Convergence and dynamics of improved Chebyshev-Secant-type methods for non differentiable operators

[EN] In this paper, the convergence and dynamics of improved Chebyshev-Secant-type iterative methods are studied for solving nonlinear equations in Banach space settings. Their semilocal convergence is established using recurrence relations under weaker continuity conditions on first-order divided differences. Convergence theorems are established for the existence-uniqueness of the solutions. Next, center-Lipschitz condition is defined on the first-order divided differences and its influence on the domain of starting iterates is compared with those corresponding to the domain of Lipschitz conditions. Several numerical examples including Automotive Steering problems and nonlinear mixed Hammerstein-type integral equations are analyzed, and the output results are compared with those obtained by some of similar existing iterative methods. It is found that improved results are obtained for all the numerical examples. Further, the dynamical analysis of the iterative method is carried out. It confirms that the proposed iterative method has better stability properties than its competitors.This research was partially supported by Ministerio de Economia y Competitividad under grant PGC2018-095896-B-C22.Kumar, A.; Gupta, DK.; Martínez Molada, E.; Hueso, JL. (2021). Convergence and dynamics of improved Chebyshev-Secant-type methods for non differentiable operators. Numerical Algorithms. 86(3):1051-1070. https://doi.org/10.1007/s11075-020-00922-9S10511070863Hernández, M.A.: Chebyshev’s approximation algorithms and applications. Comput. Math. Appl. 41(3-4), 433–445 (2001)Ezquerro, J.A., Grau-Sánchez, Miquel, Hernández, M.A.: Solving non-differentiable equations by a new one-point iterative method with memory. J. Complex. 28(1), 48–58 (2012)Ioannis , K.A., Ezquerro, J.A., Gutiérrez, J.M., hernández, M.A., saïd Hilout: On the semilocal convergence of efficient Chebyshev-Secant-type methods. J. Comput. Appl. Math. 235(10), 3195–3206 (2011)Hongmin, R., Ioannis, K.A.: Local convergence of efficient Secant-type methods for solving nonlinear equations. Appl. Math. comput. 218(14), 7655–7664 (2012)Ioannis, Ioannis K.A., Hongmin, R.: On the semilocal convergence of derivative free methods for solving nonlinear equations. J. Numer. Anal. Approx. Theory 41 (1), 3–17 (2012)Hongmin, R., Ioannis, K.A.: On the convergence of King-Werner-type methods of order $1+\sqrt {2}$ free of derivatives. Appl. Math. Comput. 256, 148–159 (2015)Kumar, A., Gupta, D.K., Martínez, E., Sukhjit, S.: Semilocal convergence of a Secant-type method under weak Lipschitz conditions in Banach spaces. J. Comput. Appl. Math. 330, 732–741 (2018)Grau-Sánchez, M., Noguera, M., Gutiérrez, J.M.: Frozen iterative methods using divided differences “à la Schmidt–Schwetlick”. J. Optim. Theory Appl. 160 (3), 931–948 (2014)Louis, B.R.: Computational Solution of Nonlinear Operator Equations. Wiley, New York (1969)Blanchard, P.: The dynamics of Newton’s method. Proc. Symp. Appl. Math. 49, 139–154 (1994)Parisa, B., Cordero, A., Taher, L., Kathayoun, M., Torregrosa, J.R.: Widening basins of attraction of optimal iterative methods. Nonlinear Dynamics 87 (2), 913–938 (2017)Chun, C., Neta, B.: The basins of attraction of Murakami’s fifth order family of methods. Appl. Numer. Math. 110, 14–25 (2016)Magreñán, Á. A.: A new tool to study real dynamics: the convergence plane. Appl. Math. Comput. 248, 215–224 (2014)Ramandeep, B., Cordero, A., Motsa, S.S., Torregrosa, J.R.: Stable high-order iterative methods for solving nonlinear models. Appl. Math. Comput. 303, 70–88 (2017)Pramanik, S.: Kinematic synthesis of a six-member mechanism for automotive steering. Trans Ame Soc. Mech. Eng. J. Mech. Des. 124(4), 642–645 (2002

Gold-Based Nanoparticles on Amino-Functionalized Mesoporous Silica Supports as Nanozymes for Glucose Oxidation

The transformation of glucose represents a topic of great interest at different levels. In the first place, glucose is currently conceived as a green feedstock for the sustainable production of chemicals. Secondly, the depletion of glucose at the cellular level is currently envisioned as a promising strategy to treat and alter the erratic metabolism of tumoral cells. The use of natural enzymes offers multiple advantages in terms of specificity towards the glucose substrate but may lack sufficient robustness and recyclability beyond the optimal operating conditions of these natural systems. In the present work, we have evaluated the potential use of an inorganic based nanohybrid containing gold nanoparticles supported onto ordered mesoporous supports. We have performed different assays that corroborate the enzyme-mimicking response of these inorganic surrogates towards the selective conversion of glucose into gluconic acid and hydrogen peroxide. Moreover, we conclude that these enzyme-like mimicking surrogates can operate at different pH ranges and under mild reaction conditions, can be recycled multiple times and maintain excellent catalytic response in comparison with other gold-based catalysts

Efficient and facile tuning of Vulcan XC72 with ultra-small Pt nanoparticles for electrocatalytic applications

In this work, we report the synergic combination of tiny, reproducible, and crystalline platinum nanoparticles with functionalized Vulcan XC72 support for the preparation of efficient electrocatalysts. The morphology and crystallinity of Pt NPs, the amination of carbon support and the Pt loading have been properly evaluated by TEM, STEM, ATR-FTIR, XPS and microwave plasma-atomic emission spectrometry MP-AES. The electrochemical performance of the as prepared powders has been evaluated and compared with the exhibited by commercial Pt-ETEK catalyst with similar Pt-loadings, i.e. 20% wt. The herein described electrocatalyst shows higher dispersion (98.2 m2 gPt-1) of Pt nanoparticles (1.5 ± 0.3 nm in size) and strong metal–support interaction than commercial Pt-ETEK counterpart, highlighting the elevated mass activity for methanol electrooxidation

Platinum-based nanodendrites as glucose oxidase-mimicking surrogates

Catalytic conversion of glucose represents an interesting field of research with multiple applications. From the biotechnology point of view, glucose conversion leads to the fabrication of different added-value by-products. In the field of nanocatalytic medicine, the reduction of glucose levels within the tumor microenvironment (TME) represents an appealing approach based on the starvation of cancer cells. Glucose typically achieves high conversion rates with the aid of glucose oxidase (GOx) enzymes or by fermentation. GOx is subjected to degradation, possesses poor recyclability and operates under very specific reaction conditions. Gold-based materials have been typically explored as inorganic catalytic alternatives to GOx in order to convert glucose into building block chemicals of interest. Still, the lack of sufficient selectivity towards certain products such as gluconolactone, the requirement of high fluxes of oxygen or the critical size dependency hinder their full potential, especially in liquid phase reactions. The present work describes the synthesis of platinum-based nanodendrites as novel enzyme-mimicking inorganic surrogates able to convert glucose into gluconolactone with outstanding selectivity values above 85%. We have also studied the enzymatic behavior of these Pt-based nanozymes using the Michaelis–Menten and Lineweaver–Burk models and used the main calculation approaches available in the literature to determine highly competitive glucose turnover rates for Pt or Pt–Au nanodendrites

Green synthesis of a Cu/SiO2 catalyst for efficient H2-SCR of NO

In this work, the synthesis of Cu/SiO2 catalysts starting frompre-formed copper nanoparticle (CuNP) colloidal suspensions was carried out. Two different protocols for the CuNP synthesis were tested: (i) a green approach using water as solvent and ascorbic acid as reducer and stabilizing agent, and (ii) a second solvothermal method involving the use of diethylene glycol as solvent, sodium hypophosphite (NaH2PO2) as reducer, and polyvinylpyrrolidone (PVP) and cetyltrimethylammonium bromide (CTAB) as stabilizing agents. In addition, and for the sake of comparison, a third catalyst was prepared by solid state conventional grinding of CuO with SiO2. The catalysts were tested in the environmentally relevant catalytic reduction of NOX with H2, in a temperature range from 300 to 500 °C. The catalysts were characterized by X-ray diffraction (XRD), temperature programmed reduction (TPR) cycles, Raman spectroscopy, and N2 adsorption for specific surface BET measurements. From these techniques CuO and Cu(0) species were detected depending on the synthesis protocol. CuNP size and size distribution in the colloid suspensions were determined by transmission electronic microscopy (TEM). The catalyst prepared from the aqueous suspension (CuAsc/SiO2) exhibited higher NO conversion (100%) and selectivity (85%) toward N2 at the lower reaction evaluated temperature (300 °C). The CuCTAB/SiO2 catalyst obtained by the solvothermal approach showed activity at high reaction temperature (400 °C) preferentially. The metal-support mechanical mixture exhibited a negligible response at low temperature and low conversion (68%) and selectivity (88%) at 500 °C. Nanoparticle size and distribution on the support, together with the metal-support interaction, were postulated as the most plausible parameters governing the catalytic performance of the different Cu/SiO2 materials

Anisotropic Au-ZnO photocatalyst for the visible-light expanded oxidation of n-hexane

In this work we present a novel metal-semiconductor heterostructure that contains triangular and prism-shaped plasmonic gold nanostructures directly synthesized and assembled onto ZnO nanostructures. Spatially-resolved (SR) high-resolution electron energy loss spectroscopy (EELS) at the local (sub-nanometer scale) level confirmed the field enhancement of the local electromagnetic fields in the surroundings of the triangular and prism-shaped Au nanostructures and in the interfacial junction between Au and ZnO. Different LED excitation sources have been systematically selected in the whole UV–vis-NIR range to evaluate the photocatalytic response of the Au-ZnO heterostructures towards the oxidation of n-hexane, selected as a model VOC present in indoor environments. The Au-ZnO exhibits visible expanded photo-response with the more energetic interband and intraband electrons and the higher LED irradiation wavelengths and it is able to outperform its ZnO plain counterpart.Financial support from the European Research Council (ERC Advanced Grant CADENCE number 742684) is gratefully acknowledged. We also acknowledge the support from CIBER-BBN and MINECO (Spain) with project CTQ2016-79419-R. R.A. acknowledges funding from the Spanish Ministerio de Ciencia, Innovación y Universidades (project grant MAT2016-79776-P (AEI/FEDER, UE)) and from the European Union H2020 program “ESTEEM3” (823717).Peer reviewe

Extraordinary sensitizing effect of co-doped carbon nanodots derived from mate herb: application to enhanced photocatalytic degradation of chlorinated wastewater compounds under visible light

tThe present work investigates the role of two types of carbon nanodots (CNDs) as novel sensitizers of TiO2to create a visible-light driven photo-catalyst that is not only efficient for solar-driven pollution abate-ment, but also inexpensive, durable and environmentally-friendly. Two widely available green organicprecursors, the Argentinean herb Mate and the Stevia plant have been selected as the carbogenic sourceto thermally induce the formation of different types of CNDs with different levels of N and P doping andtunable photoluminescence response in the UV–vis-near infrared (NIR) ranges. These CNDs have beensuccessfully assembled with TiO2to form heterogeneous photocatalysts that are highly active in thevisible-light and NIR- driven photodegradation of 2,4-dichlorophenol (2,4-DCP), a persistent chlorinatedorganic compound present in numerous pesticide formulations

In-situ preparation of ultra-small Pt nanoparticles within rod-shaped mesoporous silica particles: 3-D tomography and catalytic oxidation of n-hexane

The shape and porous configuration of supports are key parameters to design outstanding catalysts. However, the selection of a proper mesoporous support, such as SBA-15, by itself does not guarantee accessibility to catalytic sites. The distribution of the active phase and its stability are strongly related to the procedure used to deposit it on the catalytic substrate. Herein, we have prepared rod-shaped SBA-15 silica supports functionalized with amine groups to facilitate the electrostatic attraction and a good distribution of the resulting Pt-based catalytic nanoparticles along the pore walls. Additionally, the use of tetrakis-(hydroxymethyl)-phosphonium chloride (THPC) as both reductant and stabilizer is presented as a novel alternative for Pt nanoparticle synthesis. The behaviour of this catalyst in the total oxidation of n-hexane demonstrates high activity and excellent stability after 70 h on reaction stream. STEM-HAADF and 3-D tomography were used to confirm the presence of the metallic nanoparticles within the mesochannels and to corroborate their reduced sintering after reaction

Semilocal Convergence Analysis of an Iteration of Order Five Using Recurrence Relations in Banach Spaces

[EN] Semilocal convergence for an iteration of order five for solving nonlinear equations in Banach spaces is established under second-order Fr,chet derivative satisfying the Lipschitz condition. It is done by deriving a number of recurrence relations. A theorem for the existence-uniqueness along with the estimation of error bounds of the solution is established. Its R-order is shown to be equal to five. Both efficiency and computational efficiency indices are given. A variety of examples are worked out to show its applicability. In comparison to existing methods having similar R-orders, improved results in terms of computational efficiency index and error bounds are found using our methodology.The authors thank the referees for their valuable comments which have improved the presentation of the paper. The authors thankfully acknowledge the financial assistance provided by Council of Scientific and Industrial Research (CSIR), New Delhi, India.Singh, S.; Gupta, D.; Martínez Molada, E.; Hueso Pagoaga, JL. (2016). Semilocal Convergence Analysis of an Iteration of Order Five Using Recurrence Relations in Banach Spaces. Mediterranean Journal of Mathematics. 13(6):4219-4235. doi:10.1007/s00009-016-0741-5S42194235136Cordero A., Hueso J.L., Martinez E., Torregrosa J.R.: Increasing the convergence order of an iterative method for nonlinear systems. Appl. Math. Lett. 25, 2369–2374 (2012)Chen, L., Gu, C., Ma Y.: Semilocal convergence for a fifth order Newton’s method using Recurrence relations in Banach spaces. J. Appl. Math. 2011, 1–15 (2011)Wang X., Kou J., Gu C.: Semilocal convergence of a sixth order Jarrat method in Banach spaces. Numer. Algorithms 57, 441–456 (2011)Zheng L., Gu C.: Semilocal convergence of a sixth order method in Banach spaces. Numer. Algorithms 61, 413–427 (2012)Zheng L., Gu C.: Recurrence relations for semilocal convergence of a fifth order method in Banach spaces. Numer. Algorithms 59, 623–638 (2012)Proinov P.D., Ivanov S.I.: On the convergence of Halley’s method for multiple polynomial zeros. Mediterr. J. Math. 12, 555–572 (2015)Ezquerro, J.A., Hernández-Verón M.A.: On the domain of starting points of Newton’s method under center lipschitz conditions. Mediterr. J. Math. (2015). doi: 10.1007/s00009-015-0596-1Cordero A., Hernández-Verón M.A., Romero N., Torregrosa J.R.: Semilocal convergence by using recurrence relations for a fifth-order method in Banach spaces. J. Comput. Appl. Math. 273, 205–213 (2015)Parida P.K., Gupta D.K.: Recurrence relations for a Newton-like method in Banach spaces. J. Comput. Appl. Math. 206, 873–887 (2007)Hueso J.L., Martínez E.: Semilocal convergence of a family of iterative methods in Banach spaces. Numer. Algorithms 67, 365–384 (2014)Argyros, I.K., Hilout S.: Numerical methods in nonlinear analysis. World Scientific Publ. Comp., New Jersey (2013)Argyros, I.K., Hilout, S., Tabatabai, M.A.: Mathematical modelling with applications in biosciences and engineering. Nova Publishers, New York (2011)Argyros I.K., Khattri S.K.: Local convergence for a family of third order methods in Banach spaces. J. Math. 46, 53–62 (2004)Argyros I.K., Hilout A.S.: On the local convergence of fast two-step Newton-like methods for solving nonlinear equations. J. Comput. Appl. Math. 245, 1–9 (2013)Kantorovich, L.V., Akilov G.P.: Functional analysis. Pergamon Press, Oxford (1982)Argyros I.K., George S., Magreñán A.A.: Local convergence for multi-point-parametric Chebyshev-Halley-type methods of higher convergence order. J. Comput. Appl. Math. 282, 215–224 (2015)Argyros I.K., Magreñán A.A.: A study on the local convergence and the dynamics of Chebyshev-Halley-type methods free from second derivative. Numer. Algorithms 71, 1–23 (2015)Amat S., Hernández M.A., Romero N.: A modified Chebyshev’s iterative method with at least sixth order of convergence. Appl. Math. Comput. 206, 164–174 (2008)Chun, C., St $${\breve{a}}$$ a ˘ nic $${\breve{a}}$$ a ˘ , P., Neta, B.: Third-order family of methods in Banach spaces. Comput. Math. Appl. 61, 1665–1675 (2011)Ostrowski, A.M.: Solution of equations in Euclidean and Banach spaces, 3rd edn. Academic Press, New-York (1977)Jaiswal J.P.: Semilocal convergence of an eighth-order method in Banach spaces and its computational efficiency. Numer. Algorithms 71, 933–951 (2015)Traub, J.F.: Iterative methods for the solution of equations. Prentice-Hall, Englewood Cliffs (1964