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Abstract: Abstract Radiative transfer is at the heart of the mechanism to explain the greenhouse effect based on the partial infrared opacity of carbon dioxide, methane and other greenhouse gases in the atmosphere. In absence of thermal diffusion, the mathematical model consists of a first order integro-differential equation coupled with an integral equation for the light intensity and the temperature, in the atmosphere. We revisit this much studied system from a mathematical and numerical point of view. Existence and uniqueness and implicit solutions of the Milne problem for grey atmospheres are stated. Numerical simulations are given for grey and non-grey atmospheres and applied to calculate the effect of greenhouse gases. In the context of a transparent atmosphere for sunlight, it is found that by doubling the absorption coefficient in the infrared absorption range of \(\texttt {CO}_2\) the temperature decreases by 2%. On the other hand, the same changes but in the low infrared range of the sunlight leads to an increase of temperature in the atmosphere. Several computer codes were written to cross-validate the results. The authors conclude that the radiative transfer model without thermal diffusion for an atmosphere transparent to the incident sunlight is not capable of explaining the greenhouse effect due to the greenhouse gases. A decreasing temperature due to an increasing proportion of \(\texttt {CO}_2\) has been observed in the high atmosphere (D.W.J. Thomson et al, nature11579). In the lower atmosphere thermal diffusion and convection cannot be neglected and since the absorption coefficient are highly dependent on the temperature, a full ocean–atmosphere–biosphere climate model is required. Hence, driving conclusions from this study on climate change should be cautiously avoided and a review of the hypothesis of the radiative transfer argument commonly found in textbooks should be revisited. PubDate: 2022-09-01

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Abstract: Abstract In this paper, in the absence of any constraint qualifications, we develop sequential optimality conditions for a constrained multiobjective fractional programming problem characterizing an approximate properly efficient solution. This is achieved by employing a powerful combination of conjugate analysis and the concept of approximate subdifferential. We give an example to present significance of sequential optimality conditions. PubDate: 2022-08-22

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Abstract: Abstract A nonlinear Black–Scholes-type equation is studied within counterparty risk models. The classical hypothesis on the uniform Lipschitz-continuity of the nonlinear reaction function allows for an equivalent transformation of the semilinear Black–Scholes equation into a standard parabolic problem with a monotone nonlinear reaction function and an inhomogeneous linear diffusion equation. This setting allows us to construct a scheme of monotone, increasing or decreasing, iterations that converge monotonically to the true solution. As typically any numerical solution of this problem uses most computational power for computing an approximate solution to the inhomogeneous linear diffusion equation, we discuss also this question and suggest several solution methods, including those based on Monte Carlo and finite differences/elements. PubDate: 2022-08-09

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Abstract: Abstract This paper deals with the large time behaviour of the solutions of nonlinear Vlasov–Poisson–Fokker–Planck system in presence of an external potential of confinement. The statistics of collisions we are considering here, is the Fermi–Dirac operator with the Pauli exclusion term and without the detailed balance principle. A hypocoercivity method and a notion of scalar product adapted to the presence of a Poisson coupling are used to prove the exponential convergence of the solution of our system. PubDate: 2022-08-08

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Abstract: Abstract In this paper, we consider the following nonlinear Petrovsky equation with variable exponents: $$\begin{aligned} u_{tt}+\Delta ^{2} u+a u_{t} ^{m(.)-2}u_{t}=b u ^{p(.)-2}u, \end{aligned}$$ where a, b are positive constants and the exponents m(x), p(x) are given functions. By using the Faedo-Galerkin method, the existence of a unique weak solution is established under suitable assumptions on the variable exponents m and p. We also prove a finite-time blow-up result for arbitrary negative initial energy. PubDate: 2022-08-03

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Abstract: Abstract Controlled stochastic differential equations driven by time changed Lévy noises do not enjoy the Markov property in general, but can be treated in the framework of general martingales. From the modelling point of view, time changed noises constitute a feasible way to include time dependencies at noise level and still keep a reasonably simple structure. Furthermore, they are easy to simulate, with the result that time change Lévy dynamics attract attention in various fields of application. In this work we survey an approach to stochastic control via maximum principle for time changed Lévy dynamics. We emphasise the role and use of different information flows in tackling the various control problems. We show how these techniques can be extended to include Volterra type dynamics and the control of forward–backward systems of equations. Our techniques make use of the stochastic non-anticipating (NA) derivative in a general martingale framework. PubDate: 2022-06-29 DOI: 10.1007/s40324-022-00301-5

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Abstract: Abstract We prove in this paper the Lax–Wendroff consistency of a general finite volume convection operator acting on discrete functions which are possibly not piecewise-constant over the cells of the mesh and over the time steps. It yields an extension of the Lax–Wendroff theorem for general colocated or non-colocated schemes. This result is obtained for general polygonal or polyhedral meshes, under assumptions which, for usual practical cases, essentially boil down to a flux-consistency constraint; this latter is, up to our knowledge, novel and compares the discrete flux at a face to the mean value over the adjacent cell of the continuous flux function applied to the discrete unknown function. We first briefly show how this result copes with multipoint colocated schemes on general meshes. We then apply it to prove the consistency of a finite volume discretisation of a convection operator featuring a (convected) scalar variable and a (convecting) velocity field, with a staggered approximation, i.e. with a cell-centred approximation of the scalar variable and a face-centred approximation of the velocity. PubDate: 2022-06-01 DOI: 10.1007/s40324-021-00263-0

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Abstract: Abstract We give a survey on some recent results concerning the Landau–Lifshitz equation, a fundamental nonlinear PDE with a strong geometric content, describing the dynamics of the magnetization in ferromagnetic materials. We revisit the Cauchy problem for the anisotropic LL equation, without dissipation, for smooth solutions, and also in the energy space in dimension one. We also examine two approximations of the LL equation given by of the Sine–Gordon equation and cubic Schrödinger equations, arising in certain singular limits of strong easy-plane and easy-axis anisotropy, respectively. Concerning localized solutions, we review the orbital and asymptotic stability problems for a sum of solitons in dimension one, exploiting the variational nature of the solitons in the hydrodynamical frameworkFinally, we survey results concerning the existence, uniqueness and stability of self-similar solutions (expanders and shrinkers) for the isotropic LL equation with Gilbert term. Since expanders are associated with a singular initial condition with a jump discontinuity, we also review their well-posedness in spaces linked to the BMO space. PubDate: 2022-06-01 DOI: 10.1007/s40324-021-00254-1

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Abstract: Abstract The aim of this paper is to introduce a new class of mixed contractions which allow to revise and generalize some results obtained in [6] by R. Gubran, W. M. Alfaqih and M. Imdad. We also provide an example corresponding to this class of mappings and show how the new fixed point result relates to the above-mentioned result in [6]. Further, we present an application to the solvability of a two-point boundary value problem for second order differential equations. PubDate: 2022-06-01 DOI: 10.1007/s40324-021-00261-2

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Abstract: Abstract Obtaining convergence domain is an important task in the study of iterative schemes. Analysis of local convergence of an iterative procedure provides essential information about its convergence domain around a solution. In this manuscript, we study the local analysis of the uni-parametric Kou’s class of iterative algorithms for addressing nonlinear equations. This approach expands the utility of the methods by preventing the use of Taylor expansion in convergence analysis. In the view of extending the applicability of these methods, the convergence analysis is shown using Lipschitz condition on the first derivative. Our study provides radii of convergence balls and the uniqueness of the solution along with the calculable error distances. The complex dynamical analysis of the family is also presented. Numerical examples are solved to show that our theoretical conclusions work well in the situation where the earlier analysis cannot be implemented. PubDate: 2022-06-01 DOI: 10.1007/s40324-021-00257-y

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Abstract: Abstract In this article, a new conformable fractional anisotropic diffusion model for image denoising is presented, which contains the spatial derivative along with the time-fractional derivative. This model is a generalization of the diffusion model (Welk et al. in Scale space. Springer, Berlin, pp 585–597, 2005) with forward–backward diffusivities. The proposed model is very efficient for noise removal of the noisy images in comparison to the classical anisotropic diffusion model. The numerical experiments are performed using an explicit scheme for different-different values of fractional order derivative \(\alpha \) . The experimental results are obtained in terms of peak signal to noise ratio (PSNR) as a metric. PubDate: 2022-06-01 DOI: 10.1007/s40324-021-00255-0

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Abstract: Abstract We study some qualitative properties of a misaligned journal bearing. The device consists of two cylinders closely spaced: an inner rotating cylinder (the shaft) whose symmetry axis is not parallel to the one of the outer cylinder (the bearing). We consider the load capacity of the system, defined as the force exerted by the pressure. It is given by the integral of the pressure times the normal vector to the bearing surface. We obtain finite load capacity, even in the limit case when a point contact occurs. It was also verified by numerical simulations. We used an adapted Preconditioned Conjugate Gradient Method for solving the direct problem, preserving the A-orthogonality property of the search directions, even after a restarting process. The solution of the related inverse problem is based on an interior, trust-region algorithm. To validate the numerical proposal, the predicted pressure values at the bearing mid-plane, are compared to published experimental data. PubDate: 2022-06-01 DOI: 10.1007/s40324-021-00253-2

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Abstract: Abstract A large class of physical phenomena in biophysics, chemical engineering, and physical sciences are modeled as systems of Fredhold integro-differential equations. In its simplest form, such systems are linear and analytic solutions might be obtained in some cases while numerical methods can be also used to solve such systems when analytic solutions are not possible. For more realistic and accurate study of underlying physical behavior, including nonlinear actions is useful. In this paper, we use the Chebyshev pseudo-spectral method to solve the pattern nonlinear second order systems of Fredholm integro-differential equations. The method reduces the operators to a nonlinear system of equations that can be solved alliteratively. The method is tested against the reproducing kernel Hilbert space (RKHS) method and shows good performance. The present method is easy to implement and yields very good accuracy for using a relatively small number of collocation points. PubDate: 2022-06-01 DOI: 10.1007/s40324-021-00258-x

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Abstract: Abstract Since the start of the still ongoing COVID-19 pandemic, there have been many modeling efforts to assess several issues of importance to public health. In this work, we review the theory behind some important mathematical models that have been used to answer questions raised by the development of the pandemic. We start revisiting the basic properties of simple Kermack-McKendrick type models. Then, we discuss extensions of such models and important epidemiological quantities applied to investigate the role of heterogeneity in disease transmission e.g. mixing functions and superspreading events, the impact of non-pharmaceutical interventions in the control of the pandemic, vaccine deployment, herd-immunity, viral evolution and the possibility of vaccine escape. From the perspective of mathematical epidemiology, we highlight the important properties, findings, and, of course, deficiencies, that all these models have. PubDate: 2022-06-01 DOI: 10.1007/s40324-021-00260-3

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Abstract: Abstract This research article is dedicated to solving multiple roots of real life problems. In the literature, there are numerous higher-order multiple root algorithms with derivative. But, derivative-free algorithms for multiple roots, on the other hand, are extremely rare. As a result of this, we describe a family of third-order derivative-free algorithms for calculating multiple roots that only require three function evaluations each iteration. The application of new algorithms is validated on Shokley diode and electric circuit problem, Isothermal continuous stirred tank reactor problem, Van der Waals problem and Planck law radiation problem. The presented iterative algorithms are good rivals to the existing algorithms, according to numerical results. PubDate: 2022-05-31 DOI: 10.1007/s40324-022-00300-6

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Abstract: Abstract The essentially non-oscillatory (ENO) procedure and its variant, the ENO-SR procedure, are very efficient algorithms for interpolating (reconstructing) rough functions. We prove that the ENO (and ENO-SR) procedure are equivalent to deep ReLU neural networks. This demonstrates the ability of deep ReLU neural networks to approximate rough functions to high-order of accuracy. Numerical tests for the resulting trained neural networks show excellent performance for interpolating functions, approximating solutions of nonlinear conservation laws and at data compression. PubDate: 2022-05-23 DOI: 10.1007/s40324-022-00299-w

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Abstract: Abstract Leukaemia accounts for around 3% of all cancer types diagnosed in adults, and is the most common type of cancer in children of paediatric age (typically ranging from 0 to 14 years). There is increasing interest in the use of mathematical models in oncology to draw inferences and make predictions, providing a complementary picture to experimental biomedical models. In this paper we recapitulate the state of the art of mathematical modelling of leukaemia growth dynamics, in time and response to treatment. We intend to describe the mathematical methodologies, the biological aspects taken into account in the modelling, and the conclusions of each study. This review is intended to provide researchers in the field with solid background material, in order to achieve further breakthroughs in the promising field of mathematical biology. PubDate: 2022-05-03 DOI: 10.1007/s40324-022-00296-z