physical properties as input data. The validation study has been conducted with the published experiment. The predictions are in very good agreement with the relevant experimental data, demonstrating that the present modelling approach can be used to predict upward flame spread over PMMA with reasonable accuracy. Such a fully coupled predictive tool can be used to aid fundamental studies of the flame spread phenomena such as investigating the effects of width, inclination angles and side walls on flame spread.

Tracer diffusion of hard-sphere binary mixtures under nano-confinement.

The physics of diffusion phenomena in nano- and microchannels has attracted a lot of attention in recent years, due to its close connection with many technological, medical, and industrial applications. In the present paper, we employ a kinetic approach to investigate how the confinement in nanostructured geometries affects the diffusive properties of fluid mixtures and leads to the appearance of properties different from those of bulk systems. In particular, we derive an expression for the friction tensor in the case of a bulk fluid mixture confined to a narrow slit having undulated walls. The boundary roughness leads to a new mechanism for transverse diffusion and can even lead to an effective diffusion along the channel larger than the one corresponding to a planar channel of equivalent section. Finally, we discuss a reduction of the previous equation to a one dimensional effective diffusion equation in which an entropic term encapsulates the geometrical information on the channel shape.

Controlling the conduction band offset for highly efficient ZnO nanorods based perovskite solar cell

The mechanism of charge recombination at the interface of n-type electron transport layer (n-ETL) and perovskite absorber on the carrier properties in the perovskite solar cell is theoretically studied. By solving the one dimensional diffusion equation with different boundary conditions, it reveals that the interface charge recombination in the perovskite solar cell can be suppressed by adjusting the conduction band offset (ΔEC) at ZnO ETL/perovskite absorber interface, thus leading to improvements in cell performance. Furthermore, Mg doped ZnO nanorods ETL has been designed to control the energy band levels. By optimizing the doping amount of Mg, the conduction band minimum of the Mg doped ZnO ETL has been raised up by 0.29 eV and a positive ΔEC of about 0.1 eV is obtained. The photovoltage of the cell is thus significantly increased due to the relatively low charge recombination.

Some 2+1 Dimensional Super-Integrable Systems

Abstract In the article, we make use of the binormial-residue-representation (BRR) to generate super 2+1 dimensional integrable systems. Using these systems, we can deduce a super 2+1 dimensional AKNS hierarchy, which can be reduced to a super 2+1 dimensional nonlinear Schrödinger equation. In particular, two main results are obtained. One of them is a set of super 2+1 dimensional integrable couplings. The other one is a 2+1 dimensional diffusion equation. The Hamiltonian structure of the super 2+1 dimensional hierarchy is derived by using the super-trace identity.

The PCD Method on Composite Grid

We introduce a discretization method of boundary value problems(BVP) in the case of the two dimensional diffusion equation on arectangular mesh with refined zones. The method consists inrepresenting the unknown distribution and its derivatives bypiecewise constant distributions (PCD) on distinct meshes togetherwith an appropriate approximate variational formulation of the exactBVP on this piecewise constant distributions space. This method,named the PCD method, has the advantage of producing the mostcompact possible discrete stencil. Here, we analyze and prove theconvergence of the PCD method and determine upper bounds on itsconvergence rate.

Two group, three dimensional diffusion theory coupled with single heated channel model: A study based on xenon transient simulation of Tehran research reactor

In this work xenon transient analysis of the Tehran research reactor (TRR) core from hot full power without poisoning to xenon saturation is carried out using neutronic and thermal hydraulic coupling. A complete two group, three dimensional neutronic model coupled with one dimensional single phase heat transfer model has been developed and applied to power distribution calculations. We used the WIMS-D5 cell calculation code in order to find a linear relation between important parameters (fuel temperature, moderator temperature and density, and xenon concentration) and group constants of different fuel assemblies. In this way, WIMS running will be bypassed at each time step. We developed and used the ANOMOS code that solves three dimensional diffusion equations using analytic nodal method by transverse leakage approximation in order to find the effective multiplication factor as well as flux and power distributions. Also, we developed and used a SHC code that solves the conservation equations for one dimensional axial homogeneous downward flow through channel using single heated channel model to find the assembly-wise fuel temperature, moderator temperature and density profiles versus generated power. The results have a good agreement with the safety analysis report of the TRR.

Numerical solution of fractional diffusion equation over a long time domain

In this paper, we propose a method to compute approximate solutions to one dimensional fractional diffusion equation which requires solution for a long time domain. For this, we use a set of shifted Legendre polynomials for the space domain and a set of Legendre rational functions for the time domain. The unknown solution is approximated by using these sets of orthogonal functions with unknown coefficients and the fractional derivative of the approximate solution is represented by an operational matrix, resulting in a linear system with the unknown coefficients. Numerical examples are given to demonstrate the effectiveness of the method.

A Corresponding Lie Algebra of a Reductive homogeneous Group and Its Applications* *Supported by the Innovation Team of Jiangsu Province hosted by China University of Mining and Technology (2014), the National Natural Science Foundation of China under Grant No. 11371361, the Fundamental Research Funds for the Central Universities (2013XK03), and the Natural Science Foundation of Shandong Province under Grant No.

With the help of a Lie algebra of a reductive homogeneous space G/K, where G is a Lie group and K is a resulting isotropy group, we introduce a Lax pair for which an expanding (2+1)-dimensional integrable hierarchy is obtained by applying the binormial-residue representation (BRR) method, whose Hamiltonian structure is derived from the trace identity for deducing (2+1)-dimensional integrable hierarchies, which was proposed by Tu, et al. We further consider some reductions of the expanding integrable hierarchy obtained in the paper. The first reduction is just right the (2+1)-dimensional AKNS hierarchy, the second-type reduction reveals an integrable coupling of the (2+1)-dimensional AKNS equation (also called the Davey-Stewartson hierarchy), a kind of (2+1)-dimensional Schrödinger equation, which was once reobtained by Tu, Feng and Zhang. It is interesting that a new (2+1)-dimensional integrable nonlinear coupled equation is generated from the reduction of the part of the (2+1)-dimensional integrable coupling, which is further reduced to the standard (2+1)-dimensional diffusion equation along with a parameter. In addition, the well-known (1+1)-dimensional AKNS hierarchy, the (1+1)-dimensional nonlinear Schrödinger equation are all special cases of the (2+1)-dimensional expanding integrable hierarchy. Finally, we discuss a few discrete difference equations of the diffusion equation whose stabilities are analyzed by making use of the von Neumann condition and the Fourier method. Some numerical solutions of a special stationary initial value problem of the (2+1)-dimensional diffusion equation are obtained and the resulting convergence and estimation formula are investigated.

Numerical analysis of time fractional three dimensional diffusion equation

The three dimensional diffusion equations were extended to the scope of fractional order derivative. The fractional operator used here is in Caputo sense. The resulting equation was solved using two numerical approaches: The forward in time and central in space method and the Crank-Nicholson method. The stability analysis of both methods was studied, and the study showed that the Crank-Nicholson method is unconditionally stable while the forward method is stable if some conditions are satisfied.

Two Dimensional Finite Volume Model to Study the Effect of ER on Cytosolic Calcium Distribution in Astrocytes

Astrocytes releases glio-transmitters in a Ca2+-dependent manner. The endoplasmic reticulum, major Ca2+ storage organelle in astrocytes, contains high calcium concentration level than the cytosol. The release of calcium ions from the ER is a central process that affects the cytosolic calcium concentration label in astrocytes. A mathematical model is developed in the form of two dimensional diffusion equation to study the calcium exchanges one across the ER in Astrocytes. All important parameters like diffusion coefficient and influx over [Ca2+] profile, ER fluxes like, JleakJPumpand JChan has been studied. Finite volume method (FVM) is employed to solve the problem and simulated in MATLAB to compute the numerical results.

Analytical Solutions for Ground Temperature Profiles and Stored Energy Using Meteorological Data

Analytical solutions to estimate temperature with depth and stored energy within a soil column based upon readily available meteorological data are presented in this paper, which are of particular relevance in the field of ground heat extraction and storage. The transient one dimensional heat diffusion equation is solved with second kind (Neumann) boundary conditions at the base and third kind (Robin) boundary conditions, based on a heat balance, at the soil surface. In order to describe the soil-atmosphere interactions, mathematical expressions describing the daily and annual variation of solar radiation and air temperature are proposed. The presented analytical solutions are verified against a numerical solution and applied to investigate a case-study problem based upon results of a field experiment. It is shown that the proposed analytical approach can offer a reasonable estimate of the thermal behaviour of the soil requiring no information from the soil other than its thermal properties. Comparisons of predicted and measured soil temperature profiles and stored energy transients demonstrate there is reasonable overall agreement. The research contributes a practical approach that can provide surface boundary data that are vital in the thermal analysis of many engineering problems. Applications include: inter-seasonal heat transfer, energy piles and other more established ground source heat utilization methods.

Finite Element Solution in Fluid Mechanics Using Exponential Function Based Interpolation

The wave problem of finite element solution of one dimensional stationary convection diffusion equation is analyzed. The reason for wave is the poor continuity of linear Lagrange shape function. By using exponential function based interpolation (EFBI), the results of finite element solution are compared with that of the analytical solution. The results indicate that EFBI is effective to deal with the problem of numerical wave. The shape function of three dimensional EFBI is derived and analyzed. Morphological analysis shows that EFBI has good differentiability and adaptability.

Exponential higher-order compact scheme for 3D steady convection–diffusion problem

In this paper, the exponential higher order compact (EHOC) finite difference schemes proposed by Tian and Dai (2007) [10] for solving the one and two dimensional steady convection diffusion equations with constant or variable convection coefficients are extended to the three dimensional case. The proposed EHOC scheme has the feature that it provides very accurate solution (exact in case of constant convection coefficient in 1D) of the homogeneous equation that is responsible for the fundamental singularity of the homogeneous solution while approximates the particular part of the solution by fourth order accuracy over the nineteen point compact stencil. The key properties of this scheme are its stability, accuracy and efficiency so that high gradients near the boundary layers can be effectively resolved even on coarse uniform meshes. To validate the present EHOC method, three test problems, mostly with boundary or internal layers are solved. Comparisons are made between numerical results for the present EHOC scheme and other available methods in the literature.

Solution of One –dimensional Fractional Diffusion Equations Involving Caputo Fractional Derivatives in terms of the Mittag - Leffler Functions

The object of this article is to investigate the solutions of one-dimensional linear fractional diffusion equations defined by (2.1) and (4.1). The solutions are obtained in a closed and elegant forms in terms of the H-function and generalized Mittag - Leffler functions, which are suitable for numerical computation. The derived results include the results for the one-dimentional linear fractional telegraph equation due to Orsingher and Beghin (1), and recently derived results by Saxena ,Mathai and Haubold (2).

FINITE-VOLUME SOLUTION OF DIFFUSION EQUATION AND APPLICATION TO MODEL PROBLEMS.

In the present numerical study, the two dimensional diffusion equations are converted to a system of l inear equations using finitevolume method. The system of equations is solved by a direct method. For the entire process a computer code is developed in Matlab. In the second part of the present study, the comput er codes developed for solving diffusion equation i s then applied to a series of model problems. These problems contain features fou nd in more complicated engineering situations. The problems are distinguished by their different boundary conditions, and by the variation of the source term and diffusivity in the domain.

Solution of the two dimensional diffusion and transport equations in a rectangular lattice with an elliptical fuel element using Fourier transform methods: One and two group cases

A method for solving the diffusion and transport equations in a rectangular lattice cell with an elliptical fuel element has been developed using a Fourier expansion of the neutron flux. The method is applied to a one group model with a source in the moderator. The cell flux is obtained and also the associated disadvantage factor. In addition to the one speed case, we also consider the two group equations in the cell which now become an eigenvalue problem for the lattice multiplication factor. The method of solution relies upon an efficient procedure to solve a large set of simultaneous linear equations and for this we use the IMSL library routines. Our method is compared with the results from a finite element code. The main drawback of the problem arises from the very large number of terms required in the Fourier series which taxes the storage and speed of the computer. Nevertheless, useful solutions are obtained in geometries that would normally require the use of finite element or analogous methods, for this reason the Fourier method is useful for comparison with that type of numerical approach. Extension of the method to more intricate fuel shapes, such as stars and cruciforms as well as superpositions of these, is possible.

Formula omitted] stabilization of the current profile in tokamak plasmas via an LMI approach

This paper deals with the H∞ stabilization of the spatial distribution of the current profile of tokamak plasmas using a Linear Matrix Inequality (LMI) approach. The control design is based on the one dimensional resistive diffusion equation of the magnetic flux that governs the plasma current profile evolution. The feedback control law is derived in the infinite dimensional setting without spatial discretization. The proposed distributed control is based on a proportional–integral state feedback taking into account both interior and boundary engineering actuators. Supporting numerical simulations are presented and tuning of the controller parameters attenuating uncertain disturbances is discussed.