Continuous and discrete transformations of a one-dimensional porous medium equation

We consider the one-dimensional porous medium equation $u_t=\left (u^nu_x \right )_x+\frac{\mu}{x}u^nu_x$. We derive point transformations of a general class that map this equation into itself or into equations of a similar class. In some cases this porous medium equation is connected with well known equations. With the introduction of a new dependent variable this partial differential equation can be equivalently written as a system of two equations. Point transformations are also sought for this auxiliary system. It turns out that in addition to the continuous point transformations that may be derived by Lie's method, a number of discrete transformations are obtained. In some cases the point transformations which are presented here for the single equation and for the auxiliary system form cyclic groups of finite order

Conservation Laws and Travelling Wave Solutions for Double Dispersion Equations in (1+1) and (2+1) Dimensions

In this article, we investigate two types of double dispersion equations in two different dimensions, which arise in several physical applications. Double dispersion equations are derived to describe long nonlinear wave evolution in a thin hyperelastic rod. Firstly, we obtain conservation laws for both these equations. To do this, we employ the multiplier method, which is an efficient method to derive conservation laws as it does not require the PDEs to admit a variational principle. Secondly, we obtain travelling waves and line travelling waves for these two equations. In this process, the conservation laws are used to obtain a triple reduction. Finally, a line soliton solution is found for the double dispersion equation in two dimensions

Lie point symmetries for generalised Fisher's equations describing tumour dynamics

A huge variety of phenomena are governed by ordinary differential equations (ODEs) and partial differential equations (PDEs). However, there is no general method to solve them. Obtaining solutions for differential equations is one of the greatest problem for both applied mathematics and physics. Multiple integration methods have been developed to the day to solve particular types of differential equations, specially those focused on physical or biological phenomena. In this work, we review several applications of the Lie method to obtain solutions of reaction-diffusion equations describing cell dynamics and tumour invasion.We would like to acknowledge group FQM-201 from Junta de Andalucia. We also would like to acknowledge Profs. Rita Tracina and Mariano Torrisi from the University of Catania (Italy) and Victor M. Perez Garcia from the University of Castilla-La Mancha (Spain) for discussions. This work was partially supported by the Fundacion Espanola para la Ciencia y la Tecnologia [UCA PR214], the Asociacion Pablo Ugarte (APU, Spain) and Inversion Territorial Integrada de la Provincia de Cadiz [ITI-0038-2019]

Generalized Camassa-Holm Equations: Symmetry, Conservation Laws and Regular Pulse and Front Solutions

In this paper, we consider a member of an integrable family of generalized Camassa-Holm (GCH) equations. We make an analysis of the point Lie symmetries of these equations by using the Lie method of infinitesimals. We derive nonclassical symmetries and we find new symmetries via the nonclassical method, which cannot be obtained by Lie symmetry method. We employ the multiplier method to construct conservation laws for this family of GCH equations. Using the conservation laws of the underlying equation, double reduction is also constructed. Finally, we investigate traveling waves of the GCH equations. We derive convergent series solutions both for the homoclinic and heteroclinic orbits of the traveling-wave equations, which correspond to pulse and front solutions of the original GCH equations, respectively

Classical and nonclassical symmetries of a generalized Boussinesq equation

We apply the Lie-group formalism and the nonclassical method due to Bluman and Cole to deduce symmetries of the generalized Boussinesq equation, which has the classical Boussinesq equation as an special case. We study the class of functions $f(u)$ for which this equation admit either the classical or the nonclassical method. The reductions obtained are derived. Some new exact solutions can be derived

Modifying the magnetic response of magnetotactic bacteria: incorporation of Gd and Tb ions into the magnetosome structure

Magnetotactic bacteria Magnetospirillum gryphiswaldense MSR-1 biosynthesise chains of cube–octahedral magnetosomes, which are 40 nm magnetite high quality (Fe3O4) nanoparticles. The magnetic properties of these crystalline magnetite nanoparticles, which can be modified by the addition of other elements into the magnetosome structure (doping), are of prime interest in a plethora of applications, those related to cancer therapy being some of the most promising ones. Although previous studies have focused on transition metal elements, rare earth (RE) elements are very interesting as doping agents, both from a fundamental point of view (e.g. significant differences in ionic sizes) and for the potential applications, especially in biomedicine (e.g. magnetic resonance imaging and luminescence). In this work, we have investigated the impact of Gd and Tb on the magnetic properties of magnetosomes by using different complementary techniques. X-ray diffraction, transmission electron microscopy, and X-ray absorption near edge spectroscopy analyses have revealed that a small amount of RE ions, ∼3–4%, incorporate into the Fe3O4 structure as Gd3+ and Tb3+ ions. The experimental magnetic characterisation has shown a clear Verwey transition for the RE-doped bacteria, located at T ∼ 100 K, which is slightly below the one corresponding to the undoped ones (106 K). However, we report a decrease in the coercivity and remanence of the RE-doped bacteria. Simulations based on the Stoner–Wohlfarth model have allowed us to associate these changes in the magnetic response with a reduction of the magnetocrystalline (KC) and, especially, the uniaxial (Kuni) anisotropies below the Verwey transition. In this way, Kuni reaches a value of 23 and 26 kJ m−3 for the Gd- and Tb-doped bacteria, respectively, whilst a value of 37 kJ m−3 is obtained for the undoped bacteria.This work was supported in part by the Spanish MCIN/AEI under Projects MAT2017-83631-C3-R and PID2020-115704RB-C33. The work of Elizabeth M. Jefremovas was supported by the “Concepción Arenal Grant” awarded by Gobierno de Cantabria and Universidad de Cantabria. The work of Lourdes Marcano was supported by the Postdoctoral Fellowship from the Basque Government under Grant POS-2019-2-0017. The authors would like to thank “Nanotechnology in translational hyperthermia” (HIPERNANO)-RED2018-102626-T. We thank the ALBA (CLAESS beamline) synchrotron radiation facilities and staff for the allocation of beamtime and assistance during the experiments

Elucidating the role of shape anisotropy in faceted magnetic nanoparticles using biogenic magnetosomes as a model

Shape anisotropy is of primary importance to understand the magnetic behavior of nanoparticles, but a rigorous analysis in polyhedral morphologies is missing. In this work, a model based on finite element techniques has been developed to calculate the shape anisotropy energy landscape for cubic, octahedral, and truncated octahedral morphologies. In all cases, a cubic shape anisotropy is found that evolves to quasi uniaxial anisotropy when the nanoparticle is elongated amp; 8805;2 . This model is tested on magnetosomes, amp; 8764;45 nm truncated octahedral magnetite nanoparticles forming a chain inside Magnetospirillum gryphiswaldense MSR 1 bacteria. This chain presents a slightly bent helical configuration due to a 20 tilting of the magnetic moment of each magnetosome out of chain axis. Electron cryotomography images reveal that these magnetosomes are not ideal truncated octahedrons but present amp; 8776;7.5 extrusion of one of the 001 square faces and amp; 8776;10 extrusion of an adjacent 111 hexagonal face. Our model shows that this deformation gives rise to a quasi uniaxial shape anisotropy, a result of the combination of a uniaxial Ksh u 7 kJ m amp; 8722;3 and a cubic Ksh c 1.5 kJ m amp; 8722;3 contribution, which is responsible for the 20 tilting of the magnetic moment. Finally, our results have allowed us to accurately reproduce, within the framework of the Landau Lifshitz Gilbert model, the experimental AC loops measured for these magnetotactic bacteri

Tuning the Magnetic Response of Magnetospirillum magneticum by Changing the Culture Medium A Straightforward Approach to Improve Their Hyperthermia Efficiency

Magnetotactic bacteria Magnetospirillum magneticum AMB 1 have been cultured using three different media magnetic spirillum growth medium with Wolfe s mineral solution MSGM W , magnetic spirillum growth medium without Wolfe s mineral solution MSGM W , and flask standard medium FSM . The influence of the culture medium on the structural, morphological, and magnetic characteristics of the magnetosome chains biosynthesized by these bacteria has been investigated by using transmission electron microscopy, X ray absorption spectroscopy, and X ray magnetic circular dichroism. All bacteria exhibit similar average size for magnetosomes, 40 45 nm, but FSM bacteria present slightly longer subchains. In MSGM W bacteria, Co2 ions present in the medium substitute Fe2 ions in octahedral positions with a total Co doping around 4 5 . In addition, the magnetic response of these bacteria has been thoroughly studied as functions of both the temperature and the applied magnetic field. While MSGM W and FSM bacteria exhibit similar magnetic behavior, in the case of MSGM W, the incorporation of the Co ions affects the magnetic response, in particular suppressing the Verwey amp; 8764;105 K and low temperature amp; 8764;40 K transitions and increasing the coercivity and remanence. Moreover, simulations based on a Stoner Wolhfarth model have allowed us to reproduce the experimentally obtained magnetization versus magnetic field loops, revealing clear changes in different anisotropy contributions for these bacteria depending on the employed culture medium. Finally, we have related how these magnetic changes affect their heating efficiency by using AC magnetometric measurements. The obtained AC hysteresis loops, measured with an AC magnetic field amplitude of up to 90 mT and a frequency, f, of 149 kHz, reveal the influence of the culture medium on the heating properties of these bacteria below 35 mT, MSGM W bacteria are the best heating mediators, but above 60 mT, FSM and MSGM W bacteria give the best heating results, reaching a maximum heating efficiency or specific absorption rate SAR of SAR f amp; 8776; 12 W g 1 kHz

Enhanced Group Analysis and Exact Solutions of Variable Coefficient Semilinear Diffusion Equations with a Power Source

A new approach to group classification problems and more general investigations on transformational properties of classes of differential equations is proposed. It is based on mappings between classes of differential equations, generated by families of point transformations. A class of variable coefficient (1+1)-dimensional semilinear reaction-diffusion equations of the general form $f(x)u_t=(g(x)u_x)_x+h(x)u^m$ ($m\ne0,1$) is studied from the symmetry point of view in the framework of the approach proposed. The singular subclass of the equations with $m=2$ is singled out. The group classifications of the entire class, the singular subclass and their images are performed with respect to both the corresponding (generalized extended) equivalence groups and all point transformations. The set of admissible transformations of the imaged class is exhaustively described in the general case $m\ne2$. The procedure of classification of nonclassical symmetries, which involves mappings between classes of differential equations, is discussed. Wide families of new exact solutions are also constructed for equations from the classes under consideration by the classical method of Lie reductions and by generation of new solutions from known ones for other equations with point transformations of different kinds (such as additional equivalence transformations and mappings between classes of equations).Comment: 40 pages, this is version published in Acta Applicanda Mathematica

Nanoflowers Versus Magnetosomes Comparison Between Two Promising Candidates for Magnetic Hyperthermia Therapy

Magnetic Fluid Hyperthermia mediated by iron oxide nanoparticles is one of the most promising therapies for cancer treatment. Among the different candidates, magnetite and maghemite nanoparticles have revealed to be some of the most promising candidates due to both their performance and their biocompatibility. Nonetheless, up to date, the literature comparing the heating efficiency of magnetite and maghemite nanoparticles of similar size is scarce. To fill this gap, here we provide a comparison between commercial Synomag Nanoflowers pure maghemite and bacterial magnetosomes pure magnetite synthesized by the magnetotactic bacterium Magnetospirillum gryphiswaldense of amp; 10216;D amp; 10217; amp; 8776; 40 45 nm. Both types of nanoparticles exhibit a high degree of crystallinity and an excellent degree of chemical purity and stability. The structural and magnetic properties in both nanoparticle ensembles have been studied by means of X Ray Diffraction, Transmission Electron Microscopy, X Ray Absorption Spectroscopy, and SQUID magnetometry. The heating efficiency has been analyzed in both systems using AC magnetometry at several field amplitudes 0 88 mT and frequencies 130, 300, and 530 kH