53 results
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2. The Monte Carlo Method for Solving Large Systems of Linear Ordinary Differential Equations
- Author
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M. G. Smilovitskiy and S. M. Ermakov
- Subjects
Markov chain ,Series (mathematics) ,General Mathematics ,010102 general mathematics ,Monte Carlo method ,Expected value ,01 natural sciences ,Integral equation ,010305 fluids & plasmas ,Linear differential equation ,0103 physical sciences ,Applied mathematics ,Initial value problem ,0101 mathematics ,Constant (mathematics) ,Mathematics - Abstract
The Monte Carlo method to solve the Cauchy problem for large systems of linear differential equations is proposed in this paper. Firstly, a quick overview of previously obtained results from applying the approach towards the Fredholm-type integral equations is made. In the main part of the paper, the method is applied towards a linear ODE system that is transformed into an equivalent system of the Volterra-type integral equations, which makes it possible to remove the limitations due to the conditions of convergence of the majorant series. The following key theorems are stated. Theorem 1 provides the necessary compliance conditions that should be imposed upon the transition propability and initial distribution densities that initiate the corresponding Markov chain, for which equality between the mathematical expectation of the estimate and the functional of interest would hold. Theorem 2 formulates the equation that governs the estimate’s variance. Theorem 3 states the Markov chain parameters that minimize the variance of the estimate of the functional. Proofs are given for all three theorems. In the practical part of this paper, the proposed method is used to solve a linear ODE system that describes a closed queueing system of ten conventional machines and seven conventional service persons. The solutions are obtained for systems with both constant and time-dependent matrices of coefficients, where the machine breakdown intensity is time dependent. In addition, the solutions obtained by the Monte Carlo and Runge–Kutta methods are compared. The results are presented in the corresponding tables.
- Published
- 2021
3. Exact Solutions of a Nonclassical Nonlinear Equation of the Fourth Order
- Author
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A. I. Aristov
- Subjects
Implicit function ,General Mathematics ,010102 general mathematics ,Nonlinear partial differential equation ,02 engineering and technology ,01 natural sciences ,010305 fluids & plasmas ,Nonlinear system ,020303 mechanical engineering & transports ,Fourth order ,0203 mechanical engineering ,Special functions ,Ordinary differential equation ,0103 physical sciences ,Applied mathematics ,Elementary function ,Uniqueness ,0101 mathematics ,Second derivative ,Mathematics ,Variable (mathematics) - Abstract
Since the second half of the twentieth century, wide studies of Sobolev-type equations are undertaken. These equations contain items that are derivatives with respect to time of the second order derivatives of the unknown function with respect to space variables. They can describe nonstationary processes in semiconductors, in plasm, phenomena in hydrodinamics and other ones. Notice that wide studies of qualitative properties of solutions of Sobolev-type equations exist. Namely, results about existence and uniqueness of solutions, their asymptotics and blow-up are known. But there are few results about exact solutions of Sobolev-type equations. There are books and papers about exact solutions of partial equations, but they are devoted mainly to classical equations, where the first or second order derivative with respect to time or the derivative with respect to time of the first order derivative of the unknown function with respect to the space variable is equal to a stationary expression. Therefore it is interesting to study exact solutions of Sobolev-type equations. In the present paper, a fourth order nonlinear partial equation is studied. Three classes of its exact solutions are built. They are expressed in terms of special functions (solutions of some ordinary differential equations). For two of these classes subsets that can be expressed in elementary functions are built, for the third one subsets that can be described in elementary functions and an implicit function (without a quadrature) are built.
- Published
- 2020
4. Solvability of Pseudoparabolic Equations with Non-Linear Boundary Condition
- Author
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A. S. Berdyshev, G. O. Zhumagul, and S. E. Aitzhanov
- Subjects
General Mathematics ,Weak solution ,010102 general mathematics ,Boundary (topology) ,01 natural sciences ,Domain (mathematical analysis) ,010305 fluids & plasmas ,Sobolev space ,Nonlinear system ,0103 physical sciences ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Galerkin method ,Eigenvalues and eigenvectors ,Mathematics - Abstract
The work is devoted to the fundamental problem of studying the solvability of the initial-boundary value problem for a pseudo-parabolic equation (also called Sobolev type equations) with a fairly smooth boundary. In this paper, the initial-boundary value problem for a quasilinear equation of a pseudoparabolic type with a nonlinear Neumann–Dirichlet boundary condition is studied. From a physical point of view, the initial-boundary-value problem we are considering is a mathematical model of quasi-stationary processes in semiconductors and magnetics, taking into account the most diverse physical factors. Many approximate methods are suitable for finding eigenvalues and eigenfunctions of tasks boundary conditions of which are linear with respect to the function and its derivatives. Among these methods, Galerkin’s method leads to the simplest calculations. In the paper, by means of the Galerkin method the existence of a weak solution of a pseudoparabolic equation in a bounded domain is proved. The use of the Galerkin approximations allows us to get an estimate above the time of the solution existence. Using Sobolev ’s attachment theorem, a priori solution estimates are obtained. The local theorem of the existence of the solution has been proved. The uniqueness of the weak generalized solution of the initial-boundary value problem of quasi-linear equations of pseudoparabolic type is proved on the basis of a priori estimates.A special place in the theory of nonlinear equations is taken by the range of studies of unlimited solutions, or, as they are otherwise called, modes with exacerbation. Nonlinear evolutionary problems that allow unlimited solutions are globally intractable: solutions increase indefinitely over a finite period of time. Sufficient conditions have been obtained for the destruction of its solution over finite time in a limited area with a nonlinear Neumann–Dirichle boundary condition.
- Published
- 2020
5. On Solvability of One Singular Equation of Peridynamics
- Author
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A. V. Yuldasheva
- Subjects
Partial differential equation ,Peridynamics ,General Mathematics ,Operator (physics) ,010102 general mathematics ,Function (mathematics) ,01 natural sciences ,Volterra integral equation ,010305 fluids & plasmas ,symbols.namesake ,0103 physical sciences ,Displacement field ,Solid mechanics ,symbols ,Applied mathematics ,0101 mathematics ,Laplace operator ,Mathematics - Abstract
In the classical theory of solid mechanics, the behavior of solids is described by partial differential equations (PDE) through Newton’s second law of motion. However, when spontaneous cracks and fractures exist, such PDE models are inadequate to characterize the discontinuities of physical quantities such as the displacement field. Recently, a peridynamic continuum model was proposed which only involves the integration over the differences of the displacement field. A linearized peridynamic model can be described by the integro-differential equation with initial values. In this paper, we study the well-posedness and regularity of a linearized peridynamic model with singular kernel. The novelty of the paper is that the singular kernel is represented as the Laplacian of a regular function. This let to convert the model to an operator valued Volterra integral equation. Then the existence and regularity of the solution of the peridynamics problem are established through the study of the Volterra integral equation.
- Published
- 2020
6. On the Aizerman Problem: Coefficient Conditions for the Existence of a Four-Period Cycle in a Second-Order Discrete-Time System
- Author
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T. E. Zvyagintseva
- Subjects
Automatic control ,business.industry ,General Mathematics ,010102 general mathematics ,Order (ring theory) ,Robotics ,Kalman filter ,Type (model theory) ,Motion control ,01 natural sciences ,010305 fluids & plasmas ,Nonlinear system ,Stability theory ,0103 physical sciences ,Applied mathematics ,Artificial intelligence ,0101 mathematics ,business ,Mathematics - Abstract
We consider in this paper an automatic control second-order discrete-time system whose nonlinearity satisfies the generalized Routh–Hurwitz conditions. Systems of this type are widely used in solving modern application problems that arise in engineering, theory of motion control, mechanics, physics, and robotics. Two constructed examples of discrete-time systems with nonlinearities that lie in a Hurwitz angle were presented in recent papers by W. Heath, J. Carrasco, and M. de la Sen. These examples demonstrate that in the discrete case, the Aizerman and Kalman conjectures are untrue even for second-order systems. One such system in these examples has a three-period cycle and the other system, a four-period cycle. We assume in the present paper that the nonlinearity is two-periodic and lies in a Hurwitz angle; here, we study a system for all possible parameter values. We explicitly present the conditions (for the parameters) under which it is possible to construct a two-periodic nonlinearity in such a way that a system with it is not globally asymptotically stable. Such a nonlinearity can be constructed in more than one way. We propose a method for constructing the nonlinearity in such a way that a family of four-period cycles is found in the system. The cycles are nonisolated; any solution of the system with the initial data, which lies on a certain specified ray, is a periodic solution.
- Published
- 2020
7. Higher Order Dirichlet-Type Problems in 2D Complex Quaternionic Analysis
- Author
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Baruch Schneider
- Subjects
Laplace's equation ,Helmholtz equation ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Quaternionic analysis ,Dirichlet distribution ,010305 fluids & plasmas ,Sobolev space ,symbols.namesake ,Dirac equation ,0103 physical sciences ,symbols ,Applied mathematics ,Uniqueness ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
It is well known that developing methods for solving Dirichlet problems is important and relevant for various areas of mathematical physics related to the Laplace equation, the Helmholtz equation, the Stokes equation, the Maxwell equation, the Dirac equation, and others. The author in previous papers studied the solvability of Dirichlet boundary value problems of the first and second orders in quaternionic analysis. In the present paper, we study a higher-order Dirichlet boundary value problem associated with the two-dimensional Helmholtz equation with complex potential. The existence and uniqueness of a solution to the Dirichlet boundary value problem in the two-dimensional case is proved and an appropriate representation formula for the solution of this problem is found. Most Dirichlet problems are solved for the case in three variables. Note that the case of two variables is not a simple consequence of the three-dimensional case. To solve the problem, we use the method of orthogonal decomposition of the quaternion-valued Sobolev space. This orthogonal decomposition of the space is also a tool for the study of many elliptic boundary value problems that arise in various areas of mathematics and mathematical physics. An orthogonal decomposition of the quaternion-valued Sobolev space with respect to the high-order Dirac operator is also obtained in this paper.
- Published
- 2019
8. On the Explicit Integration of Special Types of Differential Inequalities
- Author
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Yu. A. Il’in
- Subjects
General method ,Inequality ,Differential equation ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,01 natural sciences ,010305 fluids & plasmas ,Variation (linguistics) ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,0103 physical sciences ,Applied mathematics ,0101 mathematics ,Differential inequalities ,media_common ,Mathematics - Abstract
A general method was proposed in our previous paper for explicitly finding all solutions of the differential inequality, which is based on the general solution of the corresponding differential equation or, in other words, on the variation of arbitrary constants. Criteria of extendibility and characteristics of the maximally extended (full) solution of the inequality were proven. In this paper, we applied these results to specific types of inequalities most frequently encountered in applications and literature. We also compared them to other known methods in the literature.
- Published
- 2019
9. The Extension of the Monte Carlo Method for Neutron Transfer Problems Calculating to the Problems of Quantum Mechanics
- Author
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M. I. Gurevich, A. A. Danshin, Vasily Velikhov, V. A. Ilyin, and A. A. Kovalishin
- Subjects
020209 energy ,General Mathematics ,Monte Carlo method ,02 engineering and technology ,Function (mathematics) ,Space (mathematics) ,01 natural sciences ,Integral equation ,Schrödinger equation ,symbols.namesake ,0103 physical sciences ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,Applied mathematics ,010306 general physics ,Quantum ,Eigenvalues and eigenvectors ,Mathematics ,Identical particles - Abstract
There are several methods of numerical solution of eigenvalue problems by the Monte Carlo method, which are used in the calculation of nuclear reactors. This paper is devoted to the investigation of the possibility of using such methods for solving the stationary Schrodinger equation. The latter equation can easily be transformed into the form of an integral equation of the first kind, very similar to those integral equations that arise in problems of nuclear power. The Monte Carlo method for this form of the stationary Schrodinger equation looks very attractive, since it naturally parallels and is very convenient for calculations on multiprocessor systems. In addition, in this case it is necessary to operate with functions defined on a large-dimensional space. This is also natural for the Monte Carlo method. It is described how the methods long used for the calculation of nuclear reactors are transformed for this case. The main problem is that the wave function of fermions changes its sign under a permutation of identical particles, and can not be nonnegative. The proposed approach is significantly different from the known methods of applying the Monte Carlo method to quantum mechanical problems. In this paper, several examples of the successful application of the proposed new method are given.
- Published
- 2018
10. On Solvability of a Poincare–Tricomi Type Problem for an Elliptic–Hyperbolic Equation of the Second Kind
- Author
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A. A. Abdullaev, B. I. Islomov, and T. K. Yuldashev
- Subjects
Partial differential equation ,Differential equation ,General Mathematics ,010102 general mathematics ,Degenerate energy levels ,Cauchy distribution ,01 natural sciences ,010305 fluids & plasmas ,0103 physical sciences ,Applied mathematics ,Boundary value problem ,Uniqueness ,0101 mathematics ,Degeneracy (mathematics) ,Hyperbolic partial differential equation ,Mathematics - Abstract
In this paper we study a boundary value problem with the Poincare–Tricomi condition for a degenerate partial differential equation of elliptic-hyperbolic type of the second kind. In the hyperbolic part of a degenerate mixed differential equation of the second kind the line of degeneracy is a characteristic. For this type of differential equations a class of generalized solutions is introduced in the characteristic triangle. Using the properties of generalized solutions, the modified Cauchy and Dirichlet problems are studied. The solutions of these problems are found in the convenient form for further investigations. A new method has been developed for a differential equation of mixed type of the second kind, based on energy integrals. Using this method, the uniqueness of the considering problem is proved. The existence of a solution of the considering problem reduces to investigation of a singular integral equation and the unique solvability of this problem is proved by the Carleman–Vekua regularization method.
- Published
- 2021
11. On Generalizations of the Optimal Choice Problem
- Author
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Igor V. Belkov
- Subjects
Independent and identically distributed random variables ,Sequence ,Uniform distribution (continuous) ,Exponential distribution ,General Mathematics ,010102 general mathematics ,Interval (mathematics) ,Expected value ,01 natural sciences ,010305 fluids & plasmas ,0103 physical sciences ,Applied mathematics ,Point (geometry) ,0101 mathematics ,Random variable ,Mathematics - Abstract
In this paper, we consider generalizations of the optimal choice problem. There is a sequence of n identically distributed random variables on the interval [0, 1]. Sequentially obtaining the observed values of these quantities, it is necessary at some point to stop at one of them, taking it as the starting point for counting the upper or lower record values. In the optimal choice problem and its generalizations, it is required to make the correct choice of the starting point of the records in order to maximize the mathematical expectation of the sum of values or the number of upper, lower, or both record values obtained as a result of such a procedure. A review of the results on the uniform distribution of random variables and new results on the exponential distribution of random variables are presented.
- Published
- 2021
12. Reliable a Posteriori Error Estimation for Cosserat Elasticity in 3D
- Author
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M. E. Frolov
- Subjects
General Mathematics ,010102 general mathematics ,01 natural sciences ,010305 fluids & plasmas ,Orthogonality ,0103 physical sciences ,A priori and a posteriori ,Applied mathematics ,0101 mathematics ,Elasticity (economics) ,Algebra over a field ,Galerkin method ,Approximate solution ,Mathematics - Abstract
A new a posteriori error estimate for Cosserat elasticity is proposed. Paper continues implementations of functional approach to error estimation for planar problems of classical and Cosserat elasticity. The proposed majorant is reliable regardless of some additional assumptions (for instance, the Galerkin orthogonality). This property is preserved independently of methods used for solving a problem, and the estimate is valid for accuracy control of any conforming approximate solution.
- Published
- 2021
13. The Euler–Lagrange Approximation of the Mean Field Game for the Planning Problem
- Author
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V. Kornienko, V. Shaydurov, and Shuhua Zhang
- Subjects
Optimization problem ,Partial differential equation ,Discretization ,General Mathematics ,010102 general mathematics ,Field (mathematics) ,State (functional analysis) ,01 natural sciences ,010305 fluids & plasmas ,0103 physical sciences ,Applied mathematics ,Minification ,0101 mathematics ,Finite set ,Differential (mathematics) ,Mathematics - Abstract
The paper presents a finite-difference analogue of the differential problem formulated in terms of the theory of ‘‘Mean Field Games’’ for solving the planning problem of convey to a given state. Here optimization problem is formulated as coupled pair of parabolic partial differential equations of the Kolmogorov (Fokker–Planck) and Hamilton–Jacobi–Bellman type. The proposed Euler–Lagrange finite-difference analogue inherits the basic properties of an optimization differential problem at a discrete level. As a result, it can serve as an approximation of the original differential problem when the discretization steps tend to zero, or as a self-contained optimization task with a finite set of participants. For the proposed analogue, the algorithm of monotonous minimization of the value functional is constructed and illustrated on a model economic task.
- Published
- 2020
14. On Global Solvability of One-Dimensional Quasilinear Wave Equations
- Author
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D. V. Tunitsky
- Subjects
General Mathematics ,Least-upper-bound property ,010102 general mathematics ,Interval (mathematics) ,Wave equation ,01 natural sciences ,Domain (mathematical analysis) ,010305 fluids & plasmas ,Range (mathematics) ,Ordinary differential equation ,0103 physical sciences ,Applied mathematics ,Initial value problem ,Uniqueness ,0101 mathematics ,Mathematics - Abstract
The paper concerns global solvability of initial value problem for one class of hyperbolic quasilinear second order equations with two independent variables, which have a rather wide range of applications. Besides existence and uniqueness of maximal solutions of this problem it is proved that a maximal solution possess the completeness property that is an analog of the corresponding property of ordinary differential equations. Namely, a solution of an ordinary differential equation that is defined on a maximal interval leaves any compact subset of the equation domain.
- Published
- 2020
15. On a Problem of Heat Equation with Fractional Load
- Author
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L. Zh. Kasymova, M.I. Ramazanov, and M.T. Kosmakova
- Subjects
General Mathematics ,010102 general mathematics ,Generalized hypergeometric function ,01 natural sciences ,Integral equation ,Volterra integral equation ,010305 fluids & plasmas ,Fractional calculus ,symbols.namesake ,Singularity ,Kernel (statistics) ,0103 physical sciences ,symbols ,Applied mathematics ,Heat equation ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
In the paper, the solvability problems of an nonhomogeneous boundary value problem in the first quadrant for a fractionally loaded heat equation are studied. Feature of this problem is that, firstly, the loaded term is presented in the form of the Caputo fractional derivative with respect to the spatial variable, secondly, the order of the derivative in the loaded term is less than the order of the differential part and, thirdly, the point of load is moving. The problem is reduced to the Volterra integral equation of the second kind, the kernel of which contains the generalized hypergeometric series. The kernel of the obtained integral equation is estimated and it is shown that the kernel of the equation has a weak singularity (under certain restrictions on the load), this is the basis for the statement that the loaded term in the equation is a weak perturbation of its differential part. In addition, the limiting cases of the order of the fractional derivative are considered. It is proved that there is continuity in the order of the fractional derivative.
- Published
- 2020
16. A New Rusanov-Type Solver with a Local Linear Solution Reconstruction for Numerical Modeling of White Dwarf Mergers by Means Massive Parallel Supercomputers
- Author
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Igor Chernykh, Alexander V. Tutukov, Igor Kulikov, S. V. Lomakin, and Anna Sapetina
- Subjects
Polynomial ,Speedup ,General Mathematics ,010102 general mathematics ,Solver ,Supercomputer ,01 natural sciences ,010305 fluids & plasmas ,Discontinuity (linguistics) ,0103 physical sciences ,Piecewise ,Applied mathematics ,0101 mathematics ,Poisson's equation ,Eigenvalues and eigenvectors ,Mathematics - Abstract
The results of numerical modeling of white dwarf mergers on massive parallel supercomputers using a AVX-512 technique are presented. A hydrodynamic model of white dwarfs closed by a star equation of state and supplemented by a Poisson equation for the gravitational potential is constructed. This paper presents a modification based on a local linear reconstruction of the solution of the Rusanov scheme for the hydrodynamic equations. This reconstruction makes it possible to considerably decrease the numerical dissipation of the scheme for weak shock waves without any external piecewise polynomial reconstruction. The scheme is efficient for unstructured grids, when it is difficult to construct a piecewise polynomial solution, and also in parallel implementations of structured nested or adaptive grids, when the costs of interprocess interactions increase significantly. As input data, piecewise constant values of the physical variables in the left and right cells of a discontinuity are used. The smoothness of the solution is measured by the discrepancy between the maximum left and right eigenvalues. This discrepancy is used for a local piecewise polynomial reconstruction in the left and right cells. Then the solutions are integrated along the characteristics taking into account the piecewise linear representation of the physical variables. A performance of 234 gigaflops and 33-fold speedup are obtained on two Intel Skylake processors on the cluster NKS-1P of the Siberian Supercomputer Center ICM & MG SB RAS.
- Published
- 2020
17. Stochastic Mesh Method for Optimal Stopping Problems
- Author
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Igor P. Fedyaev and Yuriy N. Kashtanov
- Subjects
Discretization ,General Mathematics ,010102 general mathematics ,Stochastic game ,01 natural sciences ,010305 fluids & plasmas ,Nonlinear system ,Diffusion process ,Consistency (statistics) ,0103 physical sciences ,Applied mathematics ,Optimal stopping ,Asian option ,Gravitational singularity ,0101 mathematics ,Mathematics - Abstract
This paper considers the application of the stochastic mesh method in solving the multidimensional optimal stopping problem for a diffusion process with nonlinear payoff functions. A special discretization scheme of the diffusion process is presented to solve the problem in the case of geometric average Asian option payoff functions. This discretization scheme makes it possible to eliminate singularities in transition probabilities. Next, two estimates are given of the solution of the problem by the stochastic mesh method for the case of the stochastic mesh transition probabilities defined as averaged densities. The consistency of the estimates is proven. It is shown that the variance of the estimates is inversely proportional to the number of points in each mesh layer. The result extends the application area of the stochastic mesh method and methods for treating Asian options. A numerical example of the result of applying the obtained estimates to the call and put options compared to the obtained option prices using a regular mesh is presented.
- Published
- 2020
18. Discrete Generalized Odd Lindley–Weibull Distribution with Applications
- Author
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Andrei Volodin, Sirinapa Aryuyuen, and Winai Bodhisuwan
- Subjects
Discretization ,Distribution (number theory) ,General Mathematics ,010102 general mathematics ,Probability density function ,Variance (accounting) ,Poisson distribution ,01 natural sciences ,010305 fluids & plasmas ,symbols.namesake ,0103 physical sciences ,Probability mass function ,symbols ,Statistics::Methodology ,Applied mathematics ,0101 mathematics ,Weibull distribution ,Count data ,Mathematics - Abstract
In this paper, we introduce a new probability mass function by discretizing the continuous failure model of the generalized odd Lindley–Weibull distribution, which is called the discrete generalized odd Lindley–Weibull (DGOL-W) distribution. This new probability mass function is characterized by a very flexible probability function: reverse J-shape, right-skewed shape, left-skewed shape, and close to symmetric shape. The proposed distribution has five special models, i.e., the discrete generalized odd Lindley-exponential, discrete generalized odd Lindley–Rayleigh, discrete odd Lindley–Weibull, discrete odd Lindley-exponential, and discrete odd Lindley–Rayleigh distributions. Some properties of the proposed distribution are introduced. The maximum likelihood estimation is used to estimate the unknown parameters of the DGOL-W distribution. Applications are illustrated, which show that the model is suited for use in various data sets, i.e., the mean and variance of the count data are equal, over-dispersion count data, and under-dispersion count data. Based on the results, we have shown that the DGOL-W distribution provides a better fit compared to the Poisson, discrete Lindley and four sub-models of DGOL-W distribution for count data.
- Published
- 2020
19. New Type Super Singular Integro-Differential Equation and Its Conjugate Equation
- Author
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S. K. Zarifzoda and T. K. Yuldashev
- Subjects
General Mathematics ,010102 general mathematics ,Characteristic equation ,01 natural sciences ,Integral equation ,Regularization (mathematics) ,Volterra integral equation ,Manifold ,010305 fluids & plasmas ,symbols.namesake ,Singularity ,Integro-differential equation ,0103 physical sciences ,symbols ,Applied mathematics ,0101 mathematics ,Analytic function ,Mathematics - Abstract
In this paper for a class of model partial integro-differential equation with super singularity in the kernels, is obtained an integral representation of manifold solutions by arbitrary constants. The conjugate equation for the above-mentioned type of equations is also investigated. Such types of integro-differential equations are different from Cauchy-type singular integro-differential equations. Cauchy-type singular integro-differential equations are studied by the methods of theory of analytical functions. However, the method of analytical functions is not applicable for our case of super singular equations with integrals understanding in Riemann–Stieltjes sense. Here, we have used the method of representation the considering equation as a product of two one-dimensional singular first order integro-differential operators. Further, a complete integro-differential equation and its conjugate equation have been investigated. It is shown that in every cases of characteristic equation roots the homogeneous integro-differential equation can have a nontrivial solutions. Non-model equation is investigated by the regularization method. Regularization of non-model equation is based on selecting a model part of equation. On the basis of the analysis of a model part of equation the solution of non-model equation reduced to the solution of a second kind Volterra integral equations with super singular kernel. It is important to emphasize that in contrast to the usual theory of Volterra integral equations, the studied homogeneous integral equation has nontrivial solutions. It is easy to see that the presence of a non-model part in the equation does not affect to the general structure of the obtained solutions. From here investigation of the model equations for given class of the integro-differential equations becomes important.
- Published
- 2020
20. On a Method for Constructing the Riemann Function for Partial Differential Equations with a Singular Bessel Operator
- Author
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Sh. T. Karimov and Sh. A. Oripov
- Subjects
Variables ,Partial differential equation ,Differential equation ,General Mathematics ,Operator (physics) ,media_common.quotation_subject ,010102 general mathematics ,Function (mathematics) ,01 natural sciences ,010305 fluids & plasmas ,symbols.namesake ,Riemann hypothesis ,0103 physical sciences ,symbols ,Applied mathematics ,0101 mathematics ,Hyperbolic partial differential equation ,Bessel function ,Mathematics ,media_common - Abstract
A linear second-order hyperbolic equation of two independent variables with a singular Bessel operator is considered. For particular types of such equations, a detailed literature review of known methods for constructing Riemann functions is given. It is shown that to construct the Riemann function for equations with a singular Bessel operator, we can use the Erdelyi–Kober transmutation operator. The Riemann function for the Euler–Poisson–Darboux differential equations is found in explicit form. In this paper, we give examples and an algorithm for constructing the Riemann function for second-order hyperbolic equations with the Bessel operator.
- Published
- 2020
21. Potentials for Three-Dimensional Singular Elliptic Equation and Their Application to the Solving a Mixed Problem
- Author
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Tuhtasin Ergashev
- Subjects
General Mathematics ,010102 general mathematics ,Gaussian hypergeometric function ,01 natural sciences ,Integral equation ,Potential theory ,010305 fluids & plasmas ,Elliptic curve ,Singularity ,Simple (abstract algebra) ,0103 physical sciences ,Singular coefficients ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
In earlier research, the double- and simple layer potentials have been successfully applied in solving boundary value problems for two-dimensional elliptic equations. Despite the fact that all fundamental solutions of a three-dimensional elliptic equation with one, two and three singular coefficients were known, the potential theory was not constructed in any case of a singularity. Here, in this paper, our goal is to construct a potential theory corresponding to the three-dimensional elliptic equation with one singular coefficient. We used some properties of Gaussian hypergeometric function to prove the limiting theorems, while deriving integral equations concerning the denseness of potentials.
- Published
- 2020
22. Recovery of the Operator $$\boldsymbol{\Delta}_{\boldsymbol{B}}$$ from Its Incomplete Fourier–Bessel Image
- Author
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M. V. Polovinkina
- Subjects
General Mathematics ,010102 general mathematics ,Regular polygon ,01 natural sciences ,010305 fluids & plasmas ,symbols.namesake ,Fourier transform ,Recovery method ,0103 physical sciences ,symbols ,Applied mathematics ,Ball (mathematics) ,0101 mathematics ,Optimal methods ,Bessel function ,Mathematics - Abstract
In this paper we consider optimal recovery of the B-elliptic operator with its powers of a smooth function from the Fourier–Bessel transform given on some convex subset. We construct a series of optimal recovery methods. It turns out that information about the Fourier–Bessel transform outside of some ball with the center at zero is superfluous, optimal methods do not use it. The methods themselves differ in the way they process available information.
- Published
- 2020
23. The Cauchy Problem for the Iterated Klein–Gordon Equation with the Bessel Operator
- Author
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Sh. T. Karimov
- Subjects
Smoothness (probability theory) ,General Mathematics ,010102 general mathematics ,01 natural sciences ,010305 fluids & plasmas ,Spherical mean ,symbols.namesake ,Operator (computer programming) ,Iterated function ,0103 physical sciences ,symbols ,Order (group theory) ,Initial value problem ,Applied mathematics ,0101 mathematics ,Klein–Gordon equation ,Bessel function ,Mathematics - Abstract
An analogue of the Cauchy problem for an iterated multidimensional Klein–Gordon equation with a time-dependent Bessel operator is investigated. Applying the generalized Erdelyi–Kober operator of fractional order, we reduce the formulated problem to the Cauchy problem for the polywave equation. Applying a spherical mean method, we construct an explicit formula to solve this problem for the polywave equation; then, basing on this solution, we find an integral representation of the solution of the formulated problem. The obtained formula allows one to immediately discern the character of the dependence of the solution on the initial functions and, in particular, to establish conditions for the smoothness of the classical solution. The paper will be useful for specialists engaged in the resolving of problems of higher spin theory.
- Published
- 2020
24. Constructing c-Optimal Designs for Polynomial Regression without an Intercept
- Author
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Petr Shpilev and Viatcheslav B. Melas
- Subjects
Polynomial regression ,Optimal design ,General Mathematics ,Numerical analysis ,010102 general mathematics ,01 natural sciences ,Regression ,010305 fluids & plasmas ,0103 physical sciences ,Applied mathematics ,Point (geometry) ,0101 mathematics ,Special case ,Mathematics - Abstract
In this paper, we consider the problem of constructing c-optimal designs for polynomial regression without an intercept. The special case of c = f '(z) (i.e., the vector of derivatives of the regression functions at some point z is selected as vector c) is considered. The analytical results available in the literature are briefly reviewed. An effective numerical method for finding f '(z)-optimal designs in cases in which an analytical solution cannot be constructed is proposed.
- Published
- 2020
25. Numerical Solution of Mean Field Games Problems with Turnpike Effect
- Author
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N. V. Trusov
- Subjects
General Mathematics ,Horizon ,010102 general mathematics ,Ode ,Monotonic function ,Optimal control ,01 natural sciences ,010305 fluids & plasmas ,Reduction (complexity) ,Mean field theory ,0103 physical sciences ,Applied mathematics ,0101 mathematics ,Algebra over a field ,Finite time ,Mathematics - Abstract
We present a problem described by Mean Field Games (MFG) and Optimal Control theory on finite time horizon. This problem consists of a system of PDEs: a Kolmogorov–Fokker–Planck equation, evolving forward in time and a Hamilton–Jacobi–Bellman equation, evolving backwards in time. The numerical difficulties are based on a turnpike effect considered in this paper. We present an extremal problem whose necessary conditions of extremal satisfy the initial system of PDEs, and introduce its numerical solution at the heart of monotonic schemes. According to special assumptions, PDEs can be reduced to Riccati ODEs. We consider this reduction as a test example for the numerical solution of the extremal problem.
- Published
- 2020
26. Proposing Novel Modified Ratio Estimators by Adding an Exponential Parameter
- Author
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Emre Dünder and Tolga Zaman
- Subjects
Mean squared error ,Population mean ,General Mathematics ,Coefficient of variation ,010102 general mathematics ,Estimator ,01 natural sciences ,010305 fluids & plasmas ,Exponential function ,Auxiliary variables ,Correlation ,0103 physical sciences ,Applied mathematics ,0101 mathematics ,Algebra over a field ,Mathematics - Abstract
It is a common approach to adopt the statistical measures of the auxiliary variable such as correlation, coefficient of variation etc. for estimating the population mean. In this paper we propose novel estimators by adding an exponential parameter on the auxiliary variable. Theoretically, we obtain the mean square error (MSE) for all proposed estimators and we compare MSE equations of our proposed estimators and classical estimators. As a result of these comparisons, we observe that proposed estimators are always more efficient than classical estimators. These theoretical results are supported with the aid of a numerical and simulation examples.
- Published
- 2020
27. Boundary-value Problems with Data on Characteristics for Hyperbolic Systems of Equations
- Author
-
L. B. Mironova
- Subjects
Work (thermodynamics) ,Variables ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,01 natural sciences ,Hyperbolic systems ,010305 fluids & plasmas ,Matrix (mathematics) ,Riemann hypothesis ,symbols.namesake ,0103 physical sciences ,symbols ,Applied mathematics ,Shaping ,Boundary value problem ,0101 mathematics ,Algebra over a field ,Mathematics ,media_common - Abstract
The main subjects of the present paper are the Goursat and Darboux boundary-value problems for hyperbolic systems with two independent variables. We show that obtained by T.V. Chekmarev in terms of successive approximations formulas for solution of the Goursat problem can be built also by the Riemann method, work out an analog of the Riemann–Hadamard method for the system, and introduce its Riemann–Hadamard matrix. We solve the Darboux problem in terms of the introduced matrix.
- Published
- 2020
28. The Length-Biased Weighted Lindley Distribution with Applications
- Author
-
Yupapin Atikankul, Ampai Thongteeraparp, Winai Bodhisuwan, and Andrei Volodin
- Subjects
Distribution (number theory) ,Characteristic function (probability theory) ,Estimation theory ,General Mathematics ,010102 general mathematics ,Estimator ,Method of moments (statistics) ,Moment-generating function ,01 natural sciences ,010305 fluids & plasmas ,Lindley distribution ,0103 physical sciences ,Applied mathematics ,0101 mathematics ,Special case ,Mathematics - Abstract
In this paper, we propose a new length-biased distribution, which is a special case of weighted distributions. We derive some mathematical properties of the proposed distribution, including moment generating function, characteristic function and moments, and discuss parameter estimation by the method of moments and maximum likelihood estimation. We assess estimators via simulation, and show the potential of the proposed distribution by fitting it with some real-life data sets.
- Published
- 2020
29. Decomposition of Additive Random Fields
- Author
-
A. A. Khartov and M. Zani
- Subjects
Random field ,Covariance function ,Stochastic process ,General Mathematics ,Operator (physics) ,010102 general mathematics ,01 natural sciences ,010305 fluids & plasmas ,Arbitrarily large ,Covariance operator ,0103 physical sciences ,Applied mathematics ,0101 mathematics ,Brownian motion ,Eigenvalues and eigenvectors ,Mathematics - Abstract
We consider in this work an additive random field on [0, 1]d, which is a sum of d uncorrelated random processes. We assume that the processes have zero mean and the same continuous covariance function. There is a significant interest in the study of random fields of this type. For example, they arise in the theory of intersections and self-intersections of Brownian processes, in the problems concerning small ball probabilities, and in the finite-rank approximation problems with arbitrarily large parametric dimension d. In problems of the last kind, the spectral characteristics of the covariance operator play a key role. For a given additive random field, the dependence of eigenvalues of its covariance operator on eigenvalues of the covariance operator of the marginal processes is quite simple, provided that the identical 1 is an eigenvector of the latter operator. In the opposite case, the dependence is complex, and, therefore, it is hard to study these random fields. Here, summands of the decomposition of the random field into the sum of its integral and its centered version are orthogonal in L2([0, 1]d), but, in general, they are correlated. In the present paper, we propose another interesting decomposition for random fields (it was discovered by the authors while resolving finite-rank approximation problems in the average-case setting). In the obtained decomposition, the summands are orthogonal in L2([0, 1]d) and are uncorrelated. Moreover, for large d, they are respectively close to the integral and to the centered version of the random field with small relative mean square errors.
- Published
- 2020
30. Error of the Finite Element Approximation for a Differential Eigenvalue Problem with Nonlinear Dependence on the Spectral Parameter
- Author
-
D. M. Korosteleva, P. S. Solov’ev, A. A. Samsonov, and S. I. Solov’ev
- Subjects
General Mathematics ,010102 general mathematics ,Hilbert space ,Mathematics::Spectral Theory ,Bilinear form ,Superconvergence ,Eigenfunction ,01 natural sciences ,Finite element method ,010305 fluids & plasmas ,symbols.namesake ,Dirichlet boundary condition ,0103 physical sciences ,symbols ,Applied mathematics ,Gaussian quadrature ,0101 mathematics ,Eigenvalues and eigenvectors ,Mathematics - Abstract
The positive definite ordinary differential nonlinear eigenvalue problem of the second order with homogeneous Dirichlet boundary condition is considered. The problem is formulated as a symmetric variational eigenvalue problem with nonlinear dependence of the spectral parameter in a real infinite-dimensional Hilbert space. The variational eigenvalue problem consists in finding eigenvalues and corresponding eigenfunctions of the eigenvalue problem for a symmetric positive definite bounded bilinear form with respect to a symmetric positive definite completely continuous bilinear form in a real infinite-dimensional Hilbert space. The variational eigenvalue problem is approximated by the mesh scheme of the finite element method on the uniform grid. For constructing the mesh scheme, Lagrangian finite elements of arbitrary order are applied. Error estimates of approximate eigenvalues and error estimates of approximate eigenfunctions in the norm of initial real infinite-dimensional Hilbert space are established. These error estimates coincide in the order with error estimates of mesh scheme of the finite element method for linear eigenvalue problems. Moreover, superconvergence estimates for approximate eigenfunctions in the mesh norm with Gauss quadrature nodes are derived. Investigations of this paper generalize well known results for the eigenvalue problem with linear entrance on the spectral parameter.
- Published
- 2019
31. Approach to Solving the Inverse Problem of Filtration Based on Descriptive Regularization
- Author
-
A. I. Abdullin
- Subjects
General Mathematics ,010102 general mathematics ,Computational algorithm ,Inverse problem ,01 natural sciences ,Regularization (mathematics) ,Physics::Geophysics ,010305 fluids & plasmas ,Physics::Fluid Dynamics ,Wellbore ,Nonlinear system ,0103 physical sciences ,Pressure sensitive ,Fluid dynamics ,Applied mathematics ,0101 mathematics ,Fluid filtration ,Mathematics - Abstract
This paper presents the results of a study of inverse problem for the nonlinear parabolic equation for the fluid filtration in the fractured media. An approach to solve the inverse problem by using the descriptive regularization method is proposed. A mathematical model for the 3-D flow of a fluid through a pressure sensitive naturally fractured formation, with pseudosteady state matrix-fracture flow is developed. This model includes the effects of wellbore storage and fluid flow in the wellbore. A computational algorithm based on the proposed approach to estimate the dependence of the fractures permeability on pressure from the results of hydrodynamic studies of horizontal well is developed.
- Published
- 2019
32. Minimax-Maximin Relations for the Problem of Vector-Valued Criteria Optimization
- Author
-
Yu. A. Komarov and Alexander B. Kurzhanski
- Subjects
Statistics::Theory ,Correctness ,Inequality ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Bilinear interpolation ,Minimax ,01 natural sciences ,010305 fluids & plasmas ,0103 physical sciences ,Applied mathematics ,Always true ,0101 mathematics ,media_common ,Mathematics ,Real field ,Curse of dimensionality - Abstract
The minimax-maximin relations for vector-valued functionals over the real field are studied. An increase in the dimensionality of criteria may result in a violation of some basic relations, for example, in an inequality between maximin and minimax that is always true for classic problems. Accordingly, the conditions for its correctness or violation need to be established. This paper introduces the definitions of set-valued minimax and maximin for multidimensional criteria and with an analogue in the classic minimax inequality. Necessary and sufficient conditions for its correctness and violation are described for two particular types of vector-valued functionals: the bilinear ones and those with separated variables.
- Published
- 2020
33. Fourier Tools are Much More Powerful than Commonly Thought
- Author
-
Anry Nersessian
- Subjects
TheoryofComputation_MISCELLANEOUS ,Adaptive algorithm ,General Mathematics ,010102 general mathematics ,Acceleration (differential geometry) ,01 natural sciences ,010305 fluids & plasmas ,symbols.namesake ,Fourier transform ,0103 physical sciences ,Convergence (routing) ,symbols ,Applied mathematics ,0101 mathematics ,Algebra over a field ,Fourier series ,Mathematics - Abstract
In the proposed paper, some last autor’s results of studies devoted to the acceleration of the convergence of truncated Fourier series is presented. The corresponding universal (traditional) and special adaptive algorithms are constructed. The main result (the phenomenon of over-convergence for an non-linear adaptive algorithm) states that the use of finite Fourier coefficients leads to an exact approximation for functions from certain infinite-dimensional spaces of quasipolynomials. The corresponding summation formula of truncated Fourier series for smooth functions has unprecedented accuracy.
- Published
- 2019
34. Control of The Final State of Fuzzy Dynamical Systems
- Author
-
A. N. Sotnikov, I. A. Egereva, S. M. Dzyuba, I. I. Emelyanova, and B. V. Palyukh
- Subjects
Basis (linear algebra) ,Dynamical systems theory ,General Mathematics ,010102 general mathematics ,State (functional analysis) ,Type (model theory) ,01 natural sciences ,Fuzzy logic ,010305 fluids & plasmas ,Bellman equation ,0103 physical sciences ,Functional equation ,Applied mathematics ,0101 mathematics ,Control (linguistics) ,Mathematics - Abstract
The paper considers the problem of controlling the final state of dynamic systems characterized by classical fuzzy relations. The solution of the problem is reduced to solving a functional equation of the Bellman equation type. On the basis of modern methods of the general theory of dynamical systems, the asymptotic properties of the solutions of the obtained functional equation are studied. The problem of the existence and construction of a suboptimal autonomous control law with feedback is studied.
- Published
- 2019
35. The l-Problem of Moments for One-Dimensional Integro-Differential Equations with Erdélyi–Kober Operators
- Author
-
S. S. Postnov
- Subjects
Statement (computer science) ,Differential equation ,General Mathematics ,010102 general mathematics ,Type (model theory) ,Optimal control ,01 natural sciences ,010305 fluids & plasmas ,0103 physical sciences ,Applied mathematics ,Order (group theory) ,0101 mathematics ,Linear equation ,Differential (mathematics) ,Mathematics - Abstract
Purpose: to investigate the possibility of statement of l -problem of moments for one-dimensional linear equations of three types, which contain Erdelyi-Kober differential and integral operators of fractional order. Methods: formulation of l -problem of moments for each type of investigated equations, analytical investigation and solution of problem formulated Results. Conditions derived that determine the possibility and solvability of the problem stated. In some cases an explicit solutions of l -problem of moments obtained. Conclusions. The possibility of statement of formulated l -problem of moments shown in cases that defined by conditions obtained in paper. Some analytical solutions of investigated problem obtained.
- Published
- 2019
36. Solutions to Non-linear Euler-Poisson-Darboux Equations by Means of Generalized Separation of Variables
- Author
-
E. Orsingher, R. Garra, and Elina Shishkina
- Subjects
General Mathematics ,010102 general mathematics ,Mathematics::Analysis of PDEs ,Separation of variables ,Poisson distribution ,01 natural sciences ,010305 fluids & plasmas ,symbols.namesake ,Nonlinear system ,0103 physical sciences ,symbols ,Euler's formula ,Applied mathematics ,0101 mathematics ,Algebra over a field ,Mathematics - Abstract
This paper examines solutions to some non-linear equations which generalize well-known equations such as the Euler-Poisson-Darboux equation, the Kolmogorov-Petrovsky-Piskunov equation and telegraph-type equations. The method of generalized separation of variables is here used to derive new exact solutions to these equations.
- Published
- 2019
37. Improving an Estimate of the Convergence Rate of the Seidel Method
- Author
-
A. N. Borzykh
- Subjects
Iterative and incremental development ,General Mathematics ,010102 general mathematics ,Process (computing) ,Stability (learning theory) ,01 natural sciences ,010305 fluids & plasmas ,Algebraic equation ,Rate of convergence ,0103 physical sciences ,Convergence (routing) ,Applied mathematics ,0101 mathematics ,Equivalent system ,Mathematics - Abstract
The Seidel method for solving a system of linear algebraic equations and an estimate of the rate of its convergence are considered in this paper. It is proposed to construct an equivalent system for which the Seidel method also converges but yields a better rate of convergence. An equivalent system is constructed by a separate iterative process, where each step requires O(n) operations. The stability of this process is proved. Results of numerical experiments are presented that show an improvement in the estimate of the convergence rate.
- Published
- 2019
38. On the Rank-One Approximation of Positive Matrices Using Tropical Optimization Methods
- Author
-
Nikolai Krivulin and E. Yu. Romanova
- Subjects
Logarithmic scale ,Optimization problem ,Rank (linear algebra) ,Basis (linear algebra) ,General Mathematics ,010102 general mathematics ,01 natural sciences ,010305 fluids & plasmas ,Matrix (mathematics) ,0103 physical sciences ,Metric (mathematics) ,Idempotence ,Applied mathematics ,Nonnegative matrix ,0101 mathematics ,Mathematics - Abstract
An approach to the problem of rank-one approximation of positive matrices in the Chebyshev metric in logarithmic scale is developed in this work, based on the application of tropical optimization methods. The theory and methods of tropical optimization constitute one of the areas of tropical mathematics that deals with semirings and semifields with idempotent addition and their applications. Tropical optimization methods allow finding a complete solution to many problems of practical importance explicitly in a closed form. In this paper, the approximation problem under consideration is reduced to a multidimensional tropical optimization problem, which has a known solution in the general case. A new solution to the problem in the case when the matrix has no zero columns or rows is proposed and represented in a simpler form. On the basis of this result, a new complete solution of the problem of rank-one approximation of positive matrices is developed. To illustrate the results obtained, an example of the solution of the approximation problem for an arbitrary two-dimensional positive matrix is given in an explicit form.
- Published
- 2019
39. Strict Polynomial Separation of Two Sets
- Author
-
A. V. Plotkin and V. N. Malozemov
- Subjects
Polynomial ,Linear programming ,Plane (geometry) ,Euclidean space ,General Mathematics ,010102 general mathematics ,Zero (complex analysis) ,01 natural sciences ,010305 fluids & plasmas ,0103 physical sciences ,Binary data ,Feature (machine learning) ,Applied mathematics ,0101 mathematics ,Finite set ,Mathematics - Abstract
One of the main tasks of mathematical diagnostics is the strict separation of two finite sets in a Euclidean space. Strict linear separation is widely known and reduced to the solution of a linear programming problem. We introduce the notion of strict polynomial separation and show that the strict polynomial separation of two sets can be also reduced to the solution of a linear programming problem. The objective function of the linear programming problem proposed in this paper has the following feature: its optimal value can be only zero or one, i.e., it is zero if the sets admit strict polynomial separation and one otherwise. Some illustrative examples of the strict separation of two sets on a plane with the use of fourth degree algebraic polynomials in two variables are given. The application efficiency of strict polynomial separation to binary data classification problems is analyzed.
- Published
- 2019
40. Analysis of Dynamic Behavior of Beams with Variable Cross-section
- Author
-
V. V. Saurin
- Subjects
Timoshenko beam theory ,Energy quality ,General Mathematics ,010102 general mathematics ,01 natural sciences ,010305 fluids & plasmas ,Vibration ,symbols.namesake ,0103 physical sciences ,symbols ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Hamiltonian (quantum mechanics) ,Beam (structure) ,Mathematics - Abstract
A formulation of a boundary value problem to find natural frequencies of an inhomogeneous beam in the framework of the Euler–Bernoulli hypotheses are represented. Questions related to various classical variational formulations for a spectral problem arising in the beam theory are discussed. Particularities of the application of the Hamiltonian principles to boundary-value problems are considered. The method of integro-differential relations, which is an alternative to the classical variational approaches is discussed. Various bilateral energy quality estimates for approximate solutions that follow from the method of integro-differential relations are proposed. In the final part of the paper advantages of the variational technique in problems of free vibrations of inhomogeneous beams are discussed based on a numerical example.
- Published
- 2019
41. Stability of Singular Fractional Systems of Nonlinear Integro-Differential Equations
- Author
-
Amele Taieb
- Subjects
Mathematics::Functional Analysis ,Differential equation ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Stability (probability) ,010305 fluids & plasmas ,Nonlinear system ,Schauder fixed point theorem ,0103 physical sciences ,Applied mathematics ,Contraction mapping ,Uniqueness ,0101 mathematics ,Algebra over a field ,Mathematics - Abstract
In this paper, we study singular fractional systems of nonlinear integro-differential equations. We investigate the existence and uniqueness of solutions by means of Schauder fixed point theorem and using the contraction mapping principle. Moreover, we define and study the Ulam-Hyers stability and the generalized Ulam-Hyers stability of solutions. Some applications are presented to illustrate the main results.
- Published
- 2019
42. Mixed Solutions of Monotone Iterative Technique for Hybrid Fractional Differential Equations
- Author
-
Adem Kilicman, Rabha W. Ibrahim, and Faten H. Damag
- Subjects
Fractional differential equations ,Work (thermodynamics) ,Differential equation ,General Mathematics ,Structure (category theory) ,Fixed-point theorem ,01 natural sciences ,010305 fluids & plasmas ,Mathematics - Analysis of PDEs ,0103 physical sciences ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Applied mathematics ,mixed solutions ,0101 mathematics ,Algebra over a field ,Mathematics ,010102 general mathematics ,monotone sequences ,EXISTENCE ,Nonlinear system ,Monotone polygon ,fractional operators ,Mathematics - Classical Analysis and ODEs ,Fractional differential ,Research Subject Categories::MATHEMATICS ,Analysis of PDEs (math.AP) - Abstract
This paper concerns with a mathematical modelling of biological experiments, and its influence on our lives. Fractional hybrid iterative differential equations are equations that interested in mathematical model of biology. Our technique is based on the Dhage fixed point theorem. This tool describes mixed solutions by monotone iterative technique in the nonlinear analysis. This method is used to combine two solutions: lower and upper. It is shown an approximate result for the hybrid fractional differential equations iterative in the closed assembly formed by the lower and upper solutions., 13 pages
- Published
- 2019
43. Analytical Solution of Fractional Burgers-Huxley Equations via Residual Power Series Method
- Author
-
A. A. Freihet and M. Zuriqat
- Subjects
Power series ,Series (mathematics) ,General Mathematics ,010102 general mathematics ,Function (mathematics) ,Residual ,Software package ,01 natural sciences ,Fractional power ,010305 fluids & plasmas ,Nonlinear system ,0103 physical sciences ,Applied mathematics ,0101 mathematics ,Convergent series ,Mathematics - Abstract
This paper is aimed at constructing fractional power series (FPS) solutions of fractional Burgers-Huxley equations using residual power series method (RPSM). RPSM is combining Taylor’s formula series with residual error function. The solutions of our equation are computed in the form of rapidly convergent series with easily calculable components using Mathematica software package. Numerical simulations of the results are depicted through different graphical representations and tables showing that present scheme are reliable and powerful in finding the numerical solutions of fractional Burgers-Huxley equations. The numerical results reveal that the RPSM is very effective, convenient and quite accurate to time dependence kind of nonlinear equations. It is predicted that the RPSM can be found widely applicable in engineering.
- Published
- 2019
44. On One- and Two-Periodic Wave Solutions of the Ninth-Order KdV Equation
- Author
-
L. C. He, J. Pang, and Z. L. Zhao
- Subjects
Basis (linear algebra) ,General Mathematics ,Bilinear form ,01 natural sciences ,010305 fluids & plasmas ,Riemann hypothesis ,symbols.namesake ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,0103 physical sciences ,symbols ,Order (group theory) ,Applied mathematics ,Periodic wave ,010306 general physics ,Korteweg–de Vries equation ,Mathematics - Abstract
In this paper, periodic wave solutions of the ninth-order KdV equation are constructed and expressed explicitily in terms of bilinear forms obtained on the basis of a multidimensional Riemann theta-function. The dynamic futures of these solutions are discussed.
- Published
- 2018
45. Spectral Solution of a Boundary Value Problem for Equation of Mixed type
- Author
-
A. A. Kholomeeva and N. Kapustin
- Subjects
Matching (graph theory) ,General Mathematics ,010102 general mathematics ,Mixed type ,Type (model theory) ,01 natural sciences ,010305 fluids & plasmas ,0103 physical sciences ,Line (geometry) ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Spectral solution ,Spectral method ,Laplace operator ,Mathematics - Abstract
In this paper we apply spectral method to the Gellerstedt problem for Lavrent’ev-Bitsadze equation in a half-strip. We consider Frankl matching condition at the type change line. Using the Darboux solution formulae for mixed type equationinthe hyperbolic subdomainwe reduce problem to an auxiliary problem for the Laplace operator.
- Published
- 2019
46. Modifications of the standard vector Monte Carlo estimate for characteristics analysis of scattered polarized radiation
- Author
-
G. A. Mikhailov, S. A. Rozhenko, and Sergei M. Prigarin
- Subjects
Adaptive algorithm ,Scattering ,business.industry ,General Mathematics ,010102 general mathematics ,Statistical model ,Finite variance ,Radiation ,01 natural sciences ,010309 optics ,Matrix (mathematics) ,Optics ,Monte carlo estimate ,0103 physical sciences ,Radiative transfer ,Applied mathematics ,0101 mathematics ,business ,Mathematics - Abstract
There are two versions of weighted vector algorithms for the statistical modeling of polarized radiative transfer: a “standard” one, which is convenient for parametric analysis of results, and an “adaptive” one, which ensures finite variances of estimates. The application of the adaptive algorithm is complicated by the necessity of modeling the previously unknown transition density. An optimal version of the elimination algorithm used in this case is presented in this paper. A new combined algorithm with a finite variance and an algorithm with a mixed transition density are constructed. The comparative efficiency of the latter is numerically studied as applied to radiative transfer with a molecular scattering matrix.
- Published
- 2017
47. Weighted Zolotarev metrics and the Kantorovich metric
- Author
-
Stanislav V. Shaposhnikov, A. N. Doledenok, and Vladimir I. Bogachev
- Subjects
Mathematics::Functional Analysis ,Mathematical optimization ,High Energy Physics::Lattice ,General Mathematics ,Mathematics::History and Overview ,010102 general mathematics ,Mathematics::Optimization and Control ,Equivalence of metrics ,01 natural sciences ,Computer Science::Discrete Mathematics ,0103 physical sciences ,Metric (mathematics) ,Applied mathematics ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
In this paper we study properties of weighted Zolotarev metrics and compare them with the Kantorovich metric.
- Published
- 2017
48. Boundary-value problems for some higher-order nonclassical differential equations
- Author
-
N. R. Pinigina and A. I. Kozhanov
- Subjects
Differential equation ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,01 natural sciences ,010305 fluids & plasmas ,Elliptic operator ,0103 physical sciences ,Order (group theory) ,Applied mathematics ,Uniqueness ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
The paper consists of two parts. The first part deals with the solvability of new boundary-value problems for the model quasihyperbolic equations (−1)p D t 2p u = Au + f(x, t), where p > 1, for a self-adjoint second-order elliptic operator A. For the problems under study, the existence and uniqueness theorems are proved for regular solutions. In the second part, the results obtained in the first part are somewhat sharpened and generalized.
- Published
- 2017
49. Analogues of Feynman formulas for ill-posed problems associated with the Schrödinger equation
- Author
-
O. G. Smolyanov and V. G. Sakbaev
- Subjects
Cauchy problem ,Well-posed problem ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Cauchy distribution ,01 natural sciences ,010305 fluids & plasmas ,Schrödinger equation ,symbols.namesake ,Operator (computer programming) ,Compact space ,0103 physical sciences ,symbols ,Applied mathematics ,Initial value problem ,Feynman diagram ,0101 mathematics ,Mathematics - Abstract
Representations of Schrodinger semigroups and groups by Feynman iterations are studied. The compactness, rather than convergence, of the sequence of Feynman iterations is considered. Approximations of solutions of the Cauchy problem for the Schrodinger equation by Feynman iterations are investigated. The Cauchy problem for the Schrodinger equation under consideration is ill-posed. From the point of view of the approach of the paper, this means that the problem has no solution in the sense of integral identity for some initial data. The well-posedness of the Cauchy problem can be recovered by extending the operator to a selfadjoint one; however, there exists continuum many such extensions. Feynman iterations whose partial limits are the solutions of all Cauchy problems obtained for various self-adjoint extensions are studied.
- Published
- 2016
50. Empirical estimation of d-risks at distinguishing one-sided hypotheses
- Author
-
D. S. Simushkin
- Subjects
Characteristic function (probability theory) ,General Mathematics ,010102 general mathematics ,Conditional probability ,Estimator ,01 natural sciences ,010305 fluids & plasmas ,Normal distribution ,Rate of convergence ,Consistency (statistics) ,0103 physical sciences ,Statistics ,Prior probability ,Applied mathematics ,0101 mathematics ,Realization (probability) ,Mathematics - Abstract
This paper deals with problem of distinguishing between the two hypotheses H0: θ ≤ 0, H1: θ > 0 based on a fixed volume sample with a normal distribution N(θ, 1), θ ∈ R. It is considered by suppose that the true value θ is a realization of a random value ϑ with some unknown a priori density g(θ). An empirical estimate g(θ) based on the estimate of archive data of prior distribution characteristic function is suggested for the d-risk of the optimal criteria (conditional probability of justice of hypothesis Hj in condition that it is rejected, j = 0, 1). Consistency of empirical estimators of d-risks and appropriate critical values are studied. The rate of convergence is discovered from obtained estimates.
- Published
- 2016
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