scispace - formally typeset
Search or ask a question

Showing papers in "Ricerche Di Matematica in 2021"


Journal ArticleDOI
TL;DR: In this article, the authors considered a rarefied polyatomic gas with a non-polytropic equation of state and used the variational procedure of maximum entropy principle (MEP) to obtain the closure of the binary hierarchy of 14 moments associated with the Boltzmann equation in which the distribution function depends also on the energy of internal modes.
Abstract: In this paper, we consider a rarefied polyatomic gas with a non-polytropic equation of state. We use the variational procedure of maximum entropy principle (MEP) to obtain the closure of the binary hierarchy of 14 moments associated with the Boltzmann equation in which the distribution function depends also on the energy of internal modes. The closed partial differential system is symmetric hyperbolic and the Cauchy problem is well-posed. In the limiting case of polytropic gas in which the internal energy is a linear function of the temperature, we find, as a special case, the previous results of Pavic et al. (Physica A 392:1302–1317, 2013). This paper, therefore, completes the equivalence between the closure obtained in the phenomenological rational extended thermodynamics theory and the one obtained by the MEP for general non-polytropic gas.

16 citations


Journal ArticleDOI
TL;DR: In this paper, the authors investigated the relaxation to equilibrium of the solution of a class of one-dimensional linear Fokker-Planck type equations that have been recently considered in connection with the study of addiction phenomena in a system of individuals.
Abstract: We investigate the relaxation to equilibrium of the solution of a class of one-dimensional linear Fokker–Planck type equations that have been recently considered in connection with the study of addiction phenomena in a system of individuals. The steady states of these equations belong to the class of generalized Gamma densities. As a by-product of the relaxation analysis, we prove new weighted Poincare and logarithmic Sobolev type inequalities for this class of densities.

15 citations


Journal ArticleDOI
TL;DR: In this article, a ternary autonomous dynamical system of FitzHugh-Rinzel type is analyzed and the fundamental solution H(x, t) is explicitly determined and the initial value problem is analyzed in the whole space.
Abstract: A ternary autonomous dynamical system of FitzHugh–Rinzel type is analyzed. The system, at start, is reduced to a nonlinear integro differential equation. The fundamental solution H(x, t) is explicitly determined and the initial value problem is analyzed in the whole space. The solution is expressed by means of an integral equation involving H(x, t) . Moreover, adding an extra control term, explicit solutions are achieved.

13 citations


Journal ArticleDOI
TL;DR: In this paper, an unconditionally positive finite difference (UPFD) and the standard explicit finite difference schemes are compared to the analytical solution of the advection-diffusion reaction equation which describes the exponential traveling wave in heat and mass transfer processes.
Abstract: An unconditionally-positive finite difference (UPFD) and the standard explicit finite difference schemes are compared to the analytical solution of the advection–diffusion reaction equation which describes the exponential traveling wave in heat and mass transfer processes. It is found that although the unconditional positivity of the UPFD scheme, this scheme is less accurate than the standard explicit finite difference scheme. This is because the UPFD scheme contains additional truncation-error terms in the approximations of the first and second derivatives with respect to x, which are evaluated at different moments in time. While these terms tend to zero as the mesh is refined, the UPFD scheme nevertheless remains less accurate than its standard explicit finite difference counterpart. The presented results are important when modeling a heat and mass transfer processes using the investigated advection–diffusion reaction equation. Furthermore, current and future developers of coupled multi-species transport models may draw on the ideas of solutions methods employed in this study to further develop numerical models for various types of coupled multi-species transport problems.

13 citations


Journal ArticleDOI
TL;DR: In this paper, the past extropy is used to measure the uncertainty in a past lifetime distribution and a characterization result about the extropy of the largest order statistics is given, where the authors show that it is a measure of uncertainty in the past.
Abstract: Recently Qiu [6] have introduced residual extropy as measure of uncertainty in residual lifetime distributions analogues to residual entropy (see, e.g. [3]). Also, they obtained some properties and applications of that. In this paper, we study the extropy to measure the uncertainty in a past lifetime distribution. This measure of uncertainty is called past extropy. Also it is showed a characterization result about the past extropy of largest order statistics.

13 citations


Journal ArticleDOI
TL;DR: In particular, under suitable conditions on the operators tuple, the generalized A-joint numerical radius of a d-tuple of operators is given in this article, where the authors generalize the well-known inequalities due to Kittaneh (Studia Math 168(1):73-80, 2005).
Abstract: Let A be a positive (semidefinite) bounded linear operator on a complex Hilbert space $$\big ({\mathcal {H}}, \langle \cdot , \cdot \rangle \big )$$ . The semi-inner product induced by A is defined by $${\langle x, y\rangle }_A := \langle Ax, y\rangle $$ for all $$x, y\in {\mathcal {H}}$$ and defines a seminorm $${\Vert \cdot \Vert }_A$$ on $${\mathcal {H}}$$ . This makes $${\mathcal {H}}$$ into a semi-Hilbert space. For $$p\in [1,+\infty )$$ , the generalized A-joint numerical radius of a d-tuple of operators $${\mathbf {T}}=(T_1,\ldots ,T_d)$$ is given by $$\begin{aligned} \omega _{A,p}({\mathbf {T}}) =\sup _{\Vert x\Vert _A=1}\left( \sum _{k=1}^d|\big \langle T_kx, x\big \rangle _A|^p\right) ^{\frac{1}{p}}. \end{aligned}$$ Our aim in this paper is to establish several bounds involving $$\omega _{A,p}(\cdot )$$ . In particular, under suitable conditions on the operators tuple $${\mathbf {T}}$$ , we generalize the well-known inequalities due to Kittaneh (Studia Math 168(1):73–80, 2005).

11 citations


Journal ArticleDOI
TL;DR: The aim of this work is to provide a new instrument to describe insect pests’ population density and to consider physiological age, instead of the more widely used chronological age.
Abstract: The mathematical description of poikilothermic organisms’ life cycle, of insects in particular, is a widely discussed argument, above all for its application in decision support systems. The increasing interest among agricultural industries in obtaining products with minor quantities of chemical inputs has led entomologists and model scientists to study in greater depth not only the biology and behaviour of the insects, but also the way to translate these mechanisms into mathematical language. The aim of this work is to provide a new instrument to describe insect pests’ population density. In particular, the study analyses insects’ development through the life stages driven by environmental factors. This has led researchers to consider physiological age, instead of the more widely used chronological age. In addition to mortality and fertility rates, the possibility of improving the simulation by inserting as a boundary condition results from previous field monitoring or the links with diapause models has been considered. The work is validated in the case of the European grapevine moth Lobesia botrana.

11 citations


Journal ArticleDOI
TL;DR: In this article, similar results were obtained for one-dimensional cylindrical shock wave in a self-gravitating, rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field in the presence of conductive and radiative heat fluxes.
Abstract: Similarity solutions are obtained for one-dimensional cylindrical shock wave in a self-gravitating, rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field in the presence of conductive and radiative heat fluxes The total energy of the wave is non-constant It is obtained that the increase in the Cowling number, in the parameters of radiative as well as conductive heat transfer and the parameter of the non-idealness of the gas have a decaying effect on the shock wave however increase in the value of gravitational parameter has reverse effect on the shock strength It is manifested that the presence of azimuthal magnetic field removes the singularities which arise in some cases of the presence of axial magnetic field Also, it is observed that the effect of the parameter of non-idealness of the gas is diminished by increasing the value of the gravitational parameter

10 citations



Journal ArticleDOI
TL;DR: In this article, all central extensions of all 3-dimensional nontrivial complex Zinbiel algebras with 2-dimensional annihilator were described and a full classification of 4-dimensional and 5-dimensional NQA was given.
Abstract: We describe all central extensions of all 3-dimensional nontrivial complex Zinbiel algebras. As a corollary, we have a full classification of 4-dimensional nontrivial complex Zinbiel algebras and a full classification of 5-dimensional nontrivial complex Zinbiel algebras with 2-dimensional annihilator, which gives the principal step in the algebraic classification of 5-dimensional Zinbiel algebras.

9 citations


Journal ArticleDOI
TL;DR: In this article, the authors proposed a method to approximate the Bagley-Torvik equations by using a class of polynomials called the Bessel polynomial, which their coefficients are positive.
Abstract: The paper exhibits a practical and effective scheme to approximate the solutions of a class of fractional differential equations known as the Bagley-Torvik equations. The underlying fractional derivative is based on the Caputo definition. Both the boundary and initial conditions are considered while the domain of approximation is taken sufficiently large. The principle idea behind our approximation algorithm is to use a novel class of (orthogonal) polynomials called the Bessel polynomials, which their coefficients are positive. The convergence and error analysis of the Bessel solution are investigated in the $$L_2$$ and infinity norms. Representing the unknown solution and its derivatives in terms of basis functions together with collocation points, the Bagley-Torvik equations are reduced into an algebraic form. In particular, we present a simple but fast algorithm with linear complexity for computing the q-th order fractional derivative of the basis functions in the vectorized form. Several practical test problems are given to illustrate the utility and efficiency of the proposed approximation algorithm. Validation of the method is obtained by comparison with existing available numerical solutions. Based on experiments, the present approach produces the results of high accuracy to obtain the approximate solutions of Bagley-Torvik equations especially for long-time computations.

Journal ArticleDOI
TL;DR: In this paper, the global existence of solutions to a system of reaction cross diffusion equations appearing in the modeling of multiple sclerosis, in the one-dimensional case, was proved for general initial data, and existence, uniqueness, stability and smoothness were proven when initial data are smooth.
Abstract: We study in this work the global existence of solutions to a system of reaction cross diffusion equations appearing in the modeling of multiple sclerosis, in the one-dimensional case. Weak solutions are obtained for general initial data, and existence, uniqueness, stability and smoothness are proven when initial data are smooth.

Journal ArticleDOI
TL;DR: In this paper, the onset of natural convection in a fluid-saturated anisotropic porous layer, which rotates about the vertical axis, under the hypothesis of local thermal non-equilibrium, is analyzed.
Abstract: The onset of natural convection in a fluid-saturated anisotropic porous layer, which rotates about the vertical axis, under the hypothesis of local thermal non-equilibrium, is analysed. Since the porosity of the medium is assumed to be high, the more suitable Darcy-Brinkman model is adopted. Linear instability analysis of the conduction solution is carried out. Nonlinear stability with respect to $$L^2$$ -norm is performed in order to prove the coincidence between the linear instability and the global nonlinear stability thresholds. The effect of both rotation and thermal and mechanical anisotropies on the critical Rayleigh number for the onset of instability is discussed.

Journal ArticleDOI
TL;DR: In this paper, weakly 1-absorbing prime ideals in commutative rings were introduced and studied, and it was shown that every proper ideal is weakly one absorbing prime.
Abstract: This paper introduce and study weakly 1-absorbing prime ideals in commutative rings. Let $$A\ $$ be a commutative ring with a nonzero identity $$1 e 0.$$ A proper ideal $$P\ $$ of $$A\ $$ is said to be a weakly 1-absorbing prime ideal if for every nonunits $$x,y,z\in A\ $$ with $$0 e xyz\in P,\ $$ then $$xy\in P$$ or $$z\in P.\ $$ In addition to give many properties and characterizations of weakly 1-absorbing prime ideals, we also determine rings in which every proper ideal is weakly 1-absorbing prime. Furthermore, we investigate weakly 1-absorbing prime ideals in C(X), which is the ring of continuous functions of a topological space $$X.\ $$

Journal ArticleDOI
TL;DR: In this article, the Ricci soliton is shown to be Ricci flat and locally isometric with respect to the Euclidean distance of the potential vector field when the manifold satisfies gradient almost.
Abstract: In the present paper, we initiate the study of $$*$$ - $$\eta $$ -Ricci soliton within the framework of Kenmotsu manifolds as a characterization of Einstein metrics. Here we display that a Kenmotsu metric as a $$*$$ - $$\eta $$ -Ricci soliton is Einstein metric if the soliton vector field is contact. Further, we have developed the characterization of the Kenmotsu manifold or the nature of the potential vector field when the manifold satisfies gradient almost $$*$$ - $$\eta $$ -Ricci soliton. Next, we deliberate $$*$$ - $$\eta $$ -Ricci soliton admitting $$(\kappa ,\mu )^\prime $$ -almost Kenmotsu manifold and proved that the manifold is Ricci flat and is locally isometric to $${\mathbb {H}}^{n+1}(-4)\times {\mathbb {R}}^n$$ . Finally we present some examples to decorate the existence of $$*$$ - $$\eta $$ -Ricci soliton, gradient almost $$*$$ - $$\eta $$ -Ricci soliton on Kenmotsu manifold.

Journal ArticleDOI
TL;DR: In this article, a study related to the two-phase analysis of pulsatile blood flow through a narrowed stenosed artery with radiation and the chemical effects is presented, in which the flow of blood is assumed vertical upward and the direction of an external applied magnetic is in the radial direction of the flow.
Abstract: The paper presents a study related to the two-phase analysis of pulsatile blood flow through a narrowed stenosed artery with radiation and the chemical effects. In the model, a vertical artery is considered in which the flow of blood is assumed vertical upward and the direction of an external applied magnetic is in the radial direction of the flow. To understand the behavior of blood flow, graphs of the velocity profile, wall shear stress, flow rate, flow impedance and concentration profile are portrayed with different values of the magnetic and radiation parameters. In order to validate the results, a comparative study is presented between the single-phase and two-phase model of the blood flow, which shows that the two-phase model fits more accurately with the experimental data than the single-phase model, as mean errors are $$0.3\%$$ for the two-phase model while it is $$1\%$$ for single-phase model. For pulsatile flow, the phase difference between the pressure gradient and the flow rate is displayed with the effects of the magnetic field and different heights of the stenosis.

Journal ArticleDOI
TL;DR: In this article, it was shown that a Riemannian manifold equipped with a concurrent-recurrent vector field is of constant negative curvature when its metric is a Ricci soliton.
Abstract: In this paper, we initiate the study of impact of the existence of a unit vector $$ u $$ , called a concurrent-recurrent vector field, on the geometry of a Riemannian manifold. Some examples of these vector fields are provided on Riemannian manifolds, and basic geometric properties of these vector fields are derived. Next, we characterize Ricci solitons on 3-dimensional Riemannian manifolds and gradient Ricci almost solitons on a Riemannian manifold (of dimension n) admitting a concurrent-recurrent vector field. In particular, it is proved that the Riemannian 3-manifold equipped with a concurrent-recurrent vector field is of constant negative curvature $$-\alpha ^2$$ when its metric is a Ricci soliton. Further, it has been shown that a Riemannian manifold admitting a concurrent-recurrent vector field, whose metric is a gradient Ricci almost soliton, is Einstein.

Journal ArticleDOI
TL;DR: In this paper, the Laplace transform is used to analyze and simulate both the situations in which the input function is a Dirac delta generalized function and a box function, restricting ourselves to the Cauchy problem.
Abstract: In physics, phenomena of diffusion and wave propagation have great relevance; these physical processes are governed in the simplest cases by partial differential equations of order 1 and 2 in time, respectively. It is known that whereas the diffusion equation describes a process where the disturbance spreads infinitely fast, the propagation velocity of the disturbance is a constant for the wave equation. By replacing the time derivatives in the above standard equations with pseudo-differential operators interpreted as derivatives of non integer order (nowadays misnamed as of fractional order) we are lead to generalized processes of diffusion that may be interpreted as slow diffusion and interpolating between diffusion and wave propagation. In mathematical physics, we may refer these interpolating processes to as fractional diffusion-wave phenomena. The use of the Laplace transform in the analysis of the Cauchy and Signalling problems leads to special functions of the Wright type. In this work we analyze and simulate both the situations in which the input function is a Dirac delta generalized function and a box function, restricting ourselves to the Cauchy problem. In the first case we get the fundamental solutions (or Green functions) of the problem whereas in the latter case the solutions are obtained by a space convolution of the Green function with the input function. In order to clarify the matter for the non-specialist readers, we briefly recall the basic and essential notions of the fractional calculus (the mathematical theory that regards the integration and differentiation of non-integer order) with a look at the history of this discipline.

Journal ArticleDOI
TL;DR: In this article, the bifurcation analysis of small amplitude dust ion acoustic (DIA) waves in a multi-component dusty plasma is investigated using reductive perturbation technique (RPT).
Abstract: The dynamics of four component dusty plasma with nonthermal electrons and positrons are examined. The bifurcation analysis of small amplitude dust ion acoustic (DIA) waves in this multicomponent plasma is investigated. Employing the reductive perturbation technique (RPT), the two nonlinear equations viz. KdV and mKdV are derived and using the travelling wave transformation, respective planar dynamical systems are deduced. Based on the bifurcation theory of this dynamical system, all possible phase portraits, including nonlinear homoclinic orbit, nonlinear periodic orbit and supernonlinear periodic orbit are presented. The pseudopotential profiles for the nonlinear equations are plotted for different parametric values to illustrate and confirm the phase plane analysis. It is found that the system parameter values affect the bifurcation of the DIA waves. It is also shown that the system supports both nonlinear and supernonlinear DIA periodic waves.

Journal ArticleDOI
TL;DR: In this article, the authors investigated the thermodynamic properties of the three dimensional flow of electrically conducting water hybrid nanofluid flow past a bidirectionally stretchable melting surface and found that the heat transfer and the mass transfer rate of water hybrid flow seem to be higher than that of Al.
Abstract: Leverages of magnetic cross-field, thermal radiation, second order chemical reaction on the unsteady three dimensional flow of electrically conducting Cu–Al $$_{2}$$ O $$_{3}$$ /water hybrid nanofluid flow past a bidirectionally stretchable melting surface are investigated in the present scrutiny. A comparative inspection of Cu–Al $$_{2}$$ O $$_{3}$$ /water hybrid nanofluid and Al $$_{2}$$ O $$_{3}$$ /water nanofluid is achieved. The compatible models for the thermo-physical properties are chosen. The highly coupled nonlinear boundary layer partial differential equations and the associated initial-boundary restrictions are converted into a dimensionless structure employing the appropriate non dimensional variables. These dimension free equations are then fixed by preferring the explicit finite difference scheme. Since the current numerical procedure is conditionally stable, the convergence and stability criteria are established to justify the certainty of the outcomes. The superiority of the sundry parameters on the concentration, temperature and velocity profile in the boundary layer region are evaluated numerically and displayed through the diagrams. The responses of engineering coefficients (Nusselt number Nu, skin friction coefficients $$Cf_X,Cf_Y$$ and the Sherwood number Sh) in consequence of distinct parameters are exhibited through the tables. It is detected that the heat transfer and the mass transfer rate of Cu–Al $$_{2}$$ O $$_{3}$$ /water hybrid flow seem to be higher than that of Al $$_{2}$$ O $$_{3}$$ /water nanofluid flow. Due to the high thermal properties, hybrid nanofluids are chosen in many fields such as space, ships and defence, biomedical, nuclear system cooling and automotive.

Journal ArticleDOI
TL;DR: In this paper, a family of mean past weighted distributions of order $$\alpha $$ is introduced, and the concepts of the mean inactivity time and cumulative $$\α $$-class past entropy are used.
Abstract: In this paper, a family of mean past weighted ($$\hbox {MPW}_{\alpha }$$) distributions of order $$\alpha $$ is introduced. For the construction of this family, the concepts of the mean inactivity time and cumulative $$\alpha $$-class past entropy are used. Distributional properties and stochastic comparisons with other known weighted distributions are given. Furthermore, an upper bound for the k-order moment of the random variables associated with the new family and a characterization result are obtained. Generalized discrete mixtures that involve $$\hbox {MPW}_{\alpha }$$ distributions and other weighted distributions are also explored.

Journal ArticleDOI
TL;DR: In this article, a family of generalized generalized reversed aging intensity functions is introduced and studied, which depend on a real parameter and characterize uniquely the distribution functions of univariate positive absolutely continuous random variables, in the opposite case they characterize families of distributions.
Abstract: The reversed aging intensity function is defined as the ratio of the instantaneous reversed hazard rate to the baseline value of the reversed hazard rate. It analyzes the aging property quantitatively, the higher the reversed aging intensity, the weaker the tendency of aging. In this paper, a family of generalized reversed aging intensity functions is introduced and studied. Those functions depend on a real parameter. If the parameter is positive they characterize uniquely the distribution functions of univariate positive absolutely continuous random variables, in the opposite case they characterize families of distributions. Furthermore, the generalized reversed aging intensity orders are defined and studied. Finally, several numerical examples are given.

Journal ArticleDOI
TL;DR: In this article, an iterative construction for a common solution associated with the pseudomonotone equilibrium problems, fixed point problem of a finite family of Demimetric operators and the generalized split null point problem in Hilbert spaces is presented.
Abstract: This paper provides an iterative construction for a common solution associated with the pseudomonotone equilibrium problems, fixed point problem of a finite family $$\eta $$ -demimetric operators and the generalized split null point problem in Hilbert spaces. The sequence of approximants is a variant of the parallel shrinking extragradient algorithm with the inertial effect converging strongly to the optimal common solution under suitable set of control conditions. The viability of the approximants is demonstrated for various theoretical as well as numerical results. The results presented in this paper improve various existing results in the current literature.

Journal ArticleDOI
TL;DR: In this paper, the stability of laminar flows in a sheet of fluid (open channel) down an incline with constant slope angle was studied. And the authors showed that the basic motion is linearly stable for any Reynolds number.
Abstract: We study the stability of laminar flows in a sheet of fluid (open channel) down an incline with constant slope angle $$\beta $$ . The basic motion is the velocity field $$U(z) \mathbf{i}$$ , where z is the coordinate of the axis orthogonal to the channel, and $$\mathbf{i}$$ is the unit vector in the direction of the flow. U(z) is a parabolic function which vanishes at the bottom of the channel and whose derivative with respect to z vanishes at the top. We study the linear stability, and prove that the basic motion is linearly stable for any Reynolds number. We also study the nonlinear Lyapunov stability by solving the Orr equation for the associated maximum problem. As in Falsaperla et al. (Phys Rev E 100(1):013113, 2019. https://doi.org/10.1103/PhysRevE.100.013113 ) we finally study the nonlinear stability of tilted rolls. This work is a preliminary investigation to model debris flows down an incline (Introduction to the physics of landslides. Lecture notes on the dynamics of mass wasting. Springer, Dordrecht, 2011. https://doi.org/10.1007/978-94-007-1122-8 ).

Journal ArticleDOI
TL;DR: In this article, global optimization-based schemes have been presented to reduce fractional-order (FO) systems in the discrete-delta domain, where the delta transform theory brings about the fusion of the continuous and the discrete domains at high sampling rates.
Abstract: In this paper, global optimization-based schemes have been presented to reduce fractional-order (FO) systems in the discrete-delta domain. The delta transform theory brings about the fusion of the continuous and the discrete domains at high sampling rates. The technique can be considered a generalized approach for approximation of fractional (FO) systems, commensurate or non-commensurate. The fractional-order (FO) system is initially transformed into an integer-order (IO) system using the Oustaloup approximation. The lower-order systems of the comparable integer-order structure have been developed with the help of hybrid firefly algorithms employing applicable constraints. The proposed approach is suitably supported by two numerical problems. It is revealed from the examples that the step and Bode responses of the reduced systems generated from the considered techniques are relatively closer to that of the Oustaloup-approximate higher-order model. The efficacy of the recommended methods is also illustrated by the comparison of several performance indicators with some of the latest techniques published in academia. A handful number of techniques have been employed for comparison. The statistical measures also validate the superiority of the advocated techniques. The percentage improvement is formulated to highlight the efficacy of the suggested methods. The non-parametric tests are also performed to test the significance of the methods proposed methods. The techniques can further be extended to carry out the reduction of the fractional-order multi-input multi-output (MIMO) systems. An attempt to diminish fractional-order systems having non-rational powers will also be taken up in future.

Journal ArticleDOI
TL;DR: In this paper, the role played by magnetoelastic effects on the properties exhibited by magnetic domain walls propagating along the major axis of a thin magnetostrictive nanostrip, coupled mechanically with a thick piezoelectric actuator, is theoretically investigated.
Abstract: The role played by magnetoelastic effects on the properties exhibited by magnetic domain walls propagating along the major axis of a thin magnetostrictive nanostrip, coupled mechanically with a thick piezoelectric actuator, is theoretically investigated. The magnetostrictive layer is assumed to be a linear elastic material belonging to the cubic crystal classes $$\bar{4}$$ 3m, 432 and m $$\bar{3}$$ m and to undergo isochoric magnetostrictive deformations. The analysis is carried out in the framework of the extended Landau–Lifshitz–Gilbert equation, which allows to describe, at the mesoscale, the spatio-temporal evolution of the local magnetization vector driven by magnetic fields and electric currents, in the presence of magnetoelastic and magnetocrystalline anisotropy fields. Through the traveling-wave transformation, the explicit expression of the key features involved in both steady and precessional regimes is provided and a qualitative comparison with data from the literature is also presented.

Journal ArticleDOI
TL;DR: In this paper, two new methods to solve large-scale systems of differential equations, which are based on the Krylov method, have been proposed, and the expression of error report and numerical values to compare the two methods in terms of how long each method takes, and also compare the approaches.
Abstract: In this paper, we propose two new methods to solve large-scale systems of differential equations, which are based on the Krylov method. In the first one, the exact solution with the exponential projection technique of the matrix. In the second, we get a new problem of small size, by dropping the initial problem, and then we solve it in ways, such as the Rosenbrock and the BDF. Some theoretical results are presented such as an accurate expression of the remaining criteria. We give an expression of error report and numerical values to compare the two methods in terms of how long each method takes, and we also compare the approaches.

Journal ArticleDOI
TL;DR: In this paper, a new class of nonexpansive mappings is defined and some fixed point theorems for such newly type are proved in the setting of bounded metric spaces without using neither the compactness nor the so-called uniform convexity.
Abstract: In this paper, a new class of nonexpansive mappings is defined and some fixed point theorems for such newly type are proved in the setting of bounded metric spaces without using neither the compactness nor the so-called uniform convexity. Our theorems generalize and improve many known results in the fixed point theory. Furthermore, we apply the main results to show the existence and uniqueness of a solution for a differential equation.

Journal ArticleDOI
TL;DR: Bilalov and Sadigova as discussed by the authors considered a second order elliptic equation with nonsmooth coefficients in grand-Sobolev classes and proved the corresponding theorems concerning traces, extensions, and compactness of a family of functions.
Abstract: In this paper a second order elliptic equation with nonsmooth coefficients is considered in grand-Sobolev classes $$W_{q)}^{2} \left( \varOmega \right) $$ on a bounded n-dimensional domain $$\varOmega \subset R^{n} $$ with a sufficiently smooth boundary $$\partial \varOmega $$ , generated by the norm of the grand-Lebesgue space $$L_{q)}\left( \varOmega \right) $$ . These spaces are non-separable and therefore the definition of a reasonable solution in them faces certain difficulties. For this purpose, a subspace $$N_{q)}^{2} \left( \varOmega \right) $$ is distinguished in which infinitely differentiable and finite functions are dense. The strict inclusion $$W_{q}^{2} \left( \varOmega \right) \subset N_{q)}^{2} \left( \varOmega \right) $$ holds, where $$W_{q}^{2} \left( \varOmega \right) $$ is the classical Sobolev space. This raises specific questions dictated by the theory of spaces $$W_{q}^{2} \left( \varOmega \right) $$ , for example, the characterization of the space of traces of functions from $$N_{q)}^{1} \left( \varOmega \right) $$ cannot be characterized following the classical case. In this paper, the corresponding theorems concerning traces, extensions, and compactness of a family of functions from $$N_{q)}^{k} \left( \varOmega \right) $$ are proved. These results are applied to obtain a Schauder-type estimate up to the boundary. Schauder-type estimates make it possible to establish the fredholmness of the Dirichlet problem for the considered equation in spaces $$N_{q)}^{2} \left( \varOmega \right) $$ with data from grand-Lebesgue type spaces that are different from Lebesgue spaces. Therefore, the results of this work cannot be directly obtained from the results of the $$L_{p}$$ -theory. This work is a continuation of the research carried out by the authors in articles (Bilalov and Sadigova in Complex Var Elliptic Equ, 2020. https://doi.org/10.1080/17476933.2020.1807965 ; Bilalov and Sadigova in Sahand Commun Math Anal, 2021. https://doi.org/10.22130/scma.2021.521544.893 .

Journal ArticleDOI
TL;DR: In this article, the kinematics of one-dimensional motion have been applied to construct evolution equations for non-planar weak and strong shocks propagating into a non-ideal relaxing gas.
Abstract: In this article, the kinematics of one-dimensional motion have been applied to construct evolution equations for non-planar weak and strong shocks propagating into a non-ideal relaxing gas. The approximate value of exponent of shock velocity, at the instant of shock collapse, obtained from systematic approximation method is compared with those obtained from characteristic rule and Guderley’s scheme. Computation of exponent is carried out for different values of van der Waals excluded volume. Effects of non-ideal and relaxation parameters on the wave evolution, governed by the evolution equations, are analyzed.