To solving the fractionally loaded heat equation

In this paper we consider a boundary value problem for a fractionally loaded heat equation in the class of continuous functions. Research methods are based on an approach to the study of boundary value problems, based on their reduction to integral equations. The problem is reduced to a Volterra integral equation of the second kind by inverting the differential part. We also carried out a study the limit cases for the fractional derivative order of the term with a load in the heat equation of the boundary value problem. It is shown that the existence and uniqueness of solutions to the integral equation depends on the order of the fractional derivative in the loaded term.

THE PARAMETER IDENTIFICATION PROBLEM FOR SYSTEM OF DIFFERENTIAL EQUATIONS

In this paper, the parameter identification problem for system of ordinary differential equations is considered. The parameter identification problem for system of ordinary differential equations is investigated by the Dzhumabaev’s parametrization method. At first, conditions for a unique solvability of the parameter identification problem for system of ordinary differential equations are obtained in the term of fundamental matrix of system’s differential part. Further, we establish conditions for a unique solvability of the parameter identification problem for system of ordinary differential equations in the terms of initial data. Algorithm for finding of approximate solution to a unique solvability of the parameter identification problem for system of ordinary differential equations is proposed and the conditions for its convergence are setted. Results this paper can be use for investigating of various problems with parameter and control problems for system of ordinary differential equations. The approach in this paper can be apply to the parameter identification problems for partial differential equations.

Unknown Input Observer Design for a Class of Linear Descriptor Systems

This paper addresses the problem of unknown inputs observer (UIO) design for a class of linear descriptor systems. The unknown inputs affect both state and output of the system. The basic idea of the proposed approach is based on the separation between dynamic and static relations in the descriptor model. Firstly, the method used to separate the differential part from the algebraic part is developed. Secondly, an observer design permitting the simultaneous estimation of the system state and the unknown inputs is proposed. The developed approach for the observer design is based on the synthesis of an augmented model which regroups the differential variables and unknown inputs. The exponential stability of the estimation error is studied using the Lyapunov theory and the stability condition is given in term of linear matrix inequality (LMI). Finally, to illustrate the efficiency of the proposed methodology, a heat exchanger pilot model is considered.

A Regularization Method for Nonlinear Integro-differential Equations with a Zero Operator of the Differential Part and with Several Rapidly Varying Kernels

A nonlinear integro-differential equation with a zero operator of the differential part and several rapidly changing kernels is considered. The work is a continuation of the research conducted earlier for a single rapidly changing core. The main ideas of this generalization and the subtleties that arise in the development of the corresponding algorithm for the regularization method are fully visible in the case of two rapidly changing kernels, so for the sake of reducing the calculations, this particular case is taken. A similar problem with a single spectral value of the kernel of an integral operator is analyzed in one of the authors ' papers. In this case, the singularities in the solution of the problem are described only by the spectral value of the kernel. However, the influence of the zero operator of the differential part affects the fact that in the first approximation, the asymptotics of the solution of the problem under consideration will not contain the functions of the boundary layer, and the limit operator itself will become degenerate (but not zero). The conditions for the solvability of the corresponding iterative problems, as in the linear case, will not be in the form of differential equations (as was the case in problems with a non-zero operator of the differential part), but integro-differential equations, and the formation of these equations is significantly influenced by nonlinearity. Note that, in contrast to the linear case, there is no inhomogeneity of the corresponding linear problem in the right part of the problem under study. As it was shown earlier, its presence in the problem would lead to the appearance in the asymptotic solution of terms with negative powers of a small parameter, and in the nonlinear case there would be innumerable such powers, and the corresponding formal asymptotic solution would have the form of a Laurent series. This would make the creation of an algorithm for asymptotic solutions problematic, so in this paper, wanting to remain within the framework of asymptotic solutions of the type of Taylor series, inhomogeneity is excluded. In addition, in the nonlinear case, so-called resonances may occur, which significantly complicate the development of the corresponding algorithm for the regularization method. This publication deals with the non-resonant case. It is assumed that the study of an alternative variant (a more complex resonant problem) will be carried out in the future.

The utility of a three part differential hematology analyzer with CRP for assessing malaria

BackgroundHematology analyzers display abnormal hematology parameters during malaria infection providing insightful information for suspecting and assessing malaria infection. The goal of this study is to demonstrate the potential of a three part differential hematology analyzer to assess malaria, provide information about the parasitemia, and discuss the importance of combining CRP with hematology parameters to obtain further information about the malaria infection.MethodsThe present study shows the results of a retrospective study involving comparison of raw instrument data from a three part differential hematology analyzer with CRP measurement and corresponding microscopic findings of samples obtained during the monsoon season of years 2018 and 2019 in Mumbai, India. Mann-Whitney U and Krustal-Wallis tests were applied for obtaining statistical significance of hematology parameters and CRP among P. vivax, P. falciparum, dengue and negative samples. ResultsThe study considers 1008 non-malaria febrile cases, 209 P. vivax, 31 P. falciparum positive malaria samples, five cases of mixed species malaria infection, and three co-infection of malaria and dengue. Median values of WBCs, PLTs, PCT and %LYM were lower in malaria and dengue samples compared to negative ones with statistical difference (p < 0.05). Contrary, medians of MCHC, MPV, PDW, and %MON were higher than negative samples. The greatest difference in the analyzer results of malaria and dengue cases was observed in between their medians of CRP levels, which is evidenced by the values of the first quartiles of P. vivax and P. falciparum cases, and the third quartile of dengue cases being 8.6 and 10.4 mg/L, respectively.The parameters with a statistical difference for different levels of parasitemia were WBC, PLT, %MON and CRP. An interfering abnormal peak was observed in the white blood cell histogram, below 37 fl, in malaria infected samples, especially in P. vivax cases, where the height of that peak showed a strong correlation with red blood cells infected predominantly with larger parasitic forms.ConclusionsA three differential part hematology analyzer has the potential to not only trigger malaria diagnosis confirmation but also assess the severity of the infection when CRP is considered.

The Utility of a Three Part Differential Hematology Analyzer with CRP for Assessing Malaria

BackgroundHematology analyzers display abnormal hematology parameters during malaria infection providing insightful information for suspecting and assessing malaria infection. The goal of this study is to demonstrate the potential of a three part differential hematology analyzer to assess malaria, provide information about the parasitemia, and discuss the importance of combining CRP with hematology parameters to obtain further information about the malaria infection.MethodsThe present study shows the results of a retrospective study involving comparison of raw instrument data from a three part differential hematology analyzer with CRP measurement and corresponding microscopic findings of samples obtained during the monsoon season of years 2018 and 2019 in Mumbai, India. Mann-Whitney U and Krustal-Wallis tests were applied for obtaining statistical significance of hematology parameters and CRP among P. vivax, P. falciparum, dengue and negative samples. ResultsThe study considers 1008 non-malaria febrile cases, 209 P. vivax, 31 P. falciparum positive malaria samples, five cases of mixed species malaria infection, and three co-infection of malaria and dengue. Median values of WBCs, PLTs, PCT and %LYM were lower in malaria and dengue samples compared to negative ones with statistical difference (p < 0.05). Contrary, medians of MCHC, MPV, PDW, and %MON were higher than negative samples. The greatest difference in the analyzer results of malaria and dengue cases was observed in between their medians of CRP levels, which is evidenced by the values of the first quartiles of P. vivax and P. falciparum cases, and the third quartile of dengue cases being 8.6 and 10.4 mg/L, respectively.The parameters with a statistical difference for different levels of parasitemia were WBC, PLT, %MON and CRP. An interfering abnormal peak was observed in the white blood cell histogram, below 37 fl, in malaria infected samples, especially in P. vivax cases, where the height of that peak showed a strong correlation with red blood cells infected predominantly with larger parasitic forms.ConclusionsA three differential part hematology analyzer has the potential to not only trigger malaria diagnosis confirmation but also assess the severity of the infection when CRP is considered.

Finite element approximation of an obstacle problem for a class of integro–differential operators

We study the regularity of the solution to an obstacle problem for a class of integro–differential operators. The differential part is a second order elliptic operator, whereas the nonlocal part is given by the integral fractional Laplacian. The obtained smoothness is then used to design and analyze a finite element scheme.

On Solvability in the Class of Distributions of Degenerate Integro-Differential Equations in Banach Spaces

The paper considers a new approach to constructing generalized solutions of degenerate integro-differential equations convolution type in Banach spaces. The principal idea of the method proposed implies the refusal of the condition of existence of the full Jordan set for the Fredholm operator of the higher derivative with respect to the operator bundle formed by the rest of operator coefficients of the differential part and by the operator kernel of the integral component of the equation. The conditions are superimposed upon the values of the operator function specially constructed on the basis elements of the Fredholm operator kernel. Under such an approach, the differential part of the equation may include not only the higher derivative but also any combination lower derivatives, what allows one to consider the convolution integral-differential equations from universal positions, without any special account of the structure of the structure of the operator bundle. The method proposed represents a form of generalization of the technique based on the application of Jordan sets of Fredholm operators, and in the case of existence of the latter the method coincides with this technique. The generalized solutions are constructed in the form of a convolution of the fundamental operator function, which corresponds to the equation under investigation, and a function, which includes the right-hand side of the equation and the initial data. The conditions, under which such a generalized solution does not contain any singular component, and the regular component converts the initial equation into an identity and satisfies the initial data, and the result will provide for the resolvability of the initial problem in the class of functions of characterized by the respective smoothness. In this case, the generalized solution constructed will be classical. The theorem on the form of fundamental operator function has been proved, the abstract results have been illustrated via examples of initialboundary value problems of applied character from the theory electromagnetic fields, the theory of oscillations in visco-elastic media, the theory of vibrations of thermal-elastic plates.