Second-order Diffusion Research Articles

A Fast Finite Difference Method for Two-Dimensional Space-Fractional Diffusion Equations

Fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second-order diffusion equations. Because of the nonlocal property of fractional differential operators, numerical methods for space-fractional diffusion equations generate complicated dense or full coefficient matrices. Consequently, these numerical methods were traditionally solved by Gaussian elimination, which requires computational work of $O(N^3)$ per time step and $O(N^2)$ of memory, where $N$ is the number of spatial grid points in the discretization. The significant computational work and memory requirement of the numerical methods impose a serious challenge for the numerical simulation of two- and especially three-dimensional space-fractional diffusion equations. We develop a fast and yet accurate solution method for the implicit finite difference discretization of space-fractional diffusion equations in two space dimensions by carefully analyzing the structure of the coefficient matrix of the finite difference method and delicately decomposing the coefficient matrix into a combination of sparse and structured dense matrices. The fast method has a computational work count of $O(N \log N)$ per iteration and a memory requirement of $O(N)$, while retaining the same accuracy as the underlying finite difference method solved with Gaussian elimination. Numerical experiments show that the fast method has a significant reduction of CPU time, from two months and eight days as consumed by the traditional method to less than 40 minutes, with less than one ten-thousandth of the memory required by the traditional method, in the context of a two-dimensional space-fractional diffusion equation with 512 $\times$ 512 = 262,144 spatial nodes and 512 time steps. This demonstrates the utility of the method.

A Fourth Order Parabolic Problem with a Nonlinear Diffusion

Abstract The paper investigates a thin film equation with a second-order diffusion term: ut + (unuxxx)x − (umux)x = 0 which has many important applications in biomedicine, biology and physics. For m > n − 1, the solution also converges to its mean at an exponential rate as the time.

Biosorption of Pb(II) by Biomass of KC‐2: Kinetic, Equilibrium and Characteristic Studies

Performance and characteristics of biosorption of Pb(II) had been studied in a batch system using the fungal strain biomass, KC-2. The biosorption performance was investigated by analysing the effects of such factors as the initial pH, initial Pb(II) concentration, and contact time at 303 K. The maximum Pb(II) adsorption was obtained at pH 5.0. The experimental data were described by the pseudo first-order, pseudo second-order and intraparticle diffusion kinetic models, and were closely followed the pseudo second-order kinetic model. The equilibrium experimental data were well fitted to Langmuir model and the maximum biosorption capacity was 84.03 mg g(-1). The adsorption mechanism was examined by FTIR, SEM and EDAX analysis. Results indicated that carboxylic, hydroxyl and amine groups were involved in the biosorption and ion exchange mechanism existed.

Re-weighted functional estimation of second-order diffusion processes

Second-order diffusion process can not only model integrated and differentiated diffusion processes but also overcome the difficulties associated with the nondifferentiability of the Brownian motion, so these models play an important role in econometric analysis. In this paper, we propose a re-weighted estimator of the diffusion coefficient in the second-order diffusion model. Consistence of the estimator is proved under appropriate conditions and the conditions that ensure the asymptotic normality are also stated. The performance of the proposed estimator is assessed by simulation study.

Preparation of low cost activated carbon from Myrtus communis and pomegranate and their efficient application for removal of Congo red from aqueous solution

In this research, the potential applicability of activated carbon prepared from Myrtus communis (AC–MC) and pomegranate (AC–PG) as useful adsorbents for the removal of Congo red (CR) from aqueous solutions in batch method was investigated. The effects of pH, contact time, agitation time and amount of adsorbents on removal percentage of Congo red on both adsorbents were examined. Increase in pH up to 6 for AC–MC and pH 7 for AC–PG increase the adsorption percentage (capacity) and reach equilibrium within 30 min of contact time. Fitting the experimental data to conventional isotherm models like Freundlich, Langmuir, Tempkin and Dubinin–Radushkevich show that the experimental data fitted very well to the Freundlich isotherm for AC–MC and Langmuir isotherm for AC–PG. Fitting the experimental data to different kinetic models such as pseudo first-order, pseudo second-order, Elovich and intraparticle diffusion mechanism showed the applicability of a pseudo second-order with involvement of intraparticle diffusion model for interpretation of experimental data for both adsorbents. The adsorption capacity of AC–PG and AC–MC for the removal of CR was found to be 19.231 and 10 mg g −1. These results clearly indicate the efficiency of adsorbents as a low cost adsorbent for treatment of wastewater containing CR.

New fourth-order partial differential equations for filtering in electronic speckle pattern interferometry fringes

Partial differential equations (PDEs) based methods have been demonstrated to be a powerful tool for filtering in electronic speckle pattern interferometry fringe or wrapped phase patterns. In this paper, we derive the new fourth-order partial differential equations (NFOPDE) with a better performance for filtering in electronic speckle pattern interferometry fringe patterns based on the variational methods. We test the proposed models on two computer-simulated speckle fringe patterns and an experimentally obtained fringe pattern, respectively, and compare our models with the widely used, well-known the second-order selective degenerate diffusion PDE model (SOSDPDE) and the published fourth-order PDE model (PFOPDE). The proposed NFOPDE can overcome the shortages that both the SOSDPDE and PFOPDE encounter. In all cases, our NFOPDE outperforms SOSDPDE and PFOPDE.

An O( N log 2N) alternating-direction finite difference method for two-dimensional fractional diffusion equations

Fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by the second-order diffusion equations. Because of the nonlocal property of fractional differential operators, the numerical methods for fractional diffusion equations often generate dense or even full coefficient matrices. Consequently, the numerical solution of these methods often require computational work of O( N 3) per time step and memory of O( N 2) for where N is the number of grid points. In this paper we develop a fast alternating-direction implicit finite difference method for space-fractional diffusion equations in two space dimensions. The method only requires computational work of O( N log 2 N) per time step and memory of O( N), while retaining the same accuracy and approximation property as the regular finite difference method with Gaussian elimination. Our preliminary numerical example runs for two dimensional model problem of intermediate size seem to indicate the observations: To achieve the same accuracy, the new method has a significant reduction of the CPU time from more than 2 months and 1 week consumed by a traditional finite difference method to 1.5 h, using less than one thousandth of memory the standard method does. This demonstrates the utility of the method.

Waste utilization: The removal of textile dye (Bomaplex Red CR-L) from aqueous solution on sludge waste from electrocoagulation as adsorbent

The potential of removing Bomaplex Red CR-L dye from electro coagulation sludge has been investigated for different Bomaplex Red CR-L concentrations with varying adsorbent amount, temperature, pH and stirrer speed. Removal of Bomaplex Red CR-L is found to be high at lower pH values. The maximum adsorption of the dye has been achieved in aqueous solutions using 0.5 g of calcinated sludge from electro coagulation at a pH of 2.0 within 20 min. The Langmuir and Freundlich isotherm models were tested and it was found that the Langmuir equation was fitted better than the Freundlich equation. Maximum adsorption capacity ( q m) was calculated as 192.31 mg g − 1 at 333 K. In addition, the adsorption data obtained at various temperatures were successfully applied to pseudo first-order, pseudo second-order and intraparticle diffusion models. The rates of adsorption were found to conform to pseudo second-order kinetics with good correlation ( R 2 ≥ 0.9997). This study showed that calcinated the sludge from electrocoagulation has a highly adsorbent capacity.

High Order Diffusion Tensor Imaging in Human Glioblastoma

Diffusion tensor imaging has been used to characterize tumor heterogeneity and invasion in human glioblastoma. Recently, higher order diffusion tensors have been proposed as solutions to errors associated with diffusion tensor imaging estimates of complex microstructures. The purpose of the current study was to examine higher order diffusion characteristics in human glioblastoma prior to surgical resection using the fourth-order diffusion tensor model. Twenty-five patients with newly diagnosed glioblastoma participated in the study. Diffusion-weighted images were collected in 21 directions. The second-order (traditional) and fourth-order diffusion tensors were calculated and compared in regions of contrast enhancement, T2 signal abnormality, and normal-appearing white matter. Orientation distribution functions were strikingly different between the two tensor models, particularly in regions with tumor heterogeneity and/or regions of suspected tumor invasion. Image contrast was significantly higher in fourth-order scalar measures compared to second-order scalars. Results of particular eigenvalues and scalars using the fourth-order tensor showed differences between T2 abnormal regions and contrast enhancement, whereas second-order eigenvalues and scalars did not show differences. This suggests that higher order diffusion images could potentially be more sensitive to tumor invasion. These results suggest that the fourth-order diffusion tensor has the ability to add value to second-order (traditional) diffusion tensor imaging in the evaluation of glioblastoma.

Front dynamics in fractional-order epidemic models

A number of recent studies suggest that human and animal mobility patterns exhibit scale-free, Lévy-flight dynamics. However, current reaction-diffusion epidemics models do not account for the superdiffusive spread of modern epidemics due to Lévy flights. We have developed a SIR model to simulate the spatial spread of a hypothetical epidemic driven by long-range displacements in the infective and susceptible populations. The model has been obtained by replacing the second-order diffusion operator by a fractional-order operator. Theoretical developments and numerical simulations show that fractional-order diffusion leads to an exponential acceleration of the epidemic's front and a power-law decay of the front's leading tail. Our results indicate the potential of fractional-order reaction-diffusion models to represent modern epidemics.

Link between Alginate Reaction Front Propagation and General Reaction Diffusion Theory

We provide a common theoretical framework reuniting specific models for the Ca(2+)-alginate system and general reaction diffusion theory along with experimental validation on a microfluidic chip. As a starting point, we use a set of nonlinear, partial differential equations that are traditionally solved numerically: the Mikkelsen-Elgsaeter model. Applying the traveling-wave hypothesis as a major simplification, we obtain an analytical solution. The solution indicates that the fundamental properties of the alginate reaction front are governed by a single dimensionless parameter λ. For small λ values, a large depletion zone accompanies the reaction front. For large λ values, the alginate reacts before having the time to diffuse significantly. We show that the λ parameter is of general importance beyond the alginate model system, as it can be used to classify known solutions for second-order reaction diffusion schemes, along with the novel solution presented here. For experimental validation, we develop a microchip model system, in which the alginate gel formation can be carried out in a highly controlled, essentially 1D environment. The use of a filter barrier enables us to rapidly renew the CaCl(2) solution, while maintaining flow speeds lower than 1 μm/s for the alginate compartment. This allows one to impose an exactly known bulk CaCl(2) concentration and diffusion resistance. This experimental model system, taken together with the theoretical development, enables the determination of the entire set of physicochemical parameters governing the alginate reaction front in a single experiment.

On continuous branches of very singular similarity solutions of a stable thin film equation. I – The Cauchy problem

We consider the fourth-order thin film equation, with a stable second-order diffusion term. For the first critical exponent, where N ≥ 1 is the space dimension, the Cauchy problem is shown to admit countable continuous branches of source-type self-similar very singular solutions of the form These solutions are inherently oscillatory in nature and will be shown in Part II to be the limit of appropriate free-boundary problem solutions. For p ≠ p0, the set of very singular solutions is shown to be finite and to be consisting of a countable family of branches (in the parameter p) of similarity profiles that originate at a sequence of critical exponents {pl, l ≥ 0}. At p = pl, these branches appear via a non-linear bifurcation mechanism from a countable set of similarity solutions of the second kind of the pure thin film equation Such solutions are detected by the ‘Hermitian spectral theory’, which allows an analytical n-branching approach. As such, a continuous path as n → 0+ can be constructed from the eigenfunctions of the linear rescaled operator for n = 0, i.e. for the bi-harmonic equation ut = −Δ2u. Numerics are used, wherever appropriate, to support the analysis.

On continuous branches of very singular similarity solutions of the stable thin film equation. II – Free-boundary problems

We discuss the fourth-order thin film equation with a stable second-order diffusion term, in the context of a standard free-boundary problem with zero height, zero contact angle and zero-flux conditions imposed at an interface. For the first critical exponent where N ≥ 1 is the space dimension, there are continuous sets (branches) of source-type very singular self-similar solutions of the form For p ≠ p0, the set of very singular self-similar solutions is shown to be finite and consists of a countable family of branches (in the parameter p) of similarity profiles that originate at a sequence of critical exponents {pl, l ≥ 0}. At p = pl, these branches appear via a non-linear bifurcation mechanism from a countable set of second kind similarity solutions of the pure thin film equation Such solutions are detected by a combination of linear and non-linear ‘Hermitian spectral theory’, which allows the application of an analytical n-branching approach. In order to connect with the Cauchy problem in Part I, we identify the cauchy problem solutions as suitable ‘limits’ of the free-boundary problem solutions.

Removal of Ruthenium from Aqueous Solution by Clinoptilolite

AbstractRuthenium compounds are highly toxic and carcinogenic. In the present study, clinoptilolite was used in the removal of Ru species from aqueous solutions. Clinoptilolite is a good choice of sorbents because it is naturally abundant and therefore cheap. After the process where Ru was removed from the aqueous solution, the clinoptilolite was characterized by X-ray diffraction, X-ray fluorescence, thermogravimetric analysis, and Fourier-transform infrared spectroscopy techniques. The influence of pH, contact time, and temperature on the adsorption of Ru was investigated and the optimum conditions were found to be 2 h of contact time and pH = 2. Pseudo first-order, pseudo second-order, Elovich, and intra-particle diffusion models were used to analyze the adsorption-rate data. The pseudo second-order model was found to be the best kinetics model in terms of matching the experimental results obtained. Adsorption isotherms were constructed to assess the maximum adsorption capacity of clinoptilolite. The Langmuir model fitted the data reasonably well in terms of regression coefficients. Adsorption studies were also performed at different temperatures to calculate the thermodynamic parameters. The numerical value of ΔG0 decreased with increasing temperature, indicating that adsorption is favored at higher temperatures. The positive values of ΔH0 corresponded to the endothermic nature of the adsorption processes. The proposed method of removal is applicable at an industrial scale.

Local Linear Estimation of Second-Order Diffusion Models

In this article, we present the local linear estimations for diffusion coefficient and drift coefficient in the second-order diffusion model. We show that under mild conditions, the estimators are weak consistent. We also use a Monte Carlo experiment to compare our estimators with the ones in Nicolau (2007).

INVARIANT SOLUTIONS OF NONLINEAR DIFFUSION EQUATIONS WITH MAXIMAL SYMMETRY ALGEBRA

Nonlinear n-dimensional second-order diffusion equations admitting maximal Lie algebras of point symmetries are considered. Examples of invariant solutions, as well as of solutions on invariant subspaces for some nonlinear operators, are constructed for arbitrary n. A complete description of all possible types of invariant solutions is given in the case n = 2 for the equation possessing an infinitely dimensional symmetry algebra. The results obtained are generalized for the hyperbolic and other fourth-order parabolic equations of thin film and nonlinear dispersion type.

Removal of bromate ion using powdered activated carbon

Bromate ion (BrO 3 −) removal from drinking water by powdered activated carbons (PAC S) in bath mode was evaluated under various operational conditions. Six kinds of PACs, including wood-based carbon, fruit-based carbon, coal-based carbon, and these three carbons thermally deoxidized in a nitrogen atmosphere, were selected to investigate their capacity on BrO 3 − removal. With the highest zeta potential value and being richly mesoporous, coal-based carbon had a high and an excellent BrO 3 − adsorption efficiency. The removal content of BrO 3 − by per gram of coal-based carbon was 0.45 mg within 5 hr in 100 μg/L bromate solution. The surface characteristics of PACs and bromide formation revealed that both physical and chemical PACs properties simultaneously affected the adsorption-reduction process. Under acidic conditions, PAC S possessed high zeta value and adequate basic groups and exhibited neutral or positive charges, promoting BrO 3 − adsorption-reduction on the carbon surface. Interestingly, the PAC S thermally deoxidized in N 2 atmosphere optimized their properties, e.g. increasing their zeta values and decreasing the oxygen content which accelerated the BrO 3 − removal rate. The maximum adsorption capacity of fruit-based carbon was the highest among all tested carbons (99.6 mg/g), possibly due to its highest pore volume. Remarkably, the thermal regeneration of PACs in N 2 atmosphere could completely recover the adsorption capacity of PACs. The kinetic data obtained from carbons was analyzed using pseudo second-order and intraparticle diffusion models, with results showing that the intraparticle diffusion was the more applicable model to describe adsorption of BrO 3 − onto PACs.

Optimization of Orange G dye adsorption by activated carbon of Thespesia populnea pods using response surface methodology

Thespesia populnea is a large tree found in the tropical regions and coastal forests of India. Its pods were used as a raw material for the preparation of activated carbon. The prepared activated carbon was used for the adsorptive removal of Orange G dye from aqueous system. The effects of various parameters such as agitation time, initial dye concentration and adsorbent dosage were studied using response surface methodology (RSM). RSM results show that 0.54g of activated carbon was required for the maximum adsorption of Orange G dye (17.6mgL−1) within a time period of 4.03h. Adsorption data were modeled using Freundlich and Langmuir adsorption isotherms. The adsorption of Orange G dye by activated carbon obeyed both Fruendlich and Langmuir isotherm. Adsorption kinetic data were tested using pseudo-zero, first, second-order and intraparticle diffusion models. Kinetic studies revealed that the adsorption followed pseudo-second-order reaction with regard to the intraparticle diffusion. FTIR spectral result indicated all the functional group except primary amines (3417cm−1) and CN (1618cm−1) were involved in the adsorption process. XRD data showed that Orange G dye adsorbed activated carbon might not induce the bulk phase changes. SEM results showed that the surface of the activated carbon was turned from dark to light color after dye adsorption.

Physicochemical and geochemical characteristics of raw marine sediment used in fluoride removal

The study was directed to use raw marine sediment in the removal of fluoride. The sediment was mainly composed of calcite, magnesium-calcite and aragonite. The effect of the initial fluoride concentration, pH and the contact time was studied at room temperature to determine the adsorption capacity of the sediment. The optimum adsorption capacity was observed at pH values of 5 and 6.2. The adsorption process was fast and the equilibrium was reached within 60 min. For fluoride solutions of 10 and 15 mg/L, 100% removal was obtained onto 0.1 g of raw marine sediment. Pseudo–first-order, pseudo–second-order, Elovich and intra-particle diffusion equations were used to deduce the kinetic data. The adsorption mechanism was rather complex process, and the intra-particle diffusion was not the only rate-controlling step. The equilibrium data were tested using thirteen isotherm models (Langmuir, Freundlich, Tempkin, Dubinin-Radushkevich, Erunauer-Emmett-Teller, Flory-Huggins, Non-ideal competitive adsorption, Generalized, Redlich Peterson, Khan, Sips, Koble Corrigan and Toth isotherm equations). Five different error functions were applied. For the sorption of fluoride process, the calculated activation energy and the free energy were of 0.707 and −14.491 kJ /mol, respectively.

A direct O( N log 2 N) finite difference method for fractional diffusion equations

Fractional diffusion equations model phenomena exhibiting anomalous diffusion that can not be modeled accurately by the second-order diffusion equations. Because of the nonlocal property of fractional differential operators, the numerical methods have full coefficient matrices which require storage of O( N 2) and computational cost of O( N 3) where N is the number of grid points. In this paper we develop a fast finite difference method for fractional diffusion equations, which only requires storage of O( N) and computational cost of O( N log 2 N) while retaining the same accuracy and approximation property as the regular finite difference method. Numerical experiments are presented to show the utility of the method.