System Of Nonlinear Integro-differential Equations Research Articles

On solvability of one class of third order differential equations

UDC 517.9 One-dimensional mixed problem for one class of third order partial differential equation with nonlinear right-hand side is considered. The concept of generalized solution for this problem is introduced. By the Fourier method, the problem of existence and uniqueness of generalized solution for this problem is reduced to the problem of solvability of the countable system of nonlinear integro-differential equations. Using Bellman's inequality, the uniqueness of generalized solution is proved. Under some conditions on initial functions and the right-hand side of the equation, the existence theorem for the generalized solution is proved using the method of successive approximations.

A method of solving a nonlinear boundary value problem for the Fredholm integro-differential equation

We propose a method to solve a nonlinear boundary value problem for a Fredholm integro-differential equation on a finite interval. By introducing additional parameters chosen as the values of the solution at the left-end points of the partition subintervals, the problem under consideration is transformed into an equivalent boundary value problem for a system of nonlinear integro-differential equations with parameters on the subintervals. For fixed parameters, we obtain a special Cauchy problem for this system, which is represented as a nonlinear operator equation and solved by an iterative method. By substitution of the solution to the special Cauchy problem with parameters into the boundary condition and the continuity conditions of the solution to the original problem at the interior partition points, we construct a system of nonlinear algebraic equations in parameters. It is proved that the solvability of this system provides the existence of a solution to the original boundary value problem. The algorithm for solving the special Cauchy problem includes two auxiliary problems: the Cauchy problems for ordinary differential equations and the evaluation of definite integrals. The accuracy of the method that we propose to solve the boundary value problem depends on the accuracy of methods applied to the auxiliary problems and does not depend on the number of the partition subintervals. Since iterative methods are used to solve both the constructed system of algebraic equations and the special Cauchy problem, we offer an approach to find an initial guess for the solutions to these problems.

Регуляризованные асимптотические решения интегродифференциальных уравнений с быстрыми и медленными переменными

The article considers a nonlinear integro-differential system of equations with fast and slow variables. Such systems were not considered previously from the point of view of constructing regularized (according to Lomov) asymptotic solutions. The known studies were mainly devoted to construction of the asymptotics of the Butuzov-Vasil'eva boundary layer type, which, as is known, can be applied only if the spectrum of the first variation matrix (on the degenerate solution) is located strictly in the open left-half plane of a complex variable. If the spectrum of this matrix falls on the imaginary axis, the S.A. Lomov regularization method is commonly used. However, this method was mainly developed for singularly perturbed differential systems that do not contain integral terms, or for integro-differential problems without slow variables. In this article, the regularization method is generalized for two-dimensional integro-differential equations with fast and slow variables.

Existence and uniqueness of solutions for the system ofintegro-differential equations with three-point and nonlinear integral boundary conditions

The paper examines a system of nonlinear integro-differential equations with three-point and nonlinear integral boundary conditions. The original problem demonstrated to be equivalent to integral equations by using Green function. Theorems on the existence and uniqueness of a solution to the boundary value problems for the first order nonlinear system of integro- differential equations with three-point and nonlinear integral boundary conditions are proved. A proof of uniqueness theorem of the solution is obtained by Banach fixed point principle, and the existence theorem then follows from Schaefer’s theorem.

Dynamic stability of thin-walled structure elements considering hereditary and inhomogeneous properties of the material

The paper is devoted to the study of nonlinear problems of the dynamics of thin-walled structures considering hereditary and inhomogeneous properties of the material. To describe the processes of strain in viscoelastic materials, the Boltzmann-Volterra integral model with weakly singular hereditary kernels was used. Using the Bubnov-Galerkin method, the problem under consideration was reduced to solving the systems of nonlinear integrodifferential equations considering hereditary and inhomogeneous properties of the material and the radius of curvature of the structure. A computational algorithm was developed based on the elimination of the features of integrodifferential equations with weakly singular kernels, followed by the use of quadrature formulas. The results of calculating a thin-walled structure with hereditary and inhomogeneous properties of the material streamlined by a gas flow are presented. Solutions are obtained in the form of graphs.

Octagon at finite coupling

We study a special class of four-point correlation functions of infinitely heavy half-BPS operators in planar mathcal{N} = 4 SYM which admit factorization into a product of two octagon form factors. We demonstrate that these functions satisfy a system of nonlinear integro-differential equations which are powerful enough to fully determine their dependence on the ’t Hooft coupling and two cross ratios. At weak coupling, solution to these equations yields a known series representation of the octagon in terms of ladder integrals. At strong coupling, we develop a systematic expansion of the octagon in the inverse powers of the coupling constant and calculate accompanying expansion coefficients analytically. We examine the strong coupling expansion of the correlation function in various kinematical regions and observe a perfect agreement both with the expected asymptotic behavior dictated by the OPE and with results of numerical evaluation. We find that, surprisingly enough, the strong coupling expansion is Borel summable. Applying the Borel-Padé summation method, we show that the strong coupling expansion correctly describes the correlation function over a wide region of the ’t Hooft coupling.

Numerical Approach to a Nonlocal Advection-Reaction-Diffusion Model of Cartilage Pattern Formation

We propose a numerical approach that combines a radial basis function (RBF) meshless approximation with a finite difference discretization to solve a nonlinear system of integro-differential equations. The equations are of advection-reaction-diffusion type modeling the formation of pre-cartilage condensations in embryonic chicken limbs. The computational domain is four dimensional in the sense that the cell density depends continuously on two spatial variables as well as two structure variables, namely membrane-bound counterreceptor densities. The biologically proper Dirichlet boundary conditions imposed in the semi-infinite structure variable region is in favor of a meshless method with Gaussian basis functions. Coupled with WENO5 finite difference spatial discretization and the method of integrating factors, the time integration via method of lines achieves optimal complexity. In addition, the proposed scheme can be extended to similar models with more general boundary conditions. Numerical results are provided to showcase the validity of the scheme.

On a non-linear droplet oscillation theory via the unified method

Presently, the oscillation of a liquid droplet in a dynamically negligible outer medium subject to surface tension and small viscosity is investigated. By using the potential flow assumption, the unified transform method by Fokas is employed to reduce the corresponding free boundary problem formulated on a time-dependent domain into a nonlinear system of integro-differential equations (IDEs). This new system depends on one less spatial variable and is now defined on a time-independent domain. Most importantly, the resulting set of equations governs the general droplet oscillation with arbitrarily large deviations from the spherical shape. As the nonlinearity of the above IDE system up to now prevented an analytical solution, the Poincaré expansion technique is employed, retaining terms up to the second order. By decomposing the unknowns into normal modes, these equations are uncoupled and the resulting ordinary differential equations for the mode amplitudes are solved, and the results are compared to those of previous works. It should be stressed that the present analysis is limited to small viscosity, or, in other words, for small Ohnesorge numbers. The reason for this is that, inside of the droplet, a potential flow is assumed and the viscous effect is taken into account only at the droplet surface by the jump condition of momentum. This is only reasonable for a small viscosity and a short time. Otherwise, vorticity is generated at the interface and diffuses toward the inside of the droplet.

EXISTENCE, UNIQUENESS AND STABILITY SOLUTIONS FOR NEW NONLINEAR SYSTEM OF INTEGRO-DIFFERENTIAL EQUATIONS OF VOLTERRA TYPE

In this article, we established the existence, uniqueness and stability solutions for a nonlinear system of integro-differential equations of Volterra type in Banach spaces. Krasnoselskii Fixed point theorems and Picard approximation method are the main tool used here to establish the existence and uniqueness results. A simple example of application of the main result of this paper is presented.

Solutions for nonlinear systems of integro-differential equations that contain multiple integrals of (V F) and (F V) types with isolated singular kernels.

Solutions for nonlinear systems of integro-differential equations that contain multiple integrals of (V F) and (F V) types with isolated singular kernels.

Fixed Points of Subadditive Maps with an Application to a System of Volterra–Fredholm Type Integrodifferential Equations

In this paper, some fixed-point theorems are established for strongly subadditive maps on CΩ,ϒ (where CΩ,ϒ denotes the space of ϒ-valued continuous functions on a compact Hausdorff space Ω and ϒ is a unital Banach algebra). Finally, the result is applied to prove the existence and uniqueness of a solution for a system of nonlinear integrodifferential equations.

Periodic solutions for nonlinear systems of integro-differential equations of Volterra- Friedholm type

Periodic solutions for nonlinear systems of integro-differential equations of Volterra- Friedholm type

Existence of solutions and numerical approximation of a non-local tumor growth model.

In order to model the evolution of a heterogeneous population of cancer stem cells and tumor cells, we analyse a nonlinear system of integro-differential equations. We provide an existence and uniqueness result by exploiting a suitable iterative scheme of functions which converge to the solution of the system. Then, we discretize the model and perform some numerical simulations. Numerical approximations are obtained by applying finite differences for space discretization and an exponential Runge-Kutta scheme for time integration. We exploit the numerical tool in order to investigate the effects that niches have on cancer development. In this respect, the numerical procedure is applied in the case when the function of cell redistribution is assumed to be spatially explicit. It allows for finding an approximate solution which is spatially inhomogeneous as time progresses. In this framework, numerical investigation may help in understanding the process of niche construction, which plays an important role in cancer population biology.

Approximate Solution for Nonlinear System of Integro-Differential Equations of Volterra Type with Boundary Conditions

Approximate Solution for Nonlinear System of Integro-Differential Equations of Volterra Type with Boundary Conditions

Stability of Singular Fractional Systems of Nonlinear Integro-Differential Equations

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.

Dynamic calculation of nonlinear oscillations of viscoelastic orthotropic plate with a concentrated mass

Plates, panels and shells made of composite material with fixed objects in the form of an additional mass have found a wide use due to their viscoelastic and strength properties. An analysis of their dynamic behavior indicates a significant effect of inhomogeneity of an associated mass type on their strength. The problem of oscillations of a viscoelastic orthotropic rectangular plate with an associated mass is considered according to the Kirchhoff-Love hypothesis in a geometrically nonlinear statement. This problem is reduced to solving the systems of nonlinear integro-differential equations with singular relaxation kernels, solved by the Bubnov-Galerkin method in combination with a numerical method based on the use of quadrature formulas. The numerical values of the approximate solution have been calculated in the Delphi programming environment. At wide range of changes in physicomechanical and geometrical parameters, the behavior of the plate has been studied. The effect of viscoelastic and inhomogeneous material properties, concentrated mass and their location on the oscillatory process of a rectangular plate is shown.

Quantization of the Electromagnetic Field in Three-Dimensional Photonic Structures on the Basis of the Scattering Matrix Formalism (S Quantization)

A technique for quantization of the electromagnetic field in photonic nanostructures with three-dimensional modulation of the dielectric constant is developed on the basis of the scattering matrix formalism (S quantization in the three-dimensional case). Quantization is based on equating the eigenvalues of the scattering matrix to unity, which is equivalent to equating each other the sets of Fourier expansions for the fields of the waves incident on the structure and propagating away from the structure. The spatial distribution of electromagnetic fields of the modes in a photonic nanostructure is calculated on the basis of the R and T matrices describing the reflection and transmission of the Fourier components by the structure. To calculate the reflection and transmission coefficients of individual real-space and Fourier-space components, the structure is divided into parallel layers within which the dielectric constant varies as a function of two-dimensional coordinates. Using the Fourier transform, Maxwell’s equations are written in the form of a matrix connecting the Fourier components of the electric field at the boundaries of neighboring layers. Based on the calculated reflection and transmission vectors for all polarizations and Fourier components, the scattering matrix for the entire structure is formed and quantization is carried out by equating the eigenvalues of the scattering matrix to unity. The developed method makes it possible to obtain the spatial profiles of eigenmodes without solving a system of nonlinear integro-differential equations and significantly reduces the computational resources required for calculating the probability of spontaneous emission in three-dimensional systems.

Global dynamics of an infinite dimensional epidemic model with nonlocal state structures

In this paper, we derive and analyze a state-structured epidemic model for infectious diseases in which the state structure is nonlocal. The state is a measure of infectivity of infected individuals or the intensity of viral replications in infected cells. The model gives rise to a system of nonlinear integro-differential equations with a nonlocal term. We establish the well-posedness and dissipativity of the associated nonlinear semigroup. We establish an equivalent principal spectral condition between the linearized operator and the next-generation operator and show that the basic reproduction number R0 is a sharp threshold: if R0<1, the disease-free equilibrium is globally asymptotically stable, and if R0>1, the disease-free equilibrium is unstable and a unique endemic equilibrium is globally asymptotically stable. The proof of global stability of the endemic equilibrium utilizes a global Lyapunov function whose construction was motivated by the graph-theoretic method for coupled systems on networks developed in [24].

The study of a mixed problem for one class of third order differential equations

This work is dedicated to the study of existence and uniqueness of almost everywhere solution of a multidimensional mixed problem for one class of third order differential equations with nonlinear operator on the right-hand side. The conception of almost everywhere solution for the mixed problem under consideration is introduced. After applying the Fourier method, the solution of the original problem is reduced to the solution of some countable system of nonlinear integro-differential equations in unknown Fourier coefficients of the sought solution. Besides, existence and uniqueness theorems of almost everywhere solution of the mixed problem under consideration are also proved in this work.

A Friendly Iterative Technique for Solving Nonlinear Integro-Differential and Systems of Nonlinear Integro-Differential Equations

In this work, a simple new iterative technique based on the integral operator, the inverse of the differential operator in the problem under consideration, is introduced to solve nonlinear integro-differential and systems of nonlinear integro-differential equations (IDEs). The introduced technique is simpler and shorter in its computational procedures and time than the other methods. In addition, it does not require discretization, linearization or any restrictive assumption of any form in providing analytical or approximate solution to linear and nonlinear equations. Also, this technique does not require calculating Adomian’s polynomials, Lagrange’s multiplier values or equating the terms of equal powers of the impeding parameter which need more computational procedures and time. These advantages make it reliable and its efficiency is demonstrated with numerical examples.