A Novel Technique to Control the Accuracy of a Nonlinear Fractional Order Model of COVID-19: Application of the CESTAC Method and the CADNA Library

In this paper, a nonlinear fractional order model of COVID-19 is approximated. For this aim, at first we apply the Caputo–Fabrizio fractional derivative to model the usual form of the phenomenon. In order to show the existence of a solution, the Banach fixed point theorem and the Picard–Lindelof approach are used. Additionally, the stability analysis is discussed using the fixed point theorem. The model is approximated based on Indian data and using the homotopy analysis transform method (HATM), which is among the most famous, flexible and applicable semi-analytical methods. After that, the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method and the CADNA (Control of Accuracy and Debugging for Numerical Applications) library, which are based on discrete stochastic arithmetic (DSA), are applied to validate the numerical results of the HATM. Additionally, the stopping condition in the numerical algorithm is based on two successive approximations and the main theorem of the CESTAC method can aid us analytically to apply the new terminations criterion instead of the usual absolute error that we use in the floating-point arithmetic (FPA). Finding the optimal approximations and the optimal iteration of the HATM to solve the nonlinear fractional order model of COVID-19 are the main novelties of this studyThe work of J.J.N. has been partially supported by the Xunta de Galicia under grant ED431C 2019/02, as well as by Instituto de Salud Carlos III and the Ministerio de Ciencia e Innovación of Spain, research grant COV20/00617. The work of S. Noeiaghdam has been supported by a grant from the Academic Council in the direction of the scientific school of Irkutsk National Research Technical University No. 14-NSH-RAN-2020S

On some differential equations arising in a time domain inverse scattering problem for a dissipative wave equation

The problem of identification of one spatially varying material property, defined within a slab, from boundary measurements is examined. This inverse problem is described by a functional differential equation. Uniqueness and existence of the solution of this inverse problem and the associated direct problem is proven. Of major importance in any inverse problem are the properties of the operator mapping the boundary measurements to the material property. It is shown that this operator is continuous and differentiable

The Construction and Smoothness of Invariant Manifolds by the Deformation Method

This paper proves optimal results for the invariant manifold theorems near a fixed point for a mapping (or a differential equation) by using the deformation, or Lie transform, method from singularity theory. The method was inspired by the difficulties encountered by the implicit function theorem technique in the case of the center manifold. The idea here is simply to deform the given system into its linearization and to track this deformation using the flow of a time-dependent vector field. Corresponding to the difficulties with the center manifold encountered by other techniques, we run into a "derivative loss" in this case as well, which is overcome by utilizing estimates on the differentiated equation. A survey of the other methods used in the literature is also presented

Existence and non-uniqueness of similarity solutions of a boundary layer problem

A Blasius boundary value problem with inhomogeneous lower boundary conditions f(0) = 0 and f'(0) = - lambda with lambda strictly positive was considered. The Crocco variable formulation of this problem has a key term which changes sign in the interval of interest. It is shown that solutions of the boundary value problem do not exist for values of lambda larger than a positive critical value lambda. The existence of solutions is proven for 0 lambda lambda by considering an equivalent initial value problem. It is found however that for 0 lambda lambda, solutions of the boundary value problem are nonunique. Physically, this nonuniqueness is related to multiple values of the skin friction

Stability and efficiency of waveform relaxation methods

AbstractWe investigate the behaviour of Waveform Relaxation methods (WR) for some model problems. First, it is shown how convergence (of the iteration) is related to stability of some one-step integration schemes. Then, we investigate the computational complexity of a 1-D and 2-D heat equation when WR is used in combination with nested iteration and assess its efficiency, in particular, compared to straightforward methods based on Gaussian elimination. Finally, we present some results, showing the performance of WR

Initial Value Problem For White Noise Operators And Quantum Stochastic Processes

This is the proceedings of the 2nd Japanese-German Symposium on Infinite Dimensional Harmonic Analysis held from September 20th to September 24th 1999 at the Department of Mathematics of Kyoto University.この論文集は, 1999年9月20日から9月24日の日程で京都大学理学研究科数学教室において開催された第2回日独セミナー「無限次元調和解祈」の成果をもとに編集されたものである.編集 : ハーバート・ハイヤー, 平井 武, 尾畑 信明Editors: Herbert Heyer, Takeshi Hirai, Nobuaki Obata #e