System Of Integro-differential Equations Research Articles

Effect of non-stationary external forces on vibrations of composite pipelines conveying fluid

The effect of non-stationary external forces on the vibration of pipelines made of composite materials is investigated in the paper. A mathematical model of composite pipeline vibration is developed, considering the viscosity properties of the structure and pipeline base material, axial forces, internal pressure, resistance forces, and external disturbances. A mathematical model of viscoelastic pipelines conveying fluid under vibrations is constructed based on the Boltzmann-Volterra integral model. The mathematical model to study a pipeline is based on the Euler-Bernoulli beam theory. Considering the physicomechanical properties of the pipeline material, the mathematical model of the problems under consideration presents a system of integro-differential equations (IDE) in partial derivatives with corresponding initial and boundary conditions. The nonlinear partial differential equations, obtained using the Bubnov-Galerkin method under considered boundary conditions, are reduced to solving the system of ordinary integro-differential equations. A computational algorithm is developed based on eliminating features of integro-differential equations with weakly singular kernels, followed by using quadrature formulas.

A best proximity point theorem for C-class-proximal non-self mappings and applications to an integro-differential system of equations

In this paper, we propose a best proximity point theorem for a novel class of non-self-mappings by using the definition of a pair (F , h) of upper class of type II and the concept of C-class functions. Several consequences (including the case of self-mappings) of our obtained results are suggested. We support our obtained results by concrete examples. In the end, we consider an integro-differential system of equations. We ensure the existence of an optimal solution, which turns to be an exact solution of the system when boundary conditions are equal.

THE OPTIMAL CONTROL PROBLEM FOR SYSTEMS OF INTEGRO-DIFFERENTIAL EQUATIONS ON THE HALF-AXIS

This article is devoted to exploring the optimal control problem for a system of integro-differential equations on the infinite interval. Sufficient conditions for the existence of optimal controls and trajectories have been obtained in terms of right-hand sides and the quality criterion function. Integro-differential equation systems are the mathematical models for many natural science processes, such as those in fluid dynamics and kinetic chemistry, among others. Many of these equations have the control that minimizing specific functionals related to the dynamics of these processes. This work specifically focuses on deriving sufficient optimality conditions for integro-differential systems on the half-axis. The complexity of the research is in the following aspects: Firstly, the problem at hand involves optimal control with an infinite horizon, which makes the direct application of compactness criteria like the Arzela-Ascoli theorem impossible. Secondly, the problem is considered up to the moment $\tau$ when the solution reaches the boundary of the domain. This reach moment depends on the control $\tau = \tau(u)$. Hence, the solution to the problem is essentially represented by the triplet $(u^*, x^*, \tau^*)$ — the optimal control, the optimal trajectory, and the optimal exit time. We note that a particular case of this problem is the problem of optimal speed. The main idea of proving the existence of an optimal solution relies on a compactness approach and involves the following steps: identifying a weakly convergent minimizing sequence of admissible controls, extracting a strongly convergent subsequence of corresponding trajectories, and justifying boundary transitions in equations and the quality criterion. The work provides a problem statement, formulates, and proves the main result.

Bending-torsional vibrations of aircraft wing

The problems of bending-torsional flutter of an aircraft wing are considered. Using equations of motion by the Bubnov-Galerkin method, based on the polynomial approximation of deflections and angle of twist, the problem is reduced to studying a system of ordinary integro-differential equations, where the independent variable is time. The solution of integro-differential equations with singular kernels is found by a numerical method based on quadrature formulas. The influence of physical-mechanical and geometric parameters on the flutter of the aircraft wing was studied. It was established that an account for the viscoelastic properties of the material of thin-walled aircraft structures leads to a 40-60% decrease in the critical flutter velocity. It is shown that an increase in the elongation parameter of an aircraft wing leads to a decrease in the flutter velocity.

On a common fixed point theorem in vector-valued $ b $-metric spaces: Its consequences and application

<abstract><p>We introduce a Ćirić type contraction principle in a vector-valued $ b $-metric space that generalizes Perov's contraction principle. We investigate the possible conditions on the mappings $ W, E:G\rightarrow G $ ($ G $ is a non-empty set), for which these mappings admit a unique common fixed point in $ G $ subject to a nonlinear operator $ {\bf F}:\mathbb{P}^{m} \rightarrow \mathbb{R}^{m} $. We illustrate the hypothesis of our findings with examples. We consider an infectious disease model represented by the system of delay integro-differential equations and apply the obtained fixed point theorem to show the existence of a solution to this model.</p></abstract>

Existence, uniqueness and stability results for coupled systems of fractional integro-differential equations with fixed and nonlocal anti-periodic boundary conditions

In this paper, we investigate the existence and uniqueness of solutions for a new class of coupled system of two Caputo fractional derivatives of different orders and a Riemann- Liouville type integral with fixed and nonlocal anti-periodic boundary conditions, by using the fixed point approach in generalized metric spaces. The Ulam’s type stability of the proposed coupled system is also studied. An example is provided to illustrate the obtained theory.

Dynamic buckling of plate made of glass reinforced plastic under rapidly increasing shear load

The research object of this work is a clamped rectangular plate made of glass-reinforced plastic. The dynamic problem of stability of the plate under rapidly increasing shear load is considered. Within the Kirchhoff–Love hypothesis framework, a mathematical model was built in a geometrically nonlinear formulation. By the Bubnov–Galerkin method, based on a polynomial approximation of the deflection, the problem was reduced to solving systems of nonlinear ordinary integro-differential equations. With a weakly singular Koltunov–Rzhanitsyn kernel with variable coefficients, the resulting system was solved by a numerical method based on quadrature formulas. The plate’s dynamic behavior was investigated depending on the plate’s geometric and physic parameters. The importance of considering the viscoelastic properties of the material is shown.

Nonlinear vibrations of earth structures

This article offers a detailed analysis of the current state of plane structure dynamics, addressing the complexities posed by non-homogeneous materials exhibiting nonlinear elastic and viscoelastic properties during real-life operations. The study proposes a comprehensive mathematical model and algorithm to investigate the dynamic behavior of such structures, employing the non-linear hereditary Boltzmann-Volterra theory to describe viscoelastic material properties accurately. Nonlinear oscillatory systems are analyzed using Lagrange equations based on the d’Alembert principle. The problem is approached through a multi-step process. Initially, the linear elastic problem of the structure’s natural oscillations is solved to determine its natural frequencies and modes of oscillations. Subsequently, these eigenmodes are employed as coordinate functions to address forced nonlinear oscillations in viscoelastic non-homogeneous systems. The complexity of the problem necessitates solving a Cauchy problem comprising a system of nonlinear integrodifferential equations. To illustrate the methodology, the study examines the Gissarak earth dam, considering real operational conditions and nonlinear, viscoelastic material properties near resonant modes of vibrations. Utilizing numerical methods, the dynamic behavior of the structure is analyzed, assessing displacements and stress components at different time points under non-stationary kinematic action. Stress concentration regions within the structure are identified for resonant vibrations, allowing the evaluation of its strength. Furthermore, the impact of nonlinear elasticity and viscoelasticity on the structural dynamics is quantified. This research provides valuable insights into the behavior of plane non-homogeneous structures, considering real-world scenarios and material complexities, ultimately contributing to an improved understanding of structural dynamics and facilitating the identification and mitigation of potential structural challenges.

Properties of Degenerate Systems of Linear Integro-Differential Equations and Initial Value Problems for These Equations

Properties of Degenerate Systems of Linear Integro-Differential Equations and Initial Value Problems for These Equations

Вплив поздовжньої і поперечної теплопровідності міжфазних щілин на ефективні параметри біматеріалу

The effective parameters of the bi-material with a periodic system of interfacial cracks are studied, taking into account their longitudinal and transverse thermal conductivity. The bi-material is subjected to tensile forces and uniform heat flow. The transverse and longitudinal thermal conductivity of the cracks is taken into account by the thermal resistance of the filler and the thermal conductivity of the surface films, respectively. The thermal resistance of the filler is directly proportional to the opening of the cracks and inversely proportional to the thermal conductivity of the filler. Thermal conductivity of surface films does not change under the influence of load. The thermo-elastic problem is reduced to nonlinear systems of singular integro-differential equations for an opening cracks and a temperature jump between the cracks faces. An analytical-numerical iterative procedure for solving this system is proposed. Based on the obtained solution, the effective temperature jump and the effective thermal resistance of the bi-material are determined. The dependences of the effective parameters of the bi-material on the applied load and thermal conductivity coefficients of the filler and the surface films of the cracks are analyzed.

Natural and forsed osculations of pipelines in contact with the Wincler medium

In this paper, the pipeline is modeled as a curved rod in contact with the Winkler medium. Linear oscillations of a curved viscoelastic rod lying on the Winkler base are considered. The general formulation of the problem of free oscillations of a spatially curved viscoelastic rod with variable parameters is reduced to a boundary value problem for a system of ordinary integro-differential equations of the 12th order with variable coefficients relative to eigenstates; it can be solved by the method of successive approximations. The relations allowing to present the solution of the boundary value problem for the rod in an analytical form are formulated. It is established that the dimensionless complex frequencies of natural oscillations of a spatially curved rod, while maintaining the elongation of the rod constant, do not depend on it. The Poisson's ratio has little effect on the dimensionless real and imaginary parts of the natural frequencies.

Numerical simulation of tasks of vertical viscoelastic oscillation of suspension systems of frame, engine, cabin and seat

The analysis of the structures of the springing systems of modern vehicles showed that polymer composites are used in vehicles for which the decrease in curb weight is a critical indicator. The use of springs (suspensions) from polymer composites makes it possible to reduce the weight of the elastic element and increase durability while improving the smoothness of movement, reducing noise, and increasing traffic safety. When replacing the steel suspension with a polymer composite suspension, the mass of unsprung parts of the car decreases, and the economic performance of wheeled vehicles improves. The article discusses the problem of vertical viscoelastic oscillations of the suspension systems of the skeleton, engine, cabin, and seat. When considering the suspension's rheological properties, the Boltzmann-Volterra integral model is used. The Koltunov-Rzhanitsyn nucleus is used as a nucleus, which has weakly singular features of the Abel type. The system of integro-differential equations describing the studied process is solved by a method based on the use of quadrature formulas. The effect of the parameters of the hereditary-deformable properties of the suspension on the vibration shape is shown.

An epidemic model with time delays determined by the infectivity and disease durations.

We propose an epidemiological model with distributed recovery and death rates. It represents an integrodifferential system of equations for susceptible, exposed, infectious, recovered and dead compartments. This model can be reduced to the conventional ODE model under the assumption that recovery and death rates are uniformly distributed in time during disease duration. Another limiting case, where recovery and death rates are given by the delta-function, leads to a new point-wise delay model with two time delays corresponding to the infectivity period and disease duration. Existence and positiveness of solutions for the distributed delay model and point-wise delay model are proved. The basic reproduction number and the final size of the epidemic are determined. Both, the ODE model and the delay models are used to describe COVID-19 epidemic progression. The delay model gives a better approximation of the Omicron data than the conventional ODE model from the point of view of parameter estimation.

‘Far interaction’ of small spectral perturbations of the Neumann boundary conditions for an elliptic system of differential equations in a three-dimensional domain

A formally selfadjoint system of second-order differential equations is considered in a three-dimensional domain on small parts of whose boundary an analogue of Steklov spectral conditions is set, while the Neumann boundary conditions are set on the rest of the boundary. Under certain algebraic and geometric conditions an asymptotic expression for the eigenvalues of this problem is presented and a limiting problem is put together, which produces the leading asymptotic terms and involves systems of integro-differential equations in half-spaces, interconnected by means of certain integral characteristics of vector-valued eigenfunctions. One example of a concrete problem in mathematical physics describes surface waves in several ice holes made in the ice cover of a water basin, and the asymptotic formula for eigenfrequencies shows that the local wave processes interact independently of the distance between the holes. Another series of applied problems relates to elastic fixings of bodies along small pieces of their surfaces. Possible generalizations are discussed; a number of related open questions are stated. Bibliography: 41 titles.

Numerical solution of systems of fractional order integro-differential equations with a Tau method based on monic Laguerre polynomials

In this paper, numerical technique based on monic Laguerre polynomials is proposed to obtain approximate solutions of initial value problems for systems of fractional order integro-differential equations (FIDEs). Operational fractional integral matrix is constructed. This operational matrix is applied together with the monic Laguerre Tau method to solve systems of FIDEs. This systems of FIDEs will be transformed into a system of algebraic equations which can be solved easily. Numerical results and comparisons with other methods are also presented to show the efficiency and applicability of the proposed method.

On Fractional Order Model of Tumor Growth with Cancer Stem Cell

This paper generalizes the integer-order model of the tumour growth into the fractional-order domain, where the long memory dependence of the fractional derivative can be a better fit for the cellular response. This model describes the dynamics of cancer stem cells and non-stem (ordinary) cancer cells using a coupled system of nonlinear integro-differential equations. Our analysis focuses on the existence and boundedness of the solution in correlation with the properties of Mittag-Leffler functions and the fixed point theory elucidating the proof. Some numerical examples with different fractional orders are shown using the finite difference scheme, which is easily implemented and reliably accurate. Finally, numerical simulations are employed to investigate the influence of system parameters on cancer progression and to confirm the evidence of tumour growth paradox in the presence of cancer stem cells.

A New Projection Method for a System of Fractional Cauchy Integro-Differential Equations via Vieta–Lucas Polynomials

This work presents a projection method based on Vieta–Lucas polynomials and an effective approach to solve a Cauchy-type fractional integro-differential equation system. The suggested established model overcomes two linear equation systems. We prove the existence of the problem’s approximate solution and conduct an error analysis in a weighted space. The theoretical results are numerically supported.

Inverse Problem for Viscoelastic System in a Vertically Layered Medium

In this paper, we consider a three-dimensional system of first-order viscoelasticity equations written with respect to displacement and stress tensor. This system contains convolution integrals of relaxation kernels with the solution of the direct problem. The direct problem is an initial-boundary value problem for the given system of integro-differential equations. In the inverse problem, it is required to determine the relaxation kernels if some components of the Fourier transform with respect to the variables $x_1$ and $x_2$ of the solution of the direct problem on the lateral boundaries of the region under consideration are given. At the beginning, the method of reduction to integral equations and the subsequent application of the method of successive approximations are used to study the properties of the solution of the direct problem. To ensure a continuous solution, conditions for smoothness and consistency of initial and boundary data at the corner points of the domain are obtained. To solve the inverse problem by the method of characteristics, it is reduced to an equivalent closed system of integral equations of the Volterra type of the second kind with respect to the Fourier transform in the first two spatial variables $x_1$, $x_2$, for solution to direct problem and the unknowns of inverse problem. Further, to this system, written in the form of an operator equation, the method of contraction mappings in the space of continuous functions with a weighted exponential norm is applied. It is shown that with an appropriate choice of the parameter in the exponent, this operator is contractive in some ball, which is a subset of the class of continuous functions. Thus, we prove the global existence and uniqueness theorem for the solution of the stated problem.

The adhesive contact problem for a piecewise-homogeneous orthotropic plate with an elastic patch

A piecewise-homogeneous elastic orthotropic plate, reinforced with a finite patch of the wedge-shaped, which meets the interface at a right angle and is loaded with tangential and normal forces is considered. Using methods of the theory of analytic functions, the problem is reduced to the system of singular integro-differential equations (SIDE) with fixed singularity. Under tension-compression of patch using an integral transformation a Riemann problem is obtained, the solution of which is presented in explicit form. The tangential contact stresses along the contact line are determined and their asymptotic behavior in the neighborhood of singular points is established.

NON-AUTONOMOUS FRACTIONAL EVOLUTION EQUATIONS WITH NON-INSTANTANEOUS IMPULSE CONDITIONS OF ORDER (1,2): A CAUCHY PROBLEM

The non-instantaneous condition is utilized in our study through the employment of the Cauchy problem in order to contract a system of nonlinear non-autonomous mixed-type integro-differential (ID) fractional evolution equations in infinite-dimensional Banach spaces. We reveal the existence of new mild solutions in the condition that the nonlinear function modifies approximately suitable, measure of non-compactness (MNC) form and local growth form using evolution classes along with fractional calculus (FC) theory as well as the fixed-point theorem with respect to k-set-contractive operator and MNC standard set. Consequently, as an example, we consider a fractional non-autonomous partial differential equation (PDE) with a homogeneous Dirichlet boundary condition and a non-instantaneous impulse condition. The conclusion of mild solution regarding the uniqueness and existence of a mild solution for a system with a probability density function and evolution classes is drawn with respect to the related domains.