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In this paper, a combination of boundary value problems for the nonhomogeneous analytic equations have been studied. The aim of this paper is to find a solution for the Schwarz-(Dirichlet-Neumann) problem and obtain its solvability conditions. [ABSTRACT FROM AUTHOR]

In this paper, we will find expansion formulas for 5 hypergeometric functions of two variables from Horn's list, which includes 34 functions, and for 11 of them, expansion formulas were previously known according to works of Burchnall and Chaundy. To solve the problem, we introduce operators that generalize the well-known Burchnall-Chaundy's operators, study the properties of these new operators, and apply them to finding expansion formulas. At the end of the paper, to give an example, we will show the application of the expansion formula for one of the five functions to determine the order of the singularity at the origin of the fundamental solution used in solving boundary value problem for the multidimensional Helmholtz's equation with one singular coefficient. [ABSTRACT FROM AUTHOR]

In this paper we study an optimal control problem for a linear boundary value problem with strongly degenerate coefficient in the main part of the elliptic operator and with the Nuemann boundary control. Given a cost functional, the objective is to provide the well-posedness analysis of the corresponding optimal control problem and to prove existence of the optimal solutions. [ABSTRACT FROM AUTHOR]

In this paper, we address the issue of specifying Dirichlet boundary conditions (BCs) in higher order Particle Discretization Scheme FEM (HO-PDS-FEM). HO-PDS has its roots in particle-based methods and formulated using variational principle. It approximates functions as the union of local function expansions over the elements of a chosen spatial tessellation and the derivatives over the conjugate tessellation. Unlike ordinary FEM, approximation scheme of HO-PDS-FEM do not possess Kronecker delta property. Therefore, when solving a boundary value problem (BVP) in N-dimension, the given N – 1 dimensional information along the boundary, ∂ Ω, is insufficient to evaluate all the coefficients, which include higher order gradients in N dimension, required for approximating the given BCs with HO-PDS. In previous studies, a curious solution was found to resolve this issue; the coefficients which cannot be evaluated from the N – 1 dimensional boundary value information can be left as unknowns, if the order of basis functions in set Q, which is used for approximating derivatives, is higher than that of the set P, which are used for approximating the unknown function (i.e. |Q| > |P|). In this paper, we shed some light on why this settings, which seems to disagree with calculus, resolves the issue of specifying Dirichlet BCs. Solving BVPs with strong singularities, it is shown that the setting |Q| > |P| produces solutions of expected degree of accuracy and convergence rate. Further, we explore the use of trigono-metric base functions in set Q to improve the accuracy of crack tip stress fields. BVPs with complex cracking patterns such as branching and bendng are also analyzed. Results confirm the applicability of the setting |Q| = |P| + 1 and establish the superiority of HO-PDS-FEM over original PDS-FEM. [ABSTRACT FROM AUTHOR]

In this paper, the problem of optimization of multi-revolutionary transfers of a spacecraft with a fixed angular range of auxiliary longitude and the free transfer time from a given initial orbit to geostationary orbit in a central Newtonian gravitational field is considered. The purpose of the optimization is to calculate the program of control of the vector of the thrust, which provided optimal multi-revolutionary trajectories to the final orbit with the minimum cost of the propellant. The magnitude of thrust is assumed to be constant. The optimized control is the vector control program of thrust, including its direction and periods of operation. For mathematical modeling of the spacecraft motion, differential equations in equinoctial elements are used, and as an independent variable, an auxiliary longitude is used, introduced in such a way that the differential equation for it coincides with the differential equation for true longitude in unperturbed motion. An approach based on the Pontryagin maximum principle is proposed to solve the trajectory optimization problem. Using the maximum principle, the problem of optimizing the multi-revolutionary transfer of spacecraft with limited thrust is reduced to a two-point boundary value problem and the boundary value problem is solved by the method of continuation on parameter. To get the goal, the problems of optimizing the multi-revolutionary transfer with limited power and limited thrust are considered. [ABSTRACT FROM AUTHOR]

Spacecraft insertion into halo orbits around the Luna-Earth libration point L2 is necessary for the lunar exploration programs, especially for solving problems of studying the far side of the Moon by using landing vehicles. The paper proposes a new method for calculating the transfer trajectories to such halo orbits. Considered two flight sachems: conventional transfer and transfer with ballistic capture. A numerical method has been developed for solving the problems. It consists in inserting invariant manifolds of halo orbits and calculating two-impulse trajectories from an initial circular earth orbit to the insertion point of this stable manifold. The problem of transfer orbit calculation is reduced to calculating the transfer trajectory between the LEO and a given entry point to the stable manifold. For solving the boundary value problem, the parameter continuation method is used, which based on Newtonian homotopy between the solutions of the unperturbed and perturbed boundary value problems. And a hybrid algorithm for optimizing the insertion point of the manifold for trajectories with ballistic capture is proposed, the numerical examples are given. [ABSTRACT FROM AUTHOR]

This paper concerns the description of the one-dimensional steady-state flow pattern arising from the injection of gas at supersonic velocity into a duct of uniform or diverging cross-sectional area when back pressure is applied. The considered problem is described by a boundary value problem for a singularly perturbed second order ordinary differential equation with a small parameter. But the solution of such problems is difficult, especially for small values of the parameter. Until today, there is a need to create new effective methods for solving singularly perturbed boundary value problems. The aim of the work is to find a solution to one typical aerodynamics problem. Obtained solutions will allow testing methods of numerical and asymptotic solving of this type boundary value problems. The exact solution has obtained for the constitutive equation in the case of uniform cross-sectional area. The paper found a considered problem general solution, which is divided into three independent branches. On this basis, systems of transcendental equations for finding the exact solution of the considered boundary value problem are given. The possibility of the obtained equations systems solvability and the uniqueness of the boundary value problem solution are discussed. The results obtained are in good agreement with the results of the asymptotic analysis carried out in the works of Chang, Howes, and Crocco. The form of the analytic relations confirms the possible presence of boundary and interior layers on the integral curves of the considered boundary value problem. [ABSTRACT FROM AUTHOR]

One of the most dangerous waste in the world are the End-of-Life tires (EOLT). They are waste that does not practically decompose in nature. Because of this their sound treatment is needed for environment protection. Pyrolysis process is one of the possible methods for such a treatment. Globally, around 23% of all EOLT are processed through pyrolysis, whereas in the Republic of Bulgaria only 5% are processed by pyrolysis. It is clear that for Bulgaria this method still has a good potential for usage, development and further research. The pyrolysis process, used for EOLT treatment, usually is 3D and non-stationary. Thus it is very complicated for modeling and studying. Following [3] an adequate mathematical model of the heat transfer during the pyrolysis process used for the treatment of the End-of-Life tires (EOLT) has been presented in our previous paper [1].There a numerical algorithm in MATLAB for solving the respective mathematical initial and boundary value problems has been also developed. Some results for the temperature field for several characteristic periods of operation of pyrolysis station are presented and commented in this paper [1]. In our next paper [2] we have examined the influence of the heating upon the heat transfer during the pyrolysis process used for EOLT treatment. This paper deals with studying the influence of the heater position upon the heat transfer during EOLT treatment by pyrolysis process. The results for temperature fields, temperature isolines and gradients at some specific moments of time and for two different initial heating functions are graphically presented and commented. Results from this modeling can be used in the real pyrolysis stations for more precise displacement of heathers and measurement devices and for designing of automated management of the process. [ABSTRACT FROM AUTHOR]

This paper examines the methods of similarity and dimensional analysis. The methods in question are the mathematical modeling theoretical basis in the research of the fixed connection contact of car body parts. The paper provides a detailed overview of the basis of the similarity and modeling theories with regard to solving visco-elastic boundary value problems. The mathematical modeling conditions to examine the stress-strain state of car body parts are presented as well. The analytical dependencies obtained make it possible to model the stress-strain state of car body parts under normal operating conditions. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the duality for optimal control problems with higher-order differential inclusions. Using locally adjoint mappings and the discretization method, we derive the conditions of optimality for a boundary-value problem with higher-order differential inclusions. This method connects a duality relation problem to a continuous concerning problem. Then, thanks to the dual operations of addition and infimal convolution of convex functions, we obtain duality results. In general, the construction of the duality problem with the assistance of discrete and discrete approximation problems necessitates a significant amount of effort to comprehend the computational aspects. Finally, we show that the optimal solutions to the primal convex and dual concave problems are the same. [ABSTRACT FROM AUTHOR]

The paper presents a mathematical model of the temperature field of an infinite cylinder, heated by a moving, dispersed, on a finite segment (in an extruder), a heat source. Heating of the cylinder by both internal and external heat sources is considered. The mathematical model is constructed in the form of a nonlinear boundary value problem for the heat equation with conjugation conditions in three combined domains. A method for solving the problem is proposed. It consists in introducing a moving system of dimensionless coordinates. Further, the solution of the problem is reduced to the solution of a system of one-dimensional boundary value problems for ordinary differential equations of the second order with conjugation conditions on the boundaries of the domains of definition of the equations and a regularity condition at infinity. The function of a heat source, internal or external, is written using the Heaviside unit function. In the linear case, an analytical solution is obtained and a graph of the temperature distribution is constructed. The mathematical model, which is built in this work, allows us to describe the temperature distribution in a filament during 3D printing. The temperature distributions of a filament both in the extruder and outside of it have been constructed for various conditions for controlling the process of additive manufacturing of parts. [ABSTRACT FROM AUTHOR]

In this paper some ill-posed boundary-value problems (BVPs) for three - dimensional partial differential equations are studied. The situation with them is rather surprising and there is no general understanding even more than sixty years after their statement given by Murray Protter. These problems are multidimensional analogues of classical BVPs on the plane and intuitively the initial expectations was that their properties would be similar. Unexpectedly, it turned out that unlike the two-dimensional variants, the Protter-Morawetz problems are not well posed. The generalized solution is uniquely determined, but it may have a strong singularity at an isolated boundary point even for infinitely smooth right-hand side. Also, the adjoined problem has an infinite number of smooth solution in the kernel. In the present paper such ill-posed problems for 3-D Gellerstedt equation with lower order terms under multidimensional Protter condition are studied. In addition to new results, we also make a survey of the known results concerning the Protter-Morawetz problems for both Tricomi-type equations and Keldysh-type equations. [ABSTRACT FROM AUTHOR]

End-of-Life tires (EOLT) are ones of the most dangerous waste in the world. They do not practically decompose in nature. Because of this their sound treatment is needed for the environment protection. One of the possible methods for such a treatment is pyrolysis process. It is well known [3, 4] that globally around 23% of all EOLT are processed through pyrolysis, whereas in the Republic of Bulgaria only 5% are processed by this method. Thus for Bulgaria it becomes clear that the pyrolysis method for EOLT treatment still has a good potential for usage, development and further research. This method is very complicated for modeling and studying, because it is 3D and non-stationary. In our previous work [1] we created an adequate mathematical model and numerical algorithm in MATLAB for numerical solving the mathematical initial and boundary value problems for model equations which describe pyrolysis process used for the treatment of the End-of-Life tires (EOLT). Some results for the temperature field for several characteristic periods of operation of pyrolysis station are presented and commented in the papers [1, 2, 5]. This paper deals with studying the influence of two-burner heating upon the heat transfer during EOLT treatment by pyrolysis process. The results for temperature fields, temperature isolines and gradients at some specific moments of time and for two different initial heating functions are graphically presented and commented. Results from this modeling can be used in the real pyrolysis stations for more precise displacement of heathers and measurement devices and for designing of automated management of the process. [ABSTRACT FROM AUTHOR]

In the paper, one parabolic-type boundary value problem is solved for determining the temperature field of the raw cotton and air components in drum dryers. In the proposed model, convective heat transfer is used according to Newton's law and the evaporation of moisture from the components of raw cotton (seeds, fiber) and the influence of air velocity are taken intoaccount. The resulting system of Galerkin's differential equations is solved by the finite-difference method in time. It is shown that the approximate solution is estimated according to Galerkin method in Sobolev space. The numerical results of the considered problem are obtained by the Bubnov–Galerkin method. A comparative analysis is carried out with experimental data. It is shown that the proposed mathematical model and its numerical algorithm adequately describe the drying process of raw cotton. [ABSTRACT FROM AUTHOR]

It is known that one ofathe special cases of integro-differential equations is the so-called differential equations of fractional order. In this paper, we consider a multipoint boundary value problem for an involutively transformed integro-differential equation with a conformable derivative. Using the property of the involutive transformation, the problem is reduced to a mul-tipoint boundary value problem for integro-differential equations. Further, the parameterization method proposed by Profes-sor D. Dzhumabaev is applied to the problem. New parameters are introduced, and based on these parameters, we transfer to new variables. The transition to new variables makes it possible to obtain initial conditions for the equation. The method of successive approximation determines the unique solution of the integral equation. Substituting the obtained solution into the boundary conditions, we obtain a system of linear equations with respect to the introduced parameters. A connection is estab-lished between the reversibility of the matrix of the resulting system and the unique solvability of the original problem. [ABSTRACT FROM AUTHOR]

Distribution function seems be the most full analytical description of any stochastic process. Distribution function for multiple Coulomb scattering for infinite and uniform medium had been obtained at the first half of the last century. We have used it successfully for the dynamics exploration of the beam crossing thin foil normally to its surface. The function mentioned cannot be used directly for the case of the foil installed at some angle to the beam direction of propagation. In mathematical sense, this case is already new problem and new analytical solution is required. To find out the analytical solution for this case, partial differential equation with appropriate boundary conditions must be solved. As it takes place for the majorities of the mathematical physics problems, the exact analytical solution in this specific case is unavailable. We tried to find a different approach searching for approximate solution of boundary problem based on exact solution for infinite scattering. The paper describes the flow lines model for multiple Coulomb scattering that results in analytical explanations of different phenomena taking place in the case of beam incline incidence on the border separating vacuum and scattering medium [ABSTRACT FROM AUTHOR]

In this article, a method and some results of the numerical simulation of the effective elastic properties of core samples are described. The method consists in carrying out a series of stress-strain calculations on a 3D digital model of a core sample. The digital model is a voxel structure obtained from core sample CT scan data. Based on the features of the core sample, a representative volume is determined, on which several static boundary value problems of elasticity with different boundary conditions are solved. A digital core sample model can contain up to one billion voxels and more. Therefore, the relaxation method and the finite difference method (FDM) with explicit time scheme, which do not require a large amount of RAM during the calculation, are used to solve static elasticity problems. In addition, the calculations are parallelized using the CUDA technology for high-performance GPU computing. The effective mechanical properties of the core sample are estimated by averaging over RVE generalized Hooke's law using the results of solving each static problem. The paper presents GPU calculation results for two core samples: sandstone and limestone. The calculations are carried out on full-size core samples. The comparison of results for small core fragments with CAE Fidesys is carried out for verification. [ABSTRACT FROM AUTHOR]

In this paper, oblique wave interaction with a vertical slatted surface-piercing porous screen is studied analytically using the potential flow theory. The water flow through the slatted screen is modeled by considering the inertial and quadratic drag effects. Due to the presence of the quadratic pressure drop condition across the slatted screen, the associated boundary value problem is solved using an iterative eigenfunction expansion method. The study reveals that the wave energy dissipation mainly depends on the two key parameters: the Keulegan-Carpenter number KC and submergence depth of the slatted screen. The slatted screen wave barrier with lower values of KC number dissipates maximum amount of wave energy for a wide range of incident wave frequencies. On the other hand, higher submergence depth of the slatted screen enhances the wave energy dissipation in the long wave regime. [ABSTRACT FROM AUTHOR]

In the paper, we give examples of exact solutions to two boundary value problems for plates stiffened by ribs in the form of explicit expansions in Papkovich–Fadle eigenfunctions. It is assumed that the stiffeners work only in tension-compression, and their flexural rigidity is equal to zero. The unknown expansion coefficients are determined in simple closed form with the help of biorthogonal functions. As an illustration of the method, solutions to even-symmetric boundary value problems are only considered. The final formulas describing the exact solutions are simple and can easily be used in engineering practice. [ABSTRACT FROM AUTHOR]

This paper deals with the mathematical model of a steady flow of a micropolar fluid through a spatially periodic plane profile cascade. The corresponding boundary value problem is reduced to one spatial period. We prove the existence of a weak solution of a coupled problem, with various boundary conditions on the parts of the boundary. Particularly, the condition on the outflow is a variant of the so called “do nothing” boundary condition. [ABSTRACT FROM AUTHOR]

In the work we derive regularized trace formulas which were established in papers of Kanguzhin and Tokmagambetov for the Laplace and m-Laplace operators in a punctured domain with the fixed iterating order m ∊ N. By using techniques of Sadovnichii and Lyubishkin, the authors in that papers described regularized trace formulae in the spatial dimension d = 2. In this note one claims that the formulas are also true for more general operators in the higher spatial dimensions, namely, 2 ≤ d ≤ 2m. Also, we give the further discussions on a development of the analysis associated with the operators in punctured domains. This can be done by using so called 'nonharmonic' analysis. [ABSTRACT FROM AUTHOR]

This paper presents the method of effectiveness NTU for bayonet tube heat exchangers thermal analysis, with uniform conditions along the outer tube wall surface. Using bayonet tube control volume's energy balance, linear fluid temperatures differential equations and boundary conditions are derived at steady state conditions. The governing equations and boundary conditions are expressed in non-dimensional form. The temperature is found to be dependent on four parameters Hurd number (Hu), ratio of outer tube convective heat transfer coefficient(ξ,), number of transfer unit (NTU), and flow path. The method of fourth order Runge- Kutta was used for numerical integration of the coupled linear temperature differential equations and the results are presented graphically for selected values of Hu,ξ, NTU and flow path. The effectiveness is obtained to be dependent on NTU and fluid temperature at the shell side. It was observed from the tubes temperature distribution at a low Hu value, that little heat is exchanged within the tubes. Heat transfer to shell side fluid increases with an increase in ξ value. Maximum heat transfer is achieved by reverse flow path B. [ABSTRACT FROM AUTHOR]

In this paper, the temperature distribution of porous fins is approached by the Akbari-Ganji method using a simple analytical technique. The Darcy model and energy balance were also utilised to create the heat transfer equation. An insulated tip and a limited-length fin serve as a case for exploring thermal performance. The outcomes of the Akbari-Ganji method are compared with previous results. The boundary value is also solved numerically using the MATLAB programme for validation. The outcome indicates that the Akbari-Ganji is more reliable, appropriate, and accurate than other approaches like the perturbation method, the homotopy perturbation method, and the variational iteration method. Additionally, it has been discovered that this approach is a potent mathematical instrument that may solve a wide range of non-linear and linear issues that arise in various branches of technology and science, particularly some thermal transfer equations. It is demonstrated that for solving boundary value problems in engineering applications, the AGM approach is a superior substitute to semi-analytical and conventional numerical methods, mainly when the result's efficiency is crucial. [ABSTRACT FROM AUTHOR]

Integral transforms have been used to solve certain initial and boundary value problems. Fractional Hilbert transform, a generalization of Hilbert transform has relationship with the other classical transforms and fractional transforms. In this paper, we have derived relationship between fractional Hilbert transform with Fourier transform and Fractional Fourier Transform. [ABSTRACT FROM AUTHOR]

This work proposes multistep technique for solving the nonlinear differential equations along with boundary conditions and deals with solution of the problem and stability region analysis of such methods. Some non linear differential equations are illustrated to show the efficacy of the method discussed in this paper. Stability Regions have been analyzed. The errors obtained by the methods discussed are compared to the exact solutions. Thus, the numerical solutions obtained by these methods have high accuracy. [ABSTRACT FROM AUTHOR]

This paper proposes a non-classical theory of stressed plates. Resolving equations are obtained in terms of longitudinal forces of shear forces and internal moments, and boundary conditions are formulated. The obtained solutions simultaneously satisfy the equilibrium equations and the Beltrami-Mitchell stress compatibility equations. A boundary value problem is formulated for a plate under the action of mutually balanced loads. Some applied problems have been solved. [ABSTRACT FROM AUTHOR]

Usually, the complete system of thermoelasticity equations consists of the equation of motion, the Dugamel-Neumann relations, the Cauchy relation, and the heat influx equation. In this case, usually the boundary value problem is reduced to a system of differential equations for displacements and temperature with the corresponding initial and boundary conditions. Coupled thermoelasticity problems can be formulated in both stress and temperature, but this requires the second order of smoothness of the stress tensor. In this regard, there is an additional condition regarding stress. In this paper, the coupled problems of thermoelasticity are proposed to be considered on the basis of the equations of motion, the continuity equation, the Dugamel-Neumann relation, and the heat influx equation. In this case, a system of three interrelated equations for stress, velocity and temperature was obtained. Based on this system, the problem of the stress-strain state of a homogeneous semi-infinite thermoelastic rod under the action of a dynamic thermomechanical load applied to the end of the rod is considered. The exact analytical solution was obtained by the continuation method followed by the application of the Laplace and Fourier integral transformations. The asymptotic behavior of the stress is studied for small and large values of time. The propagation velocity of a thermoelastic wave has also been studied. Numerical calculations are carried out, the qualitative and quantitative nature of the mutual influence of temperature on the distribution of stresses and velocities of the rod sections is analyzed. [ABSTRACT FROM AUTHOR]

The paper proposes a simple approximate method for solving differential equations for boundary value problems. A method is proposed for averaging differential equations over a moving volume, which allows obtaining approximate analytical solutions of differential equations. The control volume is the only one in the considered area of the boundary value problem. In this case, the control volume is considered to be moved in the area under consideration. Based on the averaging of boundary value problems over the volume being moved, an algebraic equation is obtained. When averaging over one of the variables (in the case of a two-dimensional problem), ordinary differential equations are obtained. Examples are given. [ABSTRACT FROM AUTHOR]

In this paper, in the space W 2 (m) (0, 1) (this is the space of functions, the m-th generalized derivative of which is integrable with a square in the interval [0, 1]) we consider the construction of an optimal lattice quadrature formula. Finding the extremal function of the error functional of quadrature formulas, by virtue of the Riesz representation theorem, is reduced to solving a boundary value problem for ordinary differential equations. By solving this boundary value problem, an explicit expression for the extremal function of the quadrature formulas is obtained. In addition, in the Sobolev space W 2 (m) (0, 1) the square of the norm of the error functional of quadrature formulas is explicitly calculated. We obtain a number of interesting and new results that are unattainable in the classical sense. Some theorems are presented and established in [1] theoretically support computational simplifications that reduce costs. [ABSTRACT FROM AUTHOR]

An the paper, we investigate unique solvability of a generalized solution of one semi-periodic boundary value problem for the three-dimensional Tricomi equation in an unbounded parallelepiped, by the methods of "ε–regularization" and a priori estimates using the Fourier transform. [ABSTRACT FROM AUTHOR]

In this paper, the existence of a generalized solution of the investigated boundary-value problem for a nonlinear equation of mixed type with two lines of degeneration in the weighted S. L. Sobolev space is proved. A particular case of an equation is given, in which a generalized solution exists in a weightless S. L. Sobolev space. Examples of functions, satisfying the conditions of the lemmas and theorems on the solvability of the problem, are constructed. [ABSTRACT FROM AUTHOR]

Process of propagating one-dimensional deformations in a heteromodular isotropic elastic half-space under nonstationary boundary loading is under consideration. Nonstationary boundary problems are solved with using an algorithm based on piecewise linear approximation of nonlinear boundary conditions. It is shown that this approach preserves the physical nonlinearity of model and at the same time allows us to pass to a related sequence of simpler problems with analytical solutions; all these solutions together are an approximation of a continuous solution of the original problem. Earlier the question concerned with freedom of choice of the number and position of the nodes of the approximating function was investigated for various forms of boundary conditions. The paper presents the solution of one-dimensional boundary value problem for nonstationary loading the heteromodular half-space in the "tension – compression – stop" mode. It is shown that the solution can have several non-intersecting branches due to the various positions of nodes in the piecewise linear function of boundary motion. [ABSTRACT FROM AUTHOR]

This paper deals with the numerical investigation of ellipticity of the boundary value problem for isothermal finite strain elasto-plasticity. Ellipticity can be lost when softening occurs. A discontinuity surface then appears in the considered material body and this is associated with the ill-posedness of the boundary value problem. In the paper the condition for ellipticity loss is derived using the deformation gradient and the first Piola-Kirchhoff stress tensor. Next, the obtained condition is implemented and numerically tested within symbolic-numerical tools AceGen and AceFEM using the benchmark of an elongated rectangular plate with imperfection in plane stress and plane strain conditions. [ABSTRACT FROM AUTHOR]

In the paper spectral properties of non-local boundary value problem for an equation of the parabolic-hyperbolic type is investigated. The non-local condition binds the solution values at points on boundaries of the parabolic and hyperbolic parts of the domain with each other. This problem was first formulated by T.Sh. Kal'menov and M.A. Sadybekov. They proved the unique strong solvability of the problem. One special case of this problem was investigated in more detail in the work of G. Dildabek. A boundary value problem for the heat equation with conditions of the Samarskii-Ionlin type arises in solving this problem. In this paper, we show in what case this boundary value problem does not have eigenvalues. [ABSTRACT FROM AUTHOR]

This paper deals with method of B-spline solution for a system of boundary value problems. The differential equations are useful in various fields of science and engineering. Some interesting real life problems involve more than one unknown function. These result in system of simultaneous differential equations. Such systems have been applied to many problems in mathematics, physics, engineering etc. In present paper, B-spline and B-spline with free parameter methods for the solution of a linear system of second-order boundary value problems are presented. The methods utilize the values of cubic B-spline and its derivatives at nodal points together with the equations of the given system and boundary conditions, ensuing into the linear matrix equation. [ABSTRACT FROM AUTHOR]

This paper contains a solution to a boundary value problem for an inhomogeneous wave equation in one space dimension containing a Riemann–Liouville fractional derivative with respect to a spatial variable. In solving the problem, a stationary state is distinguished, written in terms of a functions of the Mittag - Leffler type. The case is considered when the values of the required function at the ends of the interval do not coincide. This model is used to describe oscillation processes in a viscoelastic medium, in particular changes in the deformation-strength characteristics of polymer concrete (dian and dichloroanhydride-1,1-dichloro-2,2-diethylene) under constant external influence. Based on the obtained solution of the boundary value problem, the article presents two numerical examples calculated using MATLAB, high-level language for technical calculations. The initial position in the conditions of the examples is described by a linear function. The graphs of the found solutions were constructed. [ABSTRACT FROM AUTHOR]

This paper considers the construction of a unified computational scheme for solving boundary value problems of static and dynamic calculation of flexible circular solid and annular plates using the nonlinear Lyava theory [1], the development of an algorithm for dynamic and static calculation of flexible circular plates and testing of the constructed automated system and show the convergence rates of the method finite differences, to investigate the stress-strain state of flexible round plates. Based on the results obtained, it will build their graphs and calculate the calculated values. Compare them according to linear and nonlinear theories. [ABSTRACT FROM AUTHOR]

This paper deals with several boundary value problems (bvp) for the biharmonic operator in the unit ball in Rn, n ≥ 2. Some of them satisfy respectively violate Shapiro-Lopatinskii conditions. Depending on the structure of the boundary differential operators the corresponding bvp are either of Fredholm type or not. By putting some extra conditions we obtain in the latter case Fredholm bvp. In both cases subelliptic estimates for the solutions with loss of regularity 2 - 1/k+1, k ∈ N are obtained. At the end of the paper degenerate Steklov bvp is studied and subelliptic estimate with loss of regularity 2 is found. Certainly, it also violates Shapiro-Lopatinskii conditions. [ABSTRACT FROM AUTHOR]

The paper presents the analysis of the results of measurements performed on the testing ground at the Military Institute of Armour and Automotive Technology using an explosive detonated under a vehicle. The paper concerns the analysis of an injury risk in the context of significant short-term accelerations caused by explosive charges of various weights. The aim of the analysis was to assess the risk of micro-injuries in soft tissues, induced indirectly by the blast, using a numerical analysis. The measured parameters, such as acceleration and forces generated in the Hybrid III dummy, were assumed as the boundary conditions. [ABSTRACT FROM AUTHOR]

This paper presents an investigation of the effect of boundary conditions in experimental modal analysis to approximate the free-free boundary conditions in the experimental modal analysis of a slender beam. Experimental modal analysis and numerical simulation of normal mode analysis were performed to obtain the modal parameters and predicted results in term of the natural frequencies and mode shapes. Two different boundary conditions have been considered in this paper which are the suspension location at the end points of the beam and at nodal points of the beam. The experimental results indicate that there is uncorrelated pattern for the type of mode (bending and torsional modes) for the suspension location at the end points of the beam. For the conclusion, the suspension at the location of the nodal points shows a better correlation in term of mode shapes and MAC values in comparison to the end points suspension location. [ABSTRACT FROM AUTHOR]

The paper considers the flow of micropolar compressible and heat conductive fluid between two parallel plates and discusses the problem of local existence of the solution for the corresponding initial boundary value problem. [ABSTRACT FROM AUTHOR]

During the past decade, our society has become dependent on advanced mathematics for many of our daily needs. Mathematics is at the heart of the 21st century technologies and more specifically the emerging imaging technologies from thermoacoustic tomography and ultrasound computed tomography to nondestructive testing. All of these applications reconstruct the internal structure of an object from external measurements without damaging the entity under investigation. Very often the basic mathematical idea common to such reconstruction problems is based upon integral geometry. In this paper considers the problem of recovering a function from families of spheres in space. The uniqueness of the solution of the problem is proved by reducing it to the Volterra integral equation of the first and then the second kind. The methods of the theory of partial differential equations are applied. The proof of the uniqueness theorem is based on the researching of boundary value problems for auxiliary functions. Fourier transform methods are also used. Uniqueness theorems are proved for some new classes of operator equations of Volterra type in three-dimensional space. [ABSTRACT FROM AUTHOR]

In this research paper developed a three-dimensional mathematical model of the process of gas filtration in porous media in the presence of a mass transfer through the boundaries and a numerical algorithm for solving the stated initial-boundary value problem. The developed model is described using a non-linear partial differential equation with appropriate initial and boundary conditions. The proposed mathematical apparatus makes it possible to carry out hydrodynamic calculations, taking into account changes in the main factors affecting the process under consideration: permeability, porosity and thickness of formations, gas recovery coefficient, viscosity, etc. A numerical algorithm has been developed to solve the problem based on difference schemes of the second order of approximation in time and coordinates, which provide the ability to solve practical problems of analyzing and predicting the gas production process under various conditions of impact on the productive reservoir, as well as making decisions on the development of existing and design of new gas fields. Meanwhile, the use of three-dimensional models makes it possible to more accurately describe the processes occurring in reservoir porous media. Therefore, the purpose of this study is to develop a three-dimensional mathematical model of gas filtration in porous media in the presence of a mass transfer through the boundaries, as well as a numerical algorithm for solving the problem. [ABSTRACT FROM AUTHOR]

The paper is devoted to solving the Neumann boundary value problem for the Poisson equation in an elliptic body. In this case, heat transfer occurs under boundary conditions of the second kind. Based on the methods of differentiation and integration, a solution was obtained to the problem of the distribution of the temperature field of the body under study. The resulting solution has an analytical form containing hypergeometric and hyperbolic functions. The reliability of the obtained result is confirmed by the fact that the general solution of the problem coincides with the solution obtained in one of the author's works for the case of a temperature field distribution in a body with an elliptical cross section of infinite length under boundary conditions of the third kind. [ABSTRACT FROM AUTHOR]

The paper considers a model problem of the thermal diffusion process in a silicon wafer. The mathematical model of this process is an initial boundary value problem for a system of linear partial differential equations of parabolic type. In this system, one equation describes the process of heat propagation in silicon in the presence of internal heat sources, and the other describes the process of diffusion of impurities in it. Moreover, these equations are related in the same way that the diffusion coefficient of an impurity depends on temperature. A special form of the volumetric heat release function was considered, which made it possible in this particular case to write down an explicit solution in the form of a Fourier series. To find an approximate solution to the boundary value problem, an implicit difference scheme and the classical sweep method are used. A computational experiment is presented. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to introduce interval two-points fuzzy boundary value problems, in which the fuzziness course when the coefficients of the governing ordinary differential equation and/or the boundary conditions includes fuzzy numbers of either triangular or trapezoidal types. Such equations will be solved by introducing the concept of α – level sets, α ∈ [0,1] to treat the fuzzy ordinary differential equation into two nonfuzzy ordinary differential equations, which are corresponds to the lower and upper solutions of the interval fuzzy solution. The shooting method is applied to solve the resultant equations in both cases, linear and nonlinear. [ABSTRACT FROM AUTHOR]

In this paper, we consider the construction of an optimal lattice quadrature formula in the space W 2 (m) (0, 1). Further, finding the extremal function of the error functional of quadrature formulas, by virtue of the Riesz theorem on the general form of a linear continuous functional, is reduced to solving a boundary value problem for ordinary differential equations. Solving this boundary value problem, we obtain an explicit expression for the extremal function of the error functional of quadrature formulas. In addition, by virtue of the Riesz theorem, the square of the norm of the error functional is explicitly calculated and an analytical expression is obtained for the optimal coefficients of the quadrature formulas at m = 2. [ABSTRACT FROM AUTHOR]

This paper uses the Ritz method to find approximate solutions to boundary-value problems for ordinary differential equations. We present the approximate finite element solution by a linear combination of the basis functions. Here we get co-efficients of the optimal interpolation formula in W 2 (1 , 0) space as basis functions. The difference obtained in numerical results and exact solutions illustrates graphically. [ABSTRACT FROM AUTHOR]

In this paper, we study nonlocal boundary value problems for a third-order equation with an elliptic-hyperbolic operator in the principal part. Existence and uniqueness theorems for the classical solution of the stated problems are proved. The proofs of the theorem are based on energy identities, as well as on the theory of singular and Fredholm integral equations. [ABSTRACT FROM AUTHOR]

In this paper we will discuss the theoretical analysis for Bickley's method and numerical procedure for getting the approximate solution. If (sm, ck) denote the class of spline functions consisting of piecewise polynomial of degree m, joined at "knots" with at-least k continuous derivatives (k < m) then spline functions in (Sm, cm-1) are used as approximating function for two point boundary value problem in the second order differential equations. The result shows that the presented cubic spline function which interpolates the lacunary data was efficient and effective for solving such problems. [ABSTRACT FROM AUTHOR]