Raman-spektroskopische Untersuchungen an Alkali- und Erdalkalisilicatgläsern bei hohen Temperaturen

In der vorliegenden Arbeit wird die Struktur von Alkali- und Erdalkalisilicatglaesern bei hohen Temperaturen (bis 1800 K) mit Hilfe der Raman-Spektroskopie untersucht. Ein wesentlicher Teil der vorliegenden Arbeit besteht in dem Aufbau einer Hochtemperatureinrichtung, die es erlaubt, Raman-Spektren von Silicatglaesern bei sehr hohen Temperaturen zu messen. Mit der Hochtemperatur-Raman-Spektroskopie an Silicatglaesern sind erhebliche experimentelle Schwierigkeiten verbunden: Die thermische Strahlung der Probe überlagert sich mit dem Raman-Spektrum.Die Temperaturbestimmung der Glasprobe, die einen Durchmesser von nur 0,8 mm hat, erfolgt durch den Vergleich der Stokes- und Anti-Stokes-Raman-Intensitaeten einer intensiven Linie einer Referenzprobe. Die Natriumsilicatglaeser werden detailliert untersucht und die Verteilung der Struktureinheiten in den Natriumsilicatglaesern wird zwischen Zimmertemperatur und 900 K bestimmt. Aus der Verteilung der Strukturelemente wird eine Gleichgewichtskonstante K berechnet, welche die Disproportionierungsreaktion zwischen den Struktureinheiten in den Glaesern beschreibt. Der Wert für die Reaktionsenthalpie liegt im untersuchten Konzentrationsbereich zwischen 0 und 28 kJ/mol und haengt systematisch von der Zusammensetzung ab. Die Reaktionsenthalpie nimmt mit zunehmendem Natriumoxid-Gehalt zu.Die quantitative Auswertung der Raman-Spektren der Kaliumsilicatglaeser und der Bariumsilicatglaeser ist auf Grund deren Kristallisation bei hohen Temperaturen mit Problemen behaftet.

Synthesis and characterisation of hydrotalcite-like compounds for transesterification reaction

A series of templated [Mg(1-x)Alx(OH)2]x+(CO3)x/n2- with different structural properties have been synthesised using an alkali-free coprecipitation route. The macroporous materials were been obtained using two different kind of templating agents, polymeric materials, in order to cover a bigger size range (750-70 nm). All the materials have been characterized by different techniques: porosimetry, SEM-EDX, TEM-EDX, MP-AES, XRD, CO2 titration before and after the calcinations process. All the materials have been tested for transesterification reaction of C4-C8 triglycerides with methanol for biodiesel production.

The alkali element and boron geochemistry of ODP Site 169-1038 in the Escanaba Trough, East Pacific

A suite of conjugate pore fluid and sediment samples were collected during Leg 169 of the ODP from within the clastic sedimentary sequences which host massive sulphides at Central Hill, Escanaba Trough (ODP Site 1038). We report the alkali element and boron, and Li and B isotope data for these samples. Relative to a reference site (Site 1037) located outside the zone of high heat flow, pore fluids from Site 1038 show a wide variation in Cl (300-800 mM), and have far higher concentrations of Li (up to 6.2 mM), B (up to 9.7 mM), Cs (up to 5.0 mM), and Rb (up to 97 mM). We show that the pore fluids are derived from hydrothermal circulation that has extended into the basement oceanic crust, with input of the alkali elements and B as the rising hydrothermal fluids interact geochemically with the overlying clastic sediments. There is, however, no marked depletion of these elements in the conjugate sediments, suggesting that there has been advective transport of fluids away from the primary hydrothermal reaction site. This is supported by modelling of the Li and B isotope systematics of the pore fluids, which shows that they record extensive formation of secondary minerals during cooling of the fluids from ~350 to ~20ºC. Precipitation of metal-rich sulphides would have occurred prior to the formation of these minerals, thus, the pore fluid Li and B isotope data can place important constraints on the locus of sulphide deposition beneath the seafloor at Escanaba.

Estudio de la utilización de las cenizas de cenicero de centrales termoeléctricas de carbón como adición del cemento portland : análisis comparativo con las cenizas volantes

El empleo de nuevas adiciones en el cemento se plantea como una vía para que éste sea un material más sostenible. En este contexto, las cenizas de fondo o cenicero de las centrales termoeléctricas de carbón actualmente se están llevando a vertederos creando un problema medioambiental o se están empleando con usos minoritarios. Sin embargo, la presente Tesis doctoral demuestra cómo este material puede ser empleado como un componente principal de los cementos portland mezclado en una proporción optimizada con las cenizas volantes o solo. Por tanto, este estudio se puede considerar como un trabajo prenormativo que cubre las demandas de la sociedad tanto económicas como ambientales. Se han estudiado las propiedades químicas, físicas y mecánicas de las cenizas de fondo o cenicero de las centrales termoeléctricas de carbón como una adición potencial de los cementos portland (con adiciones) en comparación con los cementos portland con cenizas volantes. En consecuencia, el objeto de la presente Tesis Doctoral es el análisis de las prestaciones de morteros elaborados con clínkeres de cemento portland y cenizas de fondo o cenicero con cenizas volantes de las centrales termoeléctricas de carbón en unos porcentajes similares a los correspondientes a los CEM II/A-V, CEM II/B-V y CEM IV/A (V) de la UNE-EN 197-1:2011. La caracterización de las cenizas de fondo o cenicero y de las cenizas volantes de las centrales termoeléctricas de carbón se ha realizado mediante las técnicas analíticas de FRX, ICP, análisis químicos, DRX, densidad, granulometría láser, superficie específica Blaine, ATD, TG, puzolanicidad, MEB y EDX; mientras que la caracterización de las mezclas se ha realizado mediante análisis químico (análisis químico, FRX e ICP y), DRX, MIP, granulometría láser, puzolanicidad, MEB y EDX, agua de consistencia normal, inicio y final de fraguado, estabilidad de volumen, colorimetría, calor de hidratación, DTA y TG, asícomo ensayos de resistencia; resistencia a flexión y compresión y ensayos de durabilidad (carbonatación natural, resistencia al hielo-deshielo, resistencia a la acción de los sulfatos y resistencia a la reacción árido-álcali). Finalmente, se han comprobado las propiedades de las cenizas de cenicero y cenizas volantes en hormigones, realizando ensayos de resistencia a compresión y resistividad. Los resultados obtenidos indican que la sustitución parcial o completa de las cenizas volantes por las de cenicero no tiene un efecto tecnológicamente importante en las propiedades ni mecánicas ni durables, incluso los mejora en determinados aspectos. Por tanto, se recomienda la normalización de las cenizas de fondo o cenicero de las centrales termoeléctricas de carbón como componente principal de los cementos portland comunes de la UNE-EN 197-1:2011. Actualmente, la mayoría de las cenizas de fondo se consideran como un residuo que no tiene un posible uso. Sólo se han encontrado algunos datos relativos a la aplicación de este material combinado con cenizas volantes como un componente principal de los cementos portland. Por tanto, la realización de un estudio integrado considerando aspectos que van desde la caracterización de las cenizas hasta las mezclas de conglomerante y la hidratación de éstas, desarrollo de resistencias y demás prestaciones y durabilidad (carbonatación natural, resistencia al hielo-deshielo, resistencia a la acción de los sulfatos y resistencia a la reacción árido-álcali), así como los ensayos de resistencia en hormigón es totalmente novedoso. Como resultado final se propone incorporar estas nuevas adiciones en aplicaciones particulares y en la norma más apropiada para ello. Los resultados han demostrado que la sustitución completa o parcial de las cenizas volantes por cenizas de fondo o cenicero de las centrales termoeléctricas de carbón en cementos de los tipos CEM II/A-V, CEM II/B-V y CEM IV/A no afecta de forma significativa en la resistencia a compresión a 1, 3 ,7, 28 ni 90 días ni a la durabilidad. En parte esto se debe a que la composición química de ambas cenizas es muy similar en la mayoría de los elementos tales como Fe2O3, TiO2, P2O5, SrO2, aunque en algún caso, como en el ZnO, se encuentra alguna ligera diferencia. Por tanto, se pueden esperar unas ligeras diferencias en el mecanismo de hidratación de las diferentes mezclas estudiadas. La presencia de los óxidos mencionados afectará a la composición de la fase acuosa y, en consecuencia, podrían ser elementos lixiviables. Asimismo, influyen de distinta manera en propiedades tales como los tiempos de fraguado y en la durabilidad. New additions to the cement are needed to achieve a more sustainable construction material. Within this context, bottom ashes produced in coal-fired power stations are currently wastes which are dumped provoking an environmental problem. Only in few cases are being used in minor applications. However, the present PhD Thesis shows how this material can be used as a main constituent of Portland cement when it is mixed in an optimised proportion with fly ashes or added to the Portland clinker alone. Therefore, this study may also be considered as a pre-standardization work which covers both the environmental and economic demands of society. Chemical, physical and mechanical characteristics of pulverized coal combustion bottom ash used as a potential constituent of Portland cements (with additions) are studied in comparison to Portland cements with fly ashes. Therefore, the aim of this experimental PhD Thesis is the analysis of the performance of mortars made of clinker of Portland cement and bottom and/or fly ashes in similar proportions to those of CEM II/A-V, CEM II/B-V and CEM IV/A (V) according to EN 197-1:2011. Characterisation of bottom and fly ashes has been done by XRF, ICP, chemical analyses, XRD, density, laser granulometry, Blaine, ATD, TG, pozzolanity, SEM and EDS. Characterisation of bottom and fly ashes mixes has been perform by chemical analyses, XRF, ICP, XRD, MIP, laser granulometry, pozzolanity, SEM, EDS, setting time, soundness, colorimetric test, heat of hydration, ATD, TG, compressive strength, and durability tests (natural carbonation, frost-thaw resistance, sulphate resistance and silica-alkali resistance). In conclusion, it can be established that partial or complete replacement of fly ash by bottom ash has neither significant effect on mechanical nor durability properties. Even, they are improved in several aspects. Therefore, it is recommended to standardise the bottom ash as a main cement constituent of the European standard EN 197-1:2011. Nowadays, most bottom ashes are considered as waste without any potential re-use. Only a few papers deal with the study of this material and its use mixed with fly ashes to be employed as a main constituent of Portland cement. Therefore, the execution of an integrated study considering together aspects from the initial characterization of the ashes and blinder mixes to the hydration steps, strength achievement, leading behaviour and durability (natural carbonation, sulphate attack, aggregate-alcali reaction and freeze-thaw resistance) is totally new. As result, it is proposed to include this new addition for particular applications in the appropriate cement standard. The results have shown that with regard to the compressive strength at 1, 3, 7, 28 and 90 days, partial or complete replacement of fly ash by bottom ash in CEM II/A-V, CEM II/B-V and CEM IV/A has no more significant effects. Partially, this can be explained because the bottom ash contains a similar amount of most of the elements, Fe2O3, TiO2, P2O5, SrO2, and so on, instead of ZnO. Therefore, slight hydration differences are expected. The presence of such oxides might have a significant effect on pore solution concentration and so will be leachable constituents. They will also play an important role in the cement properties such as setting times and durability.

Synthesis and structure characterization of chromium oxide prepared by solid thermal decomposition reaction

Mesoporous chromium oxide (Cr2O3) nanocrystals were first synthesized by the thermal decomposition reaction of Cr(NO3)(3)(circle)9H(2)O using citric acid monohydrate (CA) as the mesoporous template agent. The texture and chemistry of chromium oxide nanocrystals were characterized by N-2 adsorption-desorption isotherms, FTIR, X-ray diffraction (XRD), UV-vis, and thermoanalytical methods. It was shown that the hydrate water and CA are the crucial factors in influencing the formation of mesoporous Cr2O3 nanocrystals in the mixture system. The decomposition of CA results in the formation of a mesoporous structure with wormlike pores. The hydrate water of the mixture provides surface hydroxyls that act as binders, making the nanocrystals aggregate. The pore structures and phases of chromium oxide are affected by the ratio of precursor-to-CA, thermal temperature, and time.

Evaluation of the activity and stability of alkali-doped metal oxide catalysts for application to an intensified method of biodiesel production

A series of alkali-doped metal oxide catalysts were prepared and evaluated for activity in the transesterification of rapeseed oil to biodiesel. Of those evaluated, LiNO3/CaO, NaNO3/CaO, KNO3/CaO and LiNO3/MgO exhibited >90% conversion in a standard 3 h test. There was a clear correlation between base strength and activity. These catalysts appeared to be promising candidates to replace conventional homogeneous catalysts for biodiesel production as the reaction times are low enough to be practical in continuous processes and the preparations are neither prohibitively difficult nor costly. However, metal leaching from the catalyst was detected, and this resulted in some homogeneous activity. This would have to be resolved before these catalysts would be viable for large-scale biodiesel production facilities.

Resistance of alkali activated slag concretes to chloride environments based on chloride penetration measurements

ABSTRACT

One of the binder systems with low environmental footprint is alkali activated slag concretes (AASC), made by adding alkalis such as sodium hydroxide and sodium silicate to industrial by-products such as ground granulated blast furnace slag (GGBS). Whilst they have the similar behaviour as that of traditional cement systems in terms of strength and structural behaviour, AASC do exhibit superior performance in terms of abrasion and acid resistance and fire protection.
In this article, the authors focus their attention on chloride ingress into different grades of AASC. The mix variables in AASC included water-to-binder, binder to aggregate ratio, percentage of alkali and the SiO2/Na2O ratio (silica modulus, Ms). The first challenge is to get mixes for different range of workability (with slump values from 40mm to 240mm) and reasonable early age and long term compressive strength according to each one. Then the chloride diffusion and migration in those mixes were measured and compared with same normal concretes in the existed literature based on chloride penetration depth. Comparing the chloride ingress between tradition concretes and AASCs is worthwhile to prove the possibility of increasing concrete lifetime in proximity to sea and deciding while such concretes are practical for use. Findings show that compared to the PC concretes, the AAS concretes have lower rate of chloride ingress.

Resistance of alkali activated slag concretes to chloride environments

ABSTRACT: Researchers are focusing their attention on alternative binder systems using 100% supplementary cementitious materials as it allows better control over the microstructure formation and low to moderate environmental footprint. One such system being considered is alkali activated slag concretes (AASC), made by adding alkalis such as sodium hydroxide and sodium silicate to ground granulated blast furnace slag (GGBS). Whilst they have a similar behaviour as that of traditional cement systems in terms of strength and structural behaviour, AASC are reported to exhibit superior performance in terms of abrasion,acid resistance and fire protection.
In this article, the authors investigate chloride ingress into different grades of AASC. The mix variables in AASC included water to binder, and binder to aggregate ratio, percentage of alkali and the SiO2/Na2O ratio (silica modulus, Ms). The first challenge was to develop mixes for different range of workability (with slump values from 40mm to 240mm) and reasonable early age and long term compressive strength. Further chloride ingress into those mixes were assessed and compared with the data from normal concretes based on literature. Findings show that compared to the PC concretes, the AAS concretes have lower rate of chloride ingress.

Simplified model for prediction of compressive strength of alkali-activated natural pozzolans

The assessment of pozzolanic activity is essential for estimating the reaction of a material as pozzolan. Natural pozzolans can be activated and condensed with sodium silicate in an alkaline environment to synthesize high performance cementitious construction materials with low environmental impact. In this paper, the pozzolanic activities of five natural pozzolans are studied. The correlation between type and chemical composition of natural pozzolan, which affects the formation of the geopolymer gel phase, both for the calcined and untreated natural pozzolans, have been reviewed. The improvement in pozzolanic properties was studied following heat treatment including calcinations and/or elevated curing temperature by using alkali solubility, and compressive strength tests. A model was developed to allow prediction of the alkali-activated pozzolan strength versus their chemical compositions, alkali solubility, and crystallinity.

Functional alkali tantalate 2D structures for microelectronics and related applications

Alkali tantalates and niobates, including K(Ta / Nb)O3, Li(Ta / Nb)O3 and Na(Ta / Nb)O3, are a very promising ferroic family of lead-free compounds with perovskite-like structures. Their versatile properties make them potentially interesting for current and future application in microelectronics, photocatalysis, energy and biomedics. Among them potassium tantalate, KTaO3 (KTO), has been raising interest as an alternative for the well-known strontium titanate, SrTiO3 (STO). KTO is a perovskite oxide with a quantum paraelectric behaviour when electrically stimulated and a highly polarizable lattice, giving opportunity to tailor its properties via external or internal stimuli. However problems related with the fabrication of either bulk or 2D nanostructures makes KTO not yet a viable alternative to STO. Within this context and to contribute scientifically to the leverage tantalate based compounds applications, the main goals of this thesis are: i) to produce and characterise thin films of alkali tantalates by chemical solution deposition on rigid Si based substrates, at reduced temperatures to be compatible with Si technology, ii) to fulfil scientific knowledge gaps in these relevant functional materials related to their energetics and ii) to exploit alternative applications for alkali tantalates, as photocatalysis. In what concerns the synthesis attention was given to the understanding of the phase formation in potassium tantalate synthesized via distinct routes, to control the crystallization of desired perovskite structure and to avoid low temperature pyrochlore or K-deficient phases. The phase formation process in alkali tantalates is far from being deeply analysed, as in the case of Pb-containing perovskites, therefore the work was initially focused on the process-phase relationship to identify the driving forces responsible to regulate the synthesis. Comparison of phase formation paths in conventional solid-state reaction and sol-gel method was conducted. The structural analyses revealed that intermediate pyrochlore K2Ta2O6 structure is not formed at any stage of the reaction using conventional solid-state reaction. On the other hand in the solution based processes, as alkoxide-based route, the crystallization of the perovskite occurs through the intermediate pyrochlore phase; at low temperatures pyrochlore is dominant and it is transformed to perovskite at >800 °C. The kinetic analysis carried out by using Johnson-MehlAvrami-Kolmogorow model and quantitative X-ray diffraction (XRD) demonstrated that in sol-gel derived powders the crystallization occurs in two stages: i) at early stage of the reaction dominated by primary nucleation, the mechanism is phase-boundary controlled, and ii) at the second stage the low value of Avrami exponent, n ~ 0.3, does not follow any reported category, thus not permitting an easy identification of the mechanism. Then, in collaboration with Prof. Alexandra Navrotsky group from the University of California at Davis (USA), thermodynamic studies were conducted, using high temperature oxide melt solution calorimetry. The enthalpies of formation of three structures: pyrochlore, perovskite and tetragonal tungsten bronze K6Ta10.8O30 (TTB) were calculated. The enthalpies of formation from corresponding oxides, ∆Hfox, for KTaO3, KTa2.2O6 and K6Ta10.8O30 are -203.63 ± 2.84 kJ/mol, - 358.02 ± 3.74 kJ/mol, and -1252.34 ± 10.10 kJ/mol, respectively, whereas from elements, ∆Hfel, for KTaO3, KTa2.2O6 and K6Ta10.8O30 are -1408.96 ± 3.73 kJ/mol, -2790.82 ± 6.06 kJ/mol, and -13393.04 ± 31.15 kJ/mol, respectively. The possible decomposition reactions of K-deficient KTa2.2O6 pyrochlore to KTaO3 perovskite and Ta2O5 (reaction 1) or to TTB K6Ta10.8O30 and Ta2O5 (reaction 2) were proposed, and the enthalpies were calculated to be 308.79 ± 4.41 kJ/mol and 895.79 ± 8.64 kJ/mol for reaction 1 and reaction 2, respectively. The reactions are strongly endothermic, indicating that these decompositions are energetically unfavourable, since it is unlikely that any entropy term could override such a large positive enthalpy. The energetic studies prove that pyrochlore is energetically more stable phase than perovskite at low temperature. Thus, the local order of the amorphous precipitates drives the crystallization into the most favourable structure that is the pyrochlore one with similar local organization; the distance between nearest neighbours in the amorphous or short-range ordered phase is very close to that in pyrochlore. Taking into account the stoichiometric deviation in KTO system, the selection of the most appropriate fabrication / deposition technique in thin films technology is a key issue, especially concerning complex ferroelectric oxides. Chemical solution deposition has been widely reported as a processing method to growth KTO thin films, but classical alkoxide route allows to crystallize perovskite phase at temperatures >800 °C, while the temperature endurance of platinized Si wafers is ~700 °C. Therefore, alternative diol-based routes, with distinct potassium carboxylate precursors, was developed aiming to stabilize the precursor solution, to avoid using toxic solvents and to decrease the crystallization temperature of the perovskite phase. Studies on powders revealed that in the case of KTOac (solution based on potassium acetate), a mixture of perovskite and pyrochlore phases is detected at temperature as low as 450 °C, and gradual transformation into monophasic perovskite structure occurs as temperature increases up to 750 °C, however the desired monophasic KTaO3 perovskite phase is not achieved. In the case of KTOacac (solution with potassium acetylacetonate), a broad peak is detected at temperatures <650 °C, characteristic of amorphous structures, while at higher temperatures diffraction lines from pyrochlore and perovskite phases are visible and a monophasic perovskite KTaO3 is formed at >700 °C. Infrared analysis indicated that the differences are due to a strong deformation of the carbonate-based structures upon heating. A series of thin films of alkali tantalates were spin-coated onto Si-based substrates using diol-based routes. Interestingly, monophasic perovskite KTaO3 films deposited using KTOacac solution were obtained at temperature as low as 650 °C; films were annealed in rapid thermal furnace in oxygen atmosphere for 5 min with heating rate 30 °C/sec. Other compositions of the tantalum based system as LiTaO3 (LTO) and NaTaO3 (NTO), were successfully derived as well, onto Si substrates at 650 °C as well. The ferroelectric character of LTO at room temperature was proved. Some of dielectric properties of KTO could not be measured in parallel capacitor configuration due to either substrate-film or filmelectrode interfaces. Thus, further studies have to be conducted to overcome this issue. Application-oriented studies have also been conducted; two case studies: i) photocatalytic activity of alkali tantalates and niobates for decomposition of pollutant, and ii) bioactivity of alkali tantalate ferroelectric films as functional coatings for bone regeneration. Much attention has been recently paid to develop new type of photocatalytic materials, and tantalum and niobium oxide based compositions have demonstrated to be active photocatalysts for water splitting due to high potential of the conduction bands. Thus, various powders of alkali tantalates and niobates families were tested as catalysts for methylene blue degradation. Results showed promising activities for some of the tested compounds, and KNbO3 is the most active among them, reaching over 50 % degradation of the dye after 7 h under UVA exposure. However further modifications of powders can improve the performance. In the context of bone regeneration, it is important to have platforms that with appropriate stimuli can support the attachment and direct the growth, proliferation and differentiation of the cells. In lieu of this here we exploited an alternative strategy for bone implants or repairs, based on charged mediating signals for bone regeneration. This strategy includes coating metallic 316L-type stainless steel (316L-SST) substrates with charged, functionalized via electrical charging or UV-light irradiation, ferroelectric LiTaO3 layers. It was demonstrated that the formation of surface calcium phosphates and protein adsorption is considerably enhanced for 316L-SST functionalized ferroelectric coatings. Our approach can be viewed as a set of guidelines for the development of platforms electrically functionalized that can stimulate tissue regeneration promoting direct integration of the implant in the host tissue by bone ingrowth and, hence contributing ultimately to reduce implant failure.

A-REIT seasoned equity offerings : determinants and market reaction

The paper examines the decision by Australian Real Estate Trusts (A-REITs) to issue seasoned equity offerings from 2000 - 2008 and stock market reaction to the offerings using panel data and event study methodologies, respectively. The global financial crisis has resulted in freezing of the Australian bond markets, with several A-REITs left with seasoned equity issuance and asset sales as the only viable modes of raising additional capital. The findings review that leverage and operating risk are negative significant determinants of seasoned equity offerings; profitability and growth opportunities are positive significant determinants. Of the structure and type of properties held by the A-REIT, only stapled management structure and international operations are significant determinants. Type of properties held by A-REITs show inconsistent results. Similar to previous studies of seasoned equity offerings, we find a significant negative abnormal return associated with their announcement and no evidence of excessive leakage of information. Cross-sectional regressions show that the issued amount raised and leverage are significant factors affecting abnormal returns.

Zeolite from alkali modified kaolin increases NH4+ retention by sandy soil : column experiments

Sandy soils have low water and nutrient retention capabilities so that zeolite soil amendments are used for high value land uses including turf and horticulture to reduce leaching losses of NH4+ fertilisers. MesoLite is a zeolitic material made by caustic treatment of kaolin at 80-95oC. It has a moderately low surface area (9-12m2/g) and very high cation exchange capacity (494 cmol(+)/kg). Laboratory column experiments showed that an addition of 0.4% MesoLite to a sandy soil greatly (90%) reduced leaching of added NH4+ compared to an unamended soil and MesoLite is 11 times more efficient in retaining NH4+ than natural zeolite. Furthermore, NH4+-MesoLite slowly releases NH4+ to soil solution and is likely to be an effective slow release fertiliser.

Stability and convergence of an implicit numerical method for the non-linear fractional reaction-subdiffusion process

In this paper, we consider the following non-linear fractional reaction–subdiffusion process (NFR-SubDP): Formula where f(u, x, t) is a linear function of u, the function g(u, x, t) satisfies the Lipschitz condition and 0Dt1–{gamma} is the Riemann–Liouville time fractional partial derivative of order 1 – {gamma}. We propose a new computationally efficient numerical technique to simulate the process. Firstly, the NFR-SubDP is decoupled, which is equivalent to solving a non-linear fractional reaction–subdiffusion equation (NFR-SubDE). Secondly, we propose an implicit numerical method to approximate the NFR-SubDE. Thirdly, the stability and convergence of the method are discussed using a new energy method. Finally, some numerical examples are presented to show the application of the present technique. This method and supporting theoretical results can also be applied to fractional integrodifferential equations.