The mathematical fractional modeling of TiO2 nanopowder synthesis by sol–gel method at low temperature
| dc.citation.epage | 626 | |
| dc.citation.issue | 3 | |
| dc.citation.journalTitle | Математичне моделювання та комп'ютинг | |
| dc.citation.spage | 616 | |
| dc.contributor.affiliation | Університет Шуайба Дуккалі | |
| dc.contributor.affiliation | Chouaib Doukkali University | |
| dc.contributor.author | Садек, О. | |
| dc.contributor.author | Садек, Л. | |
| dc.contributor.author | Тотух, С. | |
| dc.contributor.author | Хаджаджі, А. | |
| dc.contributor.author | Sadek, O. | |
| dc.contributor.author | Sadek, L. | |
| dc.contributor.author | Touhtouh, S. | |
| dc.contributor.author | Hajjaji, A. | |
| dc.coverage.placename | Львів | |
| dc.coverage.placename | Lviv | |
| dc.date.accessioned | 2025-03-04T11:32:58Z | |
| dc.date.created | 2022-02-28 | |
| dc.date.issued | 2022-02-28 | |
| dc.description.abstract | Діоксид титану — це сполука кисню і титану з формулою TiO2, яка є в природі та виготовляється у промислових масштабах. Він використовується в декількох галузях і сферах застосування, таких як косметика, фарби, продукти харчування, фотокаталізатор, електроди в літієвих батареях, сонячні батареї на барвнику (DSSC), біосенсори тощо. Враховуючи його важливість і різні сфери застосування, існує декілька методів синтез TiO2, наприклад, золь–гель метод, який широко використовується для отримання наночастинок. У нашому дослідженні, з одного боку, успішно синтезовано нанопорошки діоксиду титану, кристалізовані у фазі анатазу з розміром кристалів 49.25 нм, використовуючи тетраізопропоксид титану (TTIP) як попередник золь–гель метода. Отримані порошки аналізували методом рентгенівської дифракції (XRD) з CuKα випромінюванням (λ = 0.15406 нм) та інфрачервоною спектроскопією з перетворенням Фур’є (FTIR) в діапазоні хвильових чисел 4000−400 cm−1, а з іншого боку, представлено математичну модель для передбачення концентрації TiO2 як функції часу та концентрації реагентів за допомогою дробової похідної, більш точної, ніж похідна цілого порядку; досліджено існування та єдиність розв’язків. Крім того, визначено точки рівноваги. Для візуалізації ефективності цiєї моделі виконано числове моделювання та їх графічне представлення. | |
| dc.description.abstract | Titanium dioxide is a compound of oxygen and titanium with the formula TiO2 present in nature and manufactured on an industrial scale. It is used in several fields and applications such as cosmetics, paint, food, photocatalyst, electrodes in lithium batteries, dye solar cells (DSSC), biosensors, etc., given its importance and its various fields of application, there are several methods of synthesis of TiO2 such as the sol–gel method widely used to obtain nanoparticles. In our study, on the one hand we synthesized titanium dioxide nanopowders crystallized in the anatase phase at a crystal size of 49.25nm with success using titanium tetraisopropoxide (TTIP) as precursor by the sol–gel method. The powders obtained were analyzed by X-ray diffraction (XRD) with CuKα radiation (λ=0.15406 nm) and Fourier transform infrared spectroscopy (FTIR) in the wave number range 4000−400 cm−1, and on the other hand we present a mathematical model for the prediction of the TiO2 concentration as a function of time and the concentration of reactants by using the fractional order derivative more precise than the whole order derivative, we study the existence and the uniqueness of the solutions. In addition, we determine the points of equilibrium. Numerical simulations and their graphical representations are made to visualize the efficiency of this model. | |
| dc.format.extent | 616-626 | |
| dc.format.pages | 11 | |
| dc.identifier.citation | The mathematical fractional modeling of TiO2 nanopowder synthesis by sol–gel method at low temperature / O. Sadek, L. Sadek, S. Touhtouh, A. Hajjaji // Mathematical Modeling and Computing. — Lviv : Lviv Politechnic Publishing House, 2022. — Vol 9. — No 3. — P. 616–626. | |
| dc.identifier.citationen | The mathematical fractional modeling of TiO2 nanopowder synthesis by sol–gel method at low temperature / O. Sadek, L. Sadek, S. Touhtouh, A. Hajjaji // Mathematical Modeling and Computing. — Lviv : Lviv Politechnic Publishing House, 2022. — Vol 9. — No 3. — P. 616–626. | |
| dc.identifier.doi | doi.org/10.23939/mmc2022.03.616 | |
| dc.identifier.uri | https://ena.lpnu.ua/handle/ntb/63460 | |
| dc.language.iso | en | |
| dc.publisher | Видавництво Львівської політехніки | |
| dc.publisher | Lviv Politechnic Publishing House | |
| dc.relation.ispartof | Математичне моделювання та комп'ютинг, 3 (9), 2022 | |
| dc.relation.ispartof | Mathematical Modeling and Computing, 3 (9), 2022 | |
| dc.relation.references | [1] Ivanova T., Harizanova A., Koutzarova T., Vertruyen B. Optical and structural characterization of TiO2 films doped with silver nanoparticles obtained by sol–gel method. Optical Materials. 36 (2), 207–213 (2013). | |
| dc.relation.references | [2] Park H., Kim W.-R., Jeong H.-T., Lee J.-J., Kim H.-G., Choi W.-Y. Fabrication of dye-sensitized solar cells by transplanting highly ordered TiO2 nanotube arrays. Solar Energy Materials and Solar Cells. 95 (1), 184–189 (2011). | |
| dc.relation.references | [3] Ochiai T., Fujishima A. Photoelectrochemical properties of TiO2 photocatalyst and its applications for environmental purification. Journal of Photochemistry and Photobiology C: Photochemistry Reviews. 13 (4), 247–262 (2012). | |
| dc.relation.references | [4] Armstrong A. R., Armstrong G., Canales J., Bruce P. G. TiO2–B nanowires as negative electrodes for rechargeable lithium batteries. Journal of Power Sources. 146 (1–2), 501–506 (2005). | |
| dc.relation.references | [5] Macwan D. P., Dave P. N., Chaturvedi S. A review on nano-TiO2 sol–gel type syntheses and its applications. Journal of materials science. 46 (11), 3669–3686 (2011). | |
| dc.relation.references | [6] Wang S., Wu X., Qin W., Jiang Z. TiO2 films prepared by micro-plasma oxidation method for dyesensitized solar cell. Electrochimica Acta. 53 (4), 1883–1889 (2007). | |
| dc.relation.references | [7] Wu C.-I., Huang J.-W., Wen Y.-L., Wen S. B., Shen Y.-H., Yeh M.-Y. Preparation of TiO2 nanoparticles by supercritical carbon dioxide. Materials Letters. 62 (12–13), 1923–1926 (2008). | |
| dc.relation.references | [8] Kim S. J., Park S. D., Jeong Y. H., Park S. Homogeneous precipitation of TiO2 ultrafine powders from aqueous TiOCl2 solution. Journal of the American Ceramic Society. 82 (4), 927–932 (1999). | |
| dc.relation.references | [9] Neppolian B., Yamashita H., Okada Y., Nishijima H., Anpo M. Preparation of unique TiO2 nano-particle photocatalysts by a multi-gelation method for control of the physicochemical parameters and reactivity. Catalysis Letters. 105 (1), 111–117 (2005). | |
| dc.relation.references | [10] Ghorai T. K., Dhak D., Biswas S. K., Dalai S., Pramanik P. Photocatalytic oxidation of organic dyes by nano-sized metal molybdate incorporated titanium dioxide (MxMoxTi1−xO6) (M = Ni, Cu, Zn) photocatalysts. Journal of Molecular Catalysis A: Chemical. 273 (1–2), 224–229 (2007). | |
| dc.relation.references | [11] Peng F., Cai L., Yu H., Wang H., Yang J. Synthesis and characterization of substitutional and interstitial nitrogen-doped titanium dioxides with visible light photocatalytic activity. Journal of Solid State Chemistry. 181 (1), 130–136 (2008). | |
| dc.relation.references | [12] Bruns W., Ichim B., S¨oger C. The power of pyramid decomposition in Normaliz. Journal of Symbolic Computation. 74, 513–536 (2016). | |
| dc.relation.references | [13] Nachit W., Touhtouh S., Ramzi Z., Zbair M., Eddiai A., Rguiti M., Bouchikhi A., Hajjaji A., Benkhouja K. Synthesis of nanosized TiO2 powder by sol gel method at low temperature. Molecular Crystals and Liquid Crystals. 627 (1), 170–175 (2016). | |
| dc.relation.references | [14] Cri¸san M., Br˘aileanu A., R˘aileanu M., Zaharescu M., Cri¸san D., Dr˘agan N., Anastasescu M., Ianculescu A., Ni¸toi I., Marinescu V. E., Hodorogea S. M. Sol–gel S-doped TiO2 materials for environmental protection. Journal of Non-Crystalline Solids. 354 (2–9), 705–711 (2008). | |
| dc.relation.references | [15] Sadek O., Touhtouh S., Hajjaji A. The Rapid Identification of Solid Materials Using the ACP Method. Environmental Sciences Proceedings. 16 (1), 22 (2022). | |
| dc.relation.references | [16] Sadek O., Touhtouh S., Mahdi Bouabdalli E., Hajjaji A. Development of a protocol for the rapid identification of solid materials using the principal component analysis (ACP) method: Case of phosphate fertilizers. Materials Today: Proceedings (2022). | |
| dc.relation.references | [17] Bouabdalli E. M., El Jouad M., Touhtouh S., Sadek O., Hajjaji A. Structural studies on varied concentrations of europium doped strontium phosphate glasses. Materials Today: Proceedings (2022). | |
| dc.relation.references | [18] Horikawa T., Katoh M., Tomida T. Preparation and characterization of nitrogen-doped mesoporous titania with high specific surface area. Microporous and Mesoporous Materials. 110 (2–3), 397–404 (2008). | |
| dc.relation.references | [19] Kim B.-H., Lee J.-Y., Choa Y.-H., Higuchi M., Mizutani N. Preparation of TiO2 thin film by liquid sprayed mist CVD method. Materials Science and Engineering: B. 107 (3), 289–294 (2004). | |
| dc.relation.references | [20] Muscat J., Swamy V., Harrison N. M. First-principles calculations of the phase stability of TiO2. Physical Review B. 65 (22), 224112 (2002). | |
| dc.relation.references | [21] Mo S.-D., Ching W. Y. Electronic and optical properties of three phases of titanium dioxide: Rutile, anatase, and brookite. Physical Review B. 51 (19), 13023 (1995). | |
| dc.relation.references | [22] Ohno T., Akiyoshi M., Umebayashi T., Asai K., Mitsui T., Matsumura M. Preparation of S-doped TiO2 photocatalysts and their photocatalytic activities under visible light. Applied Catalysis A: General. 265 (1), 115–121 (2004). | |
| dc.relation.references | [23] Prasad K., Pinjari D. V., Pandit A. B., Mhaske S. T. Phase transformation of nanostructured titanium dioxide from anatase-to-rutile via combined ultrasound assisted sol–gel technique. Ultrasonics Sonochemistry. 17 (2), 409–415 (2010). | |
| dc.relation.references | [24] Graaf G. H., Stamhuis E. J., Beenackers A. A. C. M. Kinetics of low-pressure methanol synthesis. Chemical Engineering Science. 43 (12), 3185–3195 (1988). | |
| dc.relation.references | [25] Khajji B., Boujallal L., Elhia M., Balatif O., Rachik M. A fractional-order model for drinking alcohol behaviour leading to road accidents and violence. Mathematical Modeling and Computing. 9 (3), 501–518 (2022). | |
| dc.relation.references | [26] Gouasnouane O., Moussaid N., Boujena S., Kabli K. A nonlinear fractional partial differential equation for image inpainting. Mathematical Modeling and Computing. 9 (3), 536–546 (2022). | |
| dc.relation.references | [27] Ben-Loghfyry A., Hakim A. Time-fractional diffusion equation for signal and image smoothing. Mathematical Modeling and Computing. 9 (2), 351–364 (2022). | |
| dc.relation.references | [28] Pawar D. D., Patil W. D., Raut D. K. Fractional-order mathematical model for analysing impact of quarantine on transmission of COVID-19 in India. Mathematical Modeling and Computing. 8 (2), 253–266 (2021). | |
| dc.relation.references | [29] Fadugba S. E., Ali F., Abubakar A. B. Caputo fractional reduced differential transform method for SEIR epidemic model with fractional order. Mathematical Modeling and Computing. 8 (3), 537–548 (2021). | |
| dc.relation.references | [30] Kostrobij P. P., Markovych B. M., Ryzha I. A., Tokarchuk M. V. Generalized kinetic equation with spatiotemporal nonlocality. Mathematical Modeling and Computing. 6 (2), 289–296 (2019). | |
| dc.relation.references | [31] Kostrobij P., Markovych B., Viznovych O., Zelinska I., Tokarchuk M. Generalized Cattaneo–Maxwell diffusion equation with fractional derivatives. Dispersion relations. Mathematical Modeling and Computing. 6 (1), 58–68 (2019). | |
| dc.relation.references | [32] Odibat Z. M., Shawagfeh N. T. Generalized Taylor’s formula. Applied Mathematics and Computation. 186 (1), 286–293 (2007). | |
| dc.relation.references | [33] Samko S. G., Kilbas A. A., Marichev O. I. Fractional integrals and derivatives. Vol. 1. Yverdon-les-Bains, Switzerland: Gordon and breach science publishers, Yverdon (1993). | |
| dc.relation.references | [34] Lin W. Global existence theory and chaos control of fractional differential equations. Journal of Mathematical Analysis and Applications. 332 (1), 709–726 (2007). | |
| dc.relation.references | [35] Kim W. B., Choi S. H., Lee J. S. Quantitative Analysis of Ti-O-Si and Ti-O-Ti Bonds in Ti-Si Binary Oxides by the Linear Combination of XANES. Journal of Physical Chemistry B. 104 (36), 8670–8678 (2000). | |
| dc.relation.references | [36] Suppuraj P., Parthiban S., Swaminathan M., Muthuvel I. Hydrothermal fabrication of ternary NrGOTiO2/ZnFe2O4 nanocomposites for effective photocatalytic and fuel cell applications. Materials Today: Proceedings. 15 (3), 429–437 (2019). | |
| dc.relation.references | [37] Himabindu B., Devi N. L., Kanth B. R. Microstructural parameters from X-ray peak profile analysis by Williamson–Hall models; A review. Materials Today: Proceedings. 47 (14), 4891–4896 (2021). | |
| dc.relation.references | [38] Garrappa R. On linear stability of predictor–corrector algorithms for fractional differential equations. International Journal of Computer Mathematics. 87 (10), 2281–2290 (2010). | |
| dc.relation.referencesen | [1] Ivanova T., Harizanova A., Koutzarova T., Vertruyen B. Optical and structural characterization of TiO2 films doped with silver nanoparticles obtained by sol–gel method. Optical Materials. 36 (2), 207–213 (2013). | |
| dc.relation.referencesen | [2] Park H., Kim W.-R., Jeong H.-T., Lee J.-J., Kim H.-G., Choi W.-Y. Fabrication of dye-sensitized solar cells by transplanting highly ordered TiO2 nanotube arrays. Solar Energy Materials and Solar Cells. 95 (1), 184–189 (2011). | |
| dc.relation.referencesen | [3] Ochiai T., Fujishima A. Photoelectrochemical properties of TiO2 photocatalyst and its applications for environmental purification. Journal of Photochemistry and Photobiology C: Photochemistry Reviews. 13 (4), 247–262 (2012). | |
| dc.relation.referencesen | [4] Armstrong A. R., Armstrong G., Canales J., Bruce P. G. TiO2–B nanowires as negative electrodes for rechargeable lithium batteries. Journal of Power Sources. 146 (1–2), 501–506 (2005). | |
| dc.relation.referencesen | [5] Macwan D. P., Dave P. N., Chaturvedi S. A review on nano-TiO2 sol–gel type syntheses and its applications. Journal of materials science. 46 (11), 3669–3686 (2011). | |
| dc.relation.referencesen | [6] Wang S., Wu X., Qin W., Jiang Z. TiO2 films prepared by micro-plasma oxidation method for dyesensitized solar cell. Electrochimica Acta. 53 (4), 1883–1889 (2007). | |
| dc.relation.referencesen | [7] Wu C.-I., Huang J.-W., Wen Y.-L., Wen S. B., Shen Y.-H., Yeh M.-Y. Preparation of TiO2 nanoparticles by supercritical carbon dioxide. Materials Letters. 62 (12–13), 1923–1926 (2008). | |
| dc.relation.referencesen | [8] Kim S. J., Park S. D., Jeong Y. H., Park S. Homogeneous precipitation of TiO2 ultrafine powders from aqueous TiOCl2 solution. Journal of the American Ceramic Society. 82 (4), 927–932 (1999). | |
| dc.relation.referencesen | [9] Neppolian B., Yamashita H., Okada Y., Nishijima H., Anpo M. Preparation of unique TiO2 nano-particle photocatalysts by a multi-gelation method for control of the physicochemical parameters and reactivity. Catalysis Letters. 105 (1), 111–117 (2005). | |
| dc.relation.referencesen | [10] Ghorai T. K., Dhak D., Biswas S. K., Dalai S., Pramanik P. Photocatalytic oxidation of organic dyes by nano-sized metal molybdate incorporated titanium dioxide (MxMoxTi1−xO6) (M = Ni, Cu, Zn) photocatalysts. Journal of Molecular Catalysis A: Chemical. 273 (1–2), 224–229 (2007). | |
| dc.relation.referencesen | [11] Peng F., Cai L., Yu H., Wang H., Yang J. Synthesis and characterization of substitutional and interstitial nitrogen-doped titanium dioxides with visible light photocatalytic activity. Journal of Solid State Chemistry. 181 (1), 130–136 (2008). | |
| dc.relation.referencesen | [12] Bruns W., Ichim B., S¨oger C. The power of pyramid decomposition in Normaliz. Journal of Symbolic Computation. 74, 513–536 (2016). | |
| dc.relation.referencesen | [13] Nachit W., Touhtouh S., Ramzi Z., Zbair M., Eddiai A., Rguiti M., Bouchikhi A., Hajjaji A., Benkhouja K. Synthesis of nanosized TiO2 powder by sol gel method at low temperature. Molecular Crystals and Liquid Crystals. 627 (1), 170–175 (2016). | |
| dc.relation.referencesen | [14] Cri¸san M., Br˘aileanu A., R˘aileanu M., Zaharescu M., Cri¸san D., Dr˘agan N., Anastasescu M., Ianculescu A., Ni¸toi I., Marinescu V. E., Hodorogea S. M. Sol–gel S-doped TiO2 materials for environmental protection. Journal of Non-Crystalline Solids. 354 (2–9), 705–711 (2008). | |
| dc.relation.referencesen | [15] Sadek O., Touhtouh S., Hajjaji A. The Rapid Identification of Solid Materials Using the ACP Method. Environmental Sciences Proceedings. 16 (1), 22 (2022). | |
| dc.relation.referencesen | [16] Sadek O., Touhtouh S., Mahdi Bouabdalli E., Hajjaji A. Development of a protocol for the rapid identification of solid materials using the principal component analysis (ACP) method: Case of phosphate fertilizers. Materials Today: Proceedings (2022). | |
| dc.relation.referencesen | [17] Bouabdalli E. M., El Jouad M., Touhtouh S., Sadek O., Hajjaji A. Structural studies on varied concentrations of europium doped strontium phosphate glasses. Materials Today: Proceedings (2022). | |
| dc.relation.referencesen | [18] Horikawa T., Katoh M., Tomida T. Preparation and characterization of nitrogen-doped mesoporous titania with high specific surface area. Microporous and Mesoporous Materials. 110 (2–3), 397–404 (2008). | |
| dc.relation.referencesen | [19] Kim B.-H., Lee J.-Y., Choa Y.-H., Higuchi M., Mizutani N. Preparation of TiO2 thin film by liquid sprayed mist CVD method. Materials Science and Engineering: B. 107 (3), 289–294 (2004). | |
| dc.relation.referencesen | [20] Muscat J., Swamy V., Harrison N. M. First-principles calculations of the phase stability of TiO2. Physical Review B. 65 (22), 224112 (2002). | |
| dc.relation.referencesen | [21] Mo S.-D., Ching W. Y. Electronic and optical properties of three phases of titanium dioxide: Rutile, anatase, and brookite. Physical Review B. 51 (19), 13023 (1995). | |
| dc.relation.referencesen | [22] Ohno T., Akiyoshi M., Umebayashi T., Asai K., Mitsui T., Matsumura M. Preparation of S-doped TiO2 photocatalysts and their photocatalytic activities under visible light. Applied Catalysis A: General. 265 (1), 115–121 (2004). | |
| dc.relation.referencesen | [23] Prasad K., Pinjari D. V., Pandit A. B., Mhaske S. T. Phase transformation of nanostructured titanium dioxide from anatase-to-rutile via combined ultrasound assisted sol–gel technique. Ultrasonics Sonochemistry. 17 (2), 409–415 (2010). | |
| dc.relation.referencesen | [24] Graaf G. H., Stamhuis E. J., Beenackers A. A. C. M. Kinetics of low-pressure methanol synthesis. Chemical Engineering Science. 43 (12), 3185–3195 (1988). | |
| dc.relation.referencesen | [25] Khajji B., Boujallal L., Elhia M., Balatif O., Rachik M. A fractional-order model for drinking alcohol behaviour leading to road accidents and violence. Mathematical Modeling and Computing. 9 (3), 501–518 (2022). | |
| dc.relation.referencesen | [26] Gouasnouane O., Moussaid N., Boujena S., Kabli K. A nonlinear fractional partial differential equation for image inpainting. Mathematical Modeling and Computing. 9 (3), 536–546 (2022). | |
| dc.relation.referencesen | [27] Ben-Loghfyry A., Hakim A. Time-fractional diffusion equation for signal and image smoothing. Mathematical Modeling and Computing. 9 (2), 351–364 (2022). | |
| dc.relation.referencesen | [28] Pawar D. D., Patil W. D., Raut D. K. Fractional-order mathematical model for analysing impact of quarantine on transmission of COVID-19 in India. Mathematical Modeling and Computing. 8 (2), 253–266 (2021). | |
| dc.relation.referencesen | [29] Fadugba S. E., Ali F., Abubakar A. B. Caputo fractional reduced differential transform method for SEIR epidemic model with fractional order. Mathematical Modeling and Computing. 8 (3), 537–548 (2021). | |
| dc.relation.referencesen | [30] Kostrobij P. P., Markovych B. M., Ryzha I. A., Tokarchuk M. V. Generalized kinetic equation with spatiotemporal nonlocality. Mathematical Modeling and Computing. 6 (2), 289–296 (2019). | |
| dc.relation.referencesen | [31] Kostrobij P., Markovych B., Viznovych O., Zelinska I., Tokarchuk M. Generalized Cattaneo–Maxwell diffusion equation with fractional derivatives. Dispersion relations. Mathematical Modeling and Computing. 6 (1), 58–68 (2019). | |
| dc.relation.referencesen | [32] Odibat Z. M., Shawagfeh N. T. Generalized Taylor’s formula. Applied Mathematics and Computation. 186 (1), 286–293 (2007). | |
| dc.relation.referencesen | [33] Samko S. G., Kilbas A. A., Marichev O. I. Fractional integrals and derivatives. Vol. 1. Yverdon-les-Bains, Switzerland: Gordon and breach science publishers, Yverdon (1993). | |
| dc.relation.referencesen | [34] Lin W. Global existence theory and chaos control of fractional differential equations. Journal of Mathematical Analysis and Applications. 332 (1), 709–726 (2007). | |
| dc.relation.referencesen | [35] Kim W. B., Choi S. H., Lee J. S. Quantitative Analysis of Ti-O-Si and Ti-O-Ti Bonds in Ti-Si Binary Oxides by the Linear Combination of XANES. Journal of Physical Chemistry B. 104 (36), 8670–8678 (2000). | |
| dc.relation.referencesen | [36] Suppuraj P., Parthiban S., Swaminathan M., Muthuvel I. Hydrothermal fabrication of ternary NrGOTiO2/ZnFe2O4 nanocomposites for effective photocatalytic and fuel cell applications. Materials Today: Proceedings. 15 (3), 429–437 (2019). | |
| dc.relation.referencesen | [37] Himabindu B., Devi N. L., Kanth B. R. Microstructural parameters from X-ray peak profile analysis by Williamson–Hall models; A review. Materials Today: Proceedings. 47 (14), 4891–4896 (2021). | |
| dc.relation.referencesen | [38] Garrappa R. On linear stability of predictor–corrector algorithms for fractional differential equations. International Journal of Computer Mathematics. 87 (10), 2281–2290 (2010). | |
| dc.rights.holder | © Національний університет “Львівська політехніка”, 2022 | |
| dc.subject | діоксид титану | |
| dc.subject | золь–гель | |
| dc.subject | нанокристалізований | |
| dc.subject | анатаз | |
| dc.subject | дифракція Xпроменів | |
| dc.subject | ІЧ-Фур’є | |
| dc.subject | дробова модель | |
| dc.subject | точка рівноваги | |
| dc.subject | дробове числення | |
| dc.subject | похідні Капуто | |
| dc.subject | titanium dioxide | |
| dc.subject | sol–gel | |
| dc.subject | nanocrystallized | |
| dc.subject | anatase | |
| dc.subject | XRD | |
| dc.subject | FTIR | |
| dc.subject | fractional model | |
| dc.subject | equilibrium point | |
| dc.subject | fractional calculus | |
| dc.subject | Caputo derivatives | |
| dc.title | The mathematical fractional modeling of TiO2 nanopowder synthesis by sol–gel method at low temperature | |
| dc.title.alternative | Математичне дробове моделювання синтезу нанопорошку TiO2 золь–гель методом за низьких температур | |
| dc.type | Article |
Files
License bundle
1 - 1 of 1