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We study the valuation of callable barrier reverse convertible contracts written on one or two underlying asset prices. We assume the issuer of the contract can call early redemption at any time during a pre-specified time interval. We identify the optimal redemption policy and show, in the single underlying asset case, it is characterized by a time-dependent boundary. The boundary satisfies a nonlinear integral equation of the Volterra type. When there are two underlying assets, the boundary is a surface depending on one price in addition to time. Valuation formulas and associated integral equations are derived. Numerical experiments are performed. [ABSTRACT FROM AUTHOR]

Two hyperbolic equations with a nonlinear coefficient multiplying the highest derivative are considered. The coefficient determines the velocity of nonlinear waves and characterizes the scattering properties of the medium. For stationary traveling-wave solutions, inverse problems are set up consisting of determining a nonlinear coefficient from the dependence of the period on the amplitude of the stationary oscillations. Nonlinear integral functional equations of the inverse problems are obtained and studied, and sufficient conditions for the existence and uniqueness of solutions to the inverse problems are steady-state. Evolution-type algorithms for solving functional equations are proposed. Solutions of test inverse problems are presented. [ABSTRACT FROM AUTHOR]

Process of enhancing testing-capacity regarding COVID-19 is a topic of interest. This task of enhancing is constrained by socio-economic background of a country either in favorable or unfavorable ways. In this paper, we investigate timing of enhancing testing-capacity as an optimal problem, where the enhancement is quantified via number of tests as an instant measure and recovered portion as a long-term measure. The proposed work is structured analogous to an optimal machine replacement model based on a non-linear integral equation. Overall model is partially identifiable and compatible parameter estimations are carried out for a specific case study covering an early stage scenario. In addition, scenario development criteria on demand and effort for enhancing testing-capacity are introduced for predictions. In one numerical experiment, it is observed that frequency of enhancing testing-capacity starts decreasing after two increments indicating a favorable direction amidst effort constraints. • Enhancing testing-capacity of COVID-19 is a process that needs monitoring. • Timing for enhanced testing can be modeled as an optimal replacement model. • Parameter estimation under different scenarios paves the way for process planning. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of algebras satisfying certain conditions which has been named power algebras and we prove some fixed point theorems for a new class of contractions. We give new conditions for the existence and uniqueness of fixed points for this new class of mappings which are not a contraction in the usual algebra. Also, we introduce a new iterative algorithm for fixed-point problems in a Banach algebra. This results improve and extend the corresponding conclusions of the Ishikawa algorithm under weaker conditions and lead to stronger results. In conclusion, we give an application to nonlinear integral equations. [ABSTRACT FROM AUTHOR]

For the first time, we develop a convergent numerical method for the llinear integral equation derived by M.M. Lavrent'ev in 1964 with the goal to solve a coefficient inverse problem for a wave-like equation in 3D. The data are non overdetermined. Convergence analysis is presented along with the numerical results. An intriguing feature of the Lavrent'ev equation is that, without any linearization, it reduces a highly nonlinear coefficient inverse problem to a linear integral equation of the first kind. Nevertheless, numerical results for that equation, which use the data generated for that coefficient inverse problem, show a good reconstruction accuracy. This is similar with the classical Gelfand-Levitan equation derived in 1951, which is valid in the 1D case. [ABSTRACT FROM AUTHOR]

In this paper, we have established and proved fixed point theorems for the Boyd-Wong-type contraction in metric spaces. In particular, we have generalized the existing results for a pair of mappings that possess a fixed point but not continuous at the fixed point. We can apply this result for both continuous and discontinuous mappings. We have concluded our results by providing an illustrative example for each case and an application to the existence and uniqueness of a solution of nonlinear Volterra integral equations. [ABSTRACT FROM AUTHOR]

In this article, we provide the existence result for functional integral equations by using Petryshyn's fixed point theorem connecting the measure of noncompactness in a Banach space. The results enlarge the corresponding results of several authors. We present fascinating examples of equations. [ABSTRACT FROM AUTHOR]

The convergence of normal S-iterative method to solution of a nonlinear Fredholm integral equation with modified argument is established. The corresponding data dependence result has also been proved. An example in support of the established results is included in our analysis. [ABSTRACT FROM AUTHOR]

In this paper, a new analysis for the existence, uniqueness, and regularity of solutions to a time-dependent Kohn–Sham equation is presented. The Kohn–Sham equation is a nonlinear integral Schrödinger equation that is of great importance in many applications in physics and computational chemistry. To deal with the time-dependent, nonlinear and non-local potentials of the Kohn–Sham equation, the analysis presented in this manuscript makes use of energy estimates, fixed-point arguments, regularization techniques, and direct estimates of the non-local potential terms. The assumptions considered for the time-dependent and nonlinear potentials make the obtained theoretical results suitable to be used also in an optimal control framework. [ABSTRACT FROM AUTHOR]

In this paper, we proceed with the analysis started in the work of Braga et al. [J. Math. Phys. 60(1), 013507, 2019] by the same authors, and using the renormalization group method, we obtain logarithmic corrections to the decay of solutions for a class of nonlinear integral equations whenever the nonlinearities are classified as marginal in the renormalization group sense. [ABSTRACT FROM AUTHOR]

This research aims to investigate a novel coincidence point (cp) of generalized multivalued contraction (gmc) mapping involved a directed graph in b-metric spaces (b-ms). An example and some corollaries are derived to strengthen our main theoretical results. We end the manuscript with two important applications, one of them is interested in finding a solution to the system of nonlinear integral equations (nie) and the other one relies on the existence of a solution to fractional integral equations (fie). [ABSTRACT FROM AUTHOR]

In this paper, we present some new fixed point theorems in a locally convex space X with the so called t-Krein-Šmulian property considering the concept of Fit-measures of noncompactness, where t is a weaker Hausdorff locally convex topology of X. Further, we apply our results to discuss the existence of solutions for a nonlinear functional integral equation in the Lebesgue space L1. [ABSTRACT FROM AUTHOR]

In this paper, we study a nonlinear integral equation that arises in a model of spatial logistic dynamics. The solvability of this equation is investigated by introducing special spaces of functions that are integrable up to a constant. Sufficient conditions for the biological characteristics and the parameters of the third spatial moment closure are established that guarantee the existence of the solution of the equation described above in some ball centered at zero. In addition, it is shown that this solution is unique in the considered ball and not zero. This means that, under appropriate conditions, the equilibrium state of the population of a certain species exists and does not coincide with the state of extinction. [ABSTRACT FROM AUTHOR]

A new nonlinear integral equation (NLIE) describing the thermodynamics of the Heisenberg spin chain is derived based on the t − W relation of the quantum transfer matrices. The free energy of the system in a magnetic field is thus obtained by solving the NLIE. This method can be generalized to other lattice quantum integrable models. Taking the SU(3)-invariant quantum spin chain as an example, we construct the corre- sponding NLIEs and compute the free energy. The present results coincide exactly with those obtained via other methods previously. [ABSTRACT FROM AUTHOR]

Propagation of leaky TE-polarized electromagnetic waves in the Goubau line (a perfectly conducting cylinder covered by a concentric dielectric layer) filled with nonlinear metamaterial medium is studied. The problem is reduced to the analysis of a nonlinear integral equation with a kernel in the form of the Green function of an auxiliary boundary value problem on an interval. The existence of propagating nonlinear leaky TE waves for the chosen nonlinearity (Kerr law) is proved using the method of contraction. For the numerical solution, a method based on solving an auxiliary Cauchy problem (a version of the shooting method) is proposed. New propagation regimes are discovered. [ABSTRACT FROM AUTHOR]

Delay-differential-algebraic equations have been widely used to model some important phenomena in science and engineering. Since, in general, such equations do not admit a closed-form solution, it is necessary to solve them numerically by introducing suitable integrators. The present paper extends the class of block boundary value methods (BBVMs) to approximate the solutions of nonlinear delay-differential equations with algebraic constraint and piecewise continuous arguments. Under the classical Lipschitz conditions, convergence and stability criteria of the extended BBVMs are derived. Moreover, a couple of numerical examples are provided to illustrate computational effectiveness and accuracy of the methods. [ABSTRACT FROM AUTHOR]

Initial attempts to establish convergence HT ht and HT ht in a suitable strong sense were unsuccessful. The system HT ht represents the scaled long-time behavior of the self-exciting buffer pushed toward its stability threshold along a heavy-traffic path, as HT ht . [Extracted from the article]

The objective of this article is to introduce function weighted L-R-complete dislocated quasi-metric spaces and to present fixed point results fulfilling generalized rational type F-contraction for a multivalued mapping in these spaces. A suitable example confirms our results. We also present an application for a generalized class of nonlinear integral equations. Our results generalize and extend the results of Karapınar et al. (IEEE Access 7:89026–89032, 2019). [ABSTRACT FROM AUTHOR]

We investigate qualitative properties of solutions of special classes of convolution type nonlinear integral equations on the whole line. We study the asymptotic properties, continuity, and monotonicity of arbitrary nontrivial bounded solutions. Depending on the properties of the kernel of the equation, we find out whether there exist nontrivial bounded solutions with a finite limit at ±∞. Based on the obtained results, we establish uniqueness theorems for large classes of bounded functions. The results obtained are illustrated by examples from applications. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to study the existence and asymptotic behavior of the solutions of some functional integral equations which contain a number of classical nonlinear integral equations as special cases. The main tools used in our considerations are the concept of a measure of noncompactness and the classical Schauder fixed point principle. The investigations of the paper are placed in the space of continuous and tempered functions on the real half-line. The applicability of the results is illustrated by examples. [ABSTRACT FROM AUTHOR]

We study a class of nonlinear integral equations with a noncompact Hammerstein– Nemytskii operator on the entire line. Some special cases of such equations have specific applications in various fields of natural science. The combination of a method for constructing invariant cone segments for the corresponding nonlinear monotone operator with methods of the theory of functions of a real variable allows one to prove a constructive theorem on the existence of bounded positive solutions of equations of the class under consideration. The asymptotic behavior of the solution at is studied as well. In particular, we prove that the solution constructed in the paper is an integrable function on the negative half-line and that the difference between the limit at and the solution is integrable on the positive half-line. In one special case, we show that our solution generates a one-parameter family of bounded positive solutions. At the end of the paper, we give specific applied examples of nonlinearities to illustrate the results. [ABSTRACT FROM AUTHOR]

In this paper, we study the nonlinear Volterra integral equation satisfied by the early exercise boundary of the American put option in the one-dimensional diffusion model for a stock price with constant interest rate and constant dividend yield and with a local volatility depending on the current time t and the current stock price S. In the classical Black–Sholes model for a stock price, Theorem 4.3 of [S. D. Jacka, Optimal stopping and the American put, Math. Finance 1 1991, 2, 1–14] states that if the family of integral equations (parametrized by the variable S) holds for all S ≤ b(t) with a candidate function b(t), then this b(t) must coincide with the American put early exercise boundary c(t). We generalize Peskir's result [G. Peskir, On the American option problem, Math. Finance 15 2005, 1, 169–181] to state that if the candidate function b(t) satisfies one particular integral equation (which corresponds to the upper limit S = b(t), then all other integral equations (corresponding to S, S ≤ b(t) will be automatically satisfied by the same function b(t). [ABSTRACT FROM AUTHOR]

The aim of this work is to establish results in fixed point theory for a pair of fuzzy dominated mappings which forms a rational fuzzy dominated V-contraction in modular-like metric spaces. Some results via a partial order and using the graph concept are also developed. We apply our results to ensure the existence of a solution of nonlinear Volterra-type integral equations. [ABSTRACT FROM AUTHOR]

We examine the unique generalized solvability of the mixed problem for a higher-order nonlinear pseudoparabolic equation with two parameters in mixed derivatives. Using the Fourier variable separation method, we reduce the problem to a countable system of nonlinear integral equations whose unique solvability can be proved by the method of successive approximations. We prove the continuous dependence of a generalized solution to the mixed problem on the initial functions and the positive parameters. [ABSTRACT FROM AUTHOR]

Recently, there appeared new stability results in the inverse Sturm–Liouville theory that, unlike the usual local stability, involve uniform estimates. Besides more advanced justification of numerical simulations, the uniform stability can reveal the nature of an inverse problem to a significantly greater extent. For example, it allows obtaining global solvability of an inverse problem as a corollary of its solvability for any dense subclass of input data. In this paper, we obtain uniform stability of an inverse problem for one important class of integro-differential operators, which is a natural generalization of the classical Sturm–Liouville operator. For this purpose, we develop a new method, which is different from that used for the classical problem. Moreover, the considered situation arose relevance of the so-called full stability, i.e. the stability with respect to the full set of the input data including all a priori known components of the operator. The obtained results illustrate some essential difference from the inverse Sturm–Liouville problem. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the following nonlinear functional integral equation f(x) = Σq i=1 αi(x)f(τi(x)) + ∫ σ1(x) 0 Ψ (x, t, f(σ2(t)), ∫ σ3(t) 0 f(s)ds) dt + g(x), ∀x ∈ [0, 1], (E) where τi, σ1, σ2, σ3: [0, 1] → [0, 1]; αi, g: [0, 1] → R; Ψ: [0, 1] × [0, 1] × R2 → R are the given continuous functions and f: [0, 1] → R is an unknown function. First, two sufficient conditions for the existence and some properties of solutions of Eq. (E) are proved. By using Banach's fixed point theorem, we have the first sufficient condition yielding existence, uniqueness and stability of the solution. By applying Schauder's fixed point theorem, we have the second sufficient condition for the existence and compactness of the solution set. An example is also given in order to illustrate the results obtained here. Next, in the case of Ψ ∈ C2([0, 1] × [0, 1] × R²;R), we investigate the quadratic convergence for the solution of Eq. (E). Finally, the smoothness of the solution depending on data is established. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to investigate the solvability of infinite systems of nonlinear convolution type integral equations of N-variables in C(I1xI2x...xIN; lp) by using Hausdorff measure of noncompactness with the help of Meir-Keeler condensing operators. Finally, to credibility we propose a numerical method to find solutions of the problem with the high accuracy. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present a new and efficient numerical method to approximate the solutions of two‐dimensional nonlinear fractional Fredholm and Volterra integral equations. For this aim, the two‐variable shifted fractional‐order Jacobi polynomials are introduced and their operational matrices of fractional integration and product are derived. These operational matrices and shifted fractional‐order Jacobi collocation method are utilized to reduce the understudy equations to systems of nonlinear algebraic equations. Then, the arising systems can be solved by the Newton method. Discussion on the convergence analysis and error bound of the proposed method is presented. The efficiency, accuracy, and validity of the presented method are demonstrated by its application to three test examples and by comparing our results with the results obtained by existing numerical methods in the literature recently. [ABSTRACT FROM AUTHOR]

The purpose of this paper to establish the semilocal convergence analysis of three-step Kurchatov method under weaker conditions in Banach spaces. We construct the recurrence relations under the assumption that involved first-order divided difference operators satisfy the ω condition. Theorems are given for the existence-uniqueness balls enclosing the unique solution. The application of the iterative method is shown by solving nonlinear system of equations and nonlinear Hammerstein-type integral equations. It illustrates the theoretical development of this study. [ABSTRACT FROM AUTHOR]

A parameter‐dependent perturbation of the spectrum of the scalar Laplacian is studied for a class of nonlocal and non‐self‐adjoint rank one perturbations. A detailed description of the perturbed spectrum is obtained both for Dirichlet boundary conditions on a bounded interval as well as for the problem on the full real line. The perturbation results are applied to the study of a related parameter‐dependent nonlinear and nonlocal parabolic equation. The equation models a feedback system that admits an interpretation as a thermostat device or in the context of an agent‐based price formation model for a market. The existence and the stability of periodic self‐oscillations of the related nonlinear and nonlocal heat equation that arise from a Hopf bifurcation are proved. The bifurcation and stability results are obtained both for the nonlinear parabolic equation with Dirichlet boundary conditions and for a related problem with nonlinear Neumann boundary conditions that model feedback boundary control. They follow from a Popov criterion for integral equations after reducing the stability analysis for the nonlinear parabolic equation to the study of a related nonlinear Volterra integral equation. While the problem is studied in the scalar case only, it can be extended naturally to arbitrary Euclidean dimension and to manifolds. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear partial slip contact problem between a monoclinic half plane and a rigid punch of an arbitrary profile subjected to a normal load is considered. Applying Fourier integral transform and the appropriate boundary conditions, the mixed-boundary value problem is reduced to a set of two coupled singular integral equations, with the unknowns being the contact stresses under the punch in addition to the stick zone size. The Gauss–Chebyshev discretization method is used to convert the singular integral equations into a set of nonlinear algebraic equations, which are solved with a suitable iterative algorithm to yield the lengths of the stick zone in addition to the contact pressures. Following a validation section, an extensive parametric study is performed to illustrate the effects of material anisotropy on the contact stresses and length of the stick zone for typical monoclinic fibrous composite materials. [ABSTRACT FROM AUTHOR]

We prove the existence of a rolling odd solution to a boundary value problem for a convolution type nonlinear singular integral equation on the whole axis with power nonlinearity. We obtain the integral asymptotic formula for the solution and describe some examples. [ABSTRACT FROM AUTHOR]

This paper is devoted to the approximation of solutions for nonlinear equations by using iterative methods. We present a unified convergence analysis for some Newton-type methods. We consider both semilocal and local analysis. In the first one, the hypotheses are imposed on the initial guess and in the second on the solution. The results can be applied for smooth and non-smooth operators. In the numerical section we study two applications, first one, it is devoted to a nonlinear integral equation of Hammerstein type and in second one, we approximate the solution of a nonlinear PDE related to image denoising. [ABSTRACT FROM AUTHOR]

In this article, an idea based on moving least squares (MLS) and spectral collocation method is used to estimate the solution of nonlinear stochastic Volterra–Fredholm integral equations (NSVFIEs). The main advantage of the suggested approach is that in some parts where interpolation and integration are necessary, this approach does not require any meshes. Therefore, it is independent of the geometry of the domains, and this advantage helps us to solve the problems on irregular domains with relatively fewer computations. Another advantage of our proposed method is that with a small number of points and base functions, we were able to obtain the results with acceptable accuracy, and this is very attractive and practical. Applying the proposed method leads to the conversion of the problem into a system of algebraic equations. It is worth noting, some examples and error estimations have been provided to illustrate the accuracy and applicability of this technique. Also, we present a convergence analysis of the proposed method. [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical solution for a class of nonlinear weakly singular Volterra integral equation of the first kind. The existence and uniqueness issue of this nonlinear Volterra integral equations is studied completely. An hp -version collocation method in conjunction with Jacobi polynomials is introduced so as an appropriate numerical solution to be found. We analyze it properly and find an error estimation in L 2 -norm. The efficiency of the method is illustrated by some numerical experiments. [ABSTRACT FROM AUTHOR]

The purpose of this article is to investigate the existence of unique solution for the following mixed nonlinear Volterra Fredholm-Hammerstein integral equation considered in complex plane; (0.1) ξ (τ) = g (t) + ρ ∫ 0 τ K 1 (τ , ℘) ϝ 1 (℘ , ξ (℘)) d ℘ + ϱ ∫ 0 1 K 2 (τ , ℘) ϝ 2 (℘ , ξ (℘)) d ℘ , such that ξ = ξ 1 + ξ 2 , ξ 1 , ξ 2 ∈ (C ([ 0 , 1 ]) , R) g = g 1 + g 2 , g l : [ 0 , 1 ] → R , l = 1 , 2 , ϝ l (℘ , ξ (℘)) = ϝ l 1 * (℘ , ξ 1 *) + i ϝ l 2 * (℘ , ξ 2 *) , ϝ lj * : [ 0 , 1 ] × R → R for l , j = 1 , 2 , and ξ 1 * , ξ 2 * ∈ (C ([ 0 , 1 ]) , R) K l (t , ℘) = K l 1 * (t , ℘) + iK l 2 * (t , ℘) , for l , j = 1 , 2 and K lj * : [ 0 , 1 ] 2 → R , where ρ and ϱ are constants, g (t), the kernels Kl (τ, ℘) and the nonlinear functions ϝ1 (℘, ξ (℘)), ϝ 2 (℘, ξ (℘)) are continuous functions on the interval 0 ≤ τ ≤ 1. In this direction we apply fixed point results for self mappings with the concept of (ψ, ϕ) contractive condition in the setting of complex-valued fuzzy metric spaces. This study will be useful in the development of the theory of fuzzy fractional differential equations in a more general setting. [ABSTRACT FROM AUTHOR]

The hereditary and/or convolution integrals associated with linear viscoelastic material constitutive relations based on Prony series [1] characterization are recast into ordinary nonconvolution time integrals, which can be more efficiently evaluated analytically and numerically. Application of this protocol greatly reduces computational time, CPU usage and memory requirements used to solve linear temperature dependent and/or independent viscoelastic problems involving integral-differential equations with variable coefficients. The formulation includes temperature dependent materials with time and space dependent temperatures as well as stresses due to thermal expansions. Approximate approaches for dealing with time dependent temperatures are derived and solutions to non-convolution integral equations as well as to differential equations with variable coefficients are formulated. Relaxation time consistent relations are derived for isotropic viscoelastic materials. Applications of Galerkin and Runge-Kutta methods to viscoelastic solutions are discussed and evaluated. These protocols include solutions to IODEs and IPDEs with variable coefficients. An illustrative algorithm to be used in conjunction with differential equation solvers such as MATLAB's™ ODE45 has been developed which allows for numerical solutions simultaneously in both real and reduced time spaces without approximations of linear and nonlinear integral differential equations with variable or without coefficients. Similar protocols could be readily extended to other software such as MATHEMATICA™, MAPLE™, etc. [ABSTRACT FROM AUTHOR]

In this paper, we give a nonlinear F-contraction form of the Sadovskii fixed point theorem and we also investigate the existence of solutions for a functional integral equation of Volterra type. [ABSTRACT FROM AUTHOR]

In this paper, we introduce fixed point theorem for a general contractive condition in complex valued metric spaces. Also, some important corollaries under this contractive condition are obtained. As an application, we find a unique solution for Urysohn integral equations and some illustrative examples are given to support our obtaining results. Our results extend and generalize the results of Azam et al. [2] and some other known results in the literature. [ABSTRACT FROM AUTHOR]

We investigate the existence of solutions for a nonlinear integral inclusion of Urysohn–Stieltjes type. As applications, we give a Chandrasekhar quadratic integral equation and a nonlinear Chandrasekhar integral inclusion. [ABSTRACT FROM AUTHOR]

In this article, a numerical technique based on the Chebyshev cardinal functions (CCFs) and the Lagrange multiplier technique for the numerical approximation of the variable-order fractional integrodifferential equations are shown. The variable-order fractional derivative is considered in the sense of regularized Hilfer-Prabhakar and Hilfer-Prabhakar fractional derivatives. To solve the problem, first, we obtain the operational matrix of the regularized Hilfer-Prabhakar and Hilfer-Prabhakar fractional derivatives of CCFs. Then, this matrix and collocation method are used to reduce the solution of the nonlinear coupled variable-order fractional integrodifferential equations to a system of algebraic equations which is technically simpler for handling. Convergence and error analysis are examined. Finally, some examples are given to test the proposed numerical method to illustrate its accuracy and efficiency. [ABSTRACT FROM AUTHOR]

The aim of this manuscript is to present a new space, namely, a modified ϖ e -metric-like space, and we establish some related fixed point results using extended F ϖ e -Suzuki and generalized F ϖ e -Suzuki contractions on the mentioned space. Here, we support our theoretical consequences in two ways: the first one consists of presenting illustrative examples and the second one consists of finding analytical solutions for some integral and differential equations in the context of the mentioned space. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in an inverse problem for the biharmonic equation to recover the Robin coefficients on a non-accessible part of the boundary in simply connected planar domain from a measured Riquier-Neumann data on the accessible part of the boundary. Our work is based on a system of nonlinear and ill-posed integral equations which is solved by Tikhonov regularization method to complete missing Cauchy data. The minimization problem will be considered to solve the inverse problem, and recover the Robin coefficients on a known boundary curve. We show the feasibility of this method by numerical reconstructions. [ABSTRACT FROM AUTHOR]

• A new approach to analysis of the infection expansion based on first integrals of the mathematical model. • Painlevé analysis of the classic two-parameter SIR-model. • The new general analytical solution of the model, structurally featured through its clear relation to epidemiological data. • Re-thinking the classic SIR model with application to SARS-Cov-2 case. A classic two-parameter epidemiological SIR-model of the coronavirus propagation is considered. The first integrals of the system of non-linear equations are obtained. The Painlevé test shows that the system of equations is not integrable in the general case. However, the general solution is obtained in quadrature as an inverse time-function. Using the first integrals of the system of equations, analytical dependencies for the number of infected patients I (t) and that of recovered patients R (t) on the number of susceptible to infection S (t) are obtained. A particular attention is paid to interrelation of I (t) and R (t) both depending on α / β , where α is the contact rate in the community and β is the intensity of recovery/decease of patients. It is demonstrated that the data on particular morbidity waves in Hubei (China), Italy, Austria, South Korea, Moscow (Russia) as well some Australian territories are satisfactorily described by the expressions obtained for I (R). The variability of parameter N having been traditionally considered as a static population size is discussed. [ABSTRACT FROM AUTHOR]