System Of Integro-differential Equations Research Articles

Nonexistence Results for a System of Nonlinear Fractional Integrodifferential Equations

Nonexistence Results for a System of Nonlinear Fractional Integrodifferential Equations

APPROXIMATE AND STABILITY SOLUTION FOR NON-LINEAR SYSTEM OF INTEGRODIFFERENTIAL EQUATIONS OF VOLTERRA TYPE WITH BOUNDARY CONDITIONS

In this paper, we investigate the approximation and stability solutions of non-linear systems of integro-differential equations of Volterra type with boundary conditions, by using the numerical-analytic method which were introduced by Samoilenko. The study of such integro-differential equations leads to extend the results obtained by Butris for changing the system of non-linear integro- differential equations of Volterra type to the system of non-linear integro-differential equations of the Volterra type with boundary conditions. Theorems on a solutions are established under some necessary and sufficient conditions on compact spaces.

A developed technique of homotopy analysis method for solving nonlinear systems of Volterra integro-differential equations of fractional order

In this paper, a developed technique of homotopy analysis method (HAM) is presented to solve nonlinear system of fractional order Volterra integro-differential equations. The fractional derivative is described in the Caputo form. The main thing we have done in this paper is that we have simplified the process of evaluating the fractional order integrations in the deformation equations than the standard HAM. The proposed technique is tested by some illustrative examples in order to demonstrate its accuracy, applicability and efficiency.

Linear Boundary-Value Problems for Systems of Integrodifferential Equations with Degenerate Kernel. Resonance Case for a Weakly Perturbed Boundary-Value problem

Linear Boundary-Value Problems for Systems of Integrodifferential Equations with Degenerate Kernel. Resonance Case for a Weakly Perturbed Boundary-Value problem

MODELING OF NONLINEAR MASS FLOWS DURING WATER SOFTENING IN A MECHANICAL FILTER

A mathematical model of water softening in a mechanical filter based on the balance relations for the mass components of the thermodynamical system is constructed. The initial-boundary value problem of convective diffusion accompanied by a chemical reaction is formulated for one-dimensional case of spatial coordinates and reduced to the equivalent system of integro-differential equations. The solution of this system is constructed by the method of successive iterations by decomposing the functions of the mass concentrations of impurity in water solution and in the body skeleton into series around the solutions of the homogeneous linear initial-boundary value problem. Based on the expression for the impurity concentration in the aqueous solution the mass flows of impurity are obtained and investigated. The nonlinearity of steady-state regime, which is caused by the transfer of mass simultaneously by diffusion and convection mechanisms, is shown.

On the solvability of some systems of integro-differential equations with transport and concentrated sources

The article is devoted to the existence of solutions of a system of integro-differential equations involving the drift terms in the case of the normal diffusion and the influx/efflux terms proportional to the Dirac delta function. The proof of the existence of solutions is based on a fixed point technique. We use the solvability conditions for the non- Fredholm elliptic operators in unbounded domains. We emphasize that the study of the systems is more difficult than of the scalar case and requires to overcome more cumbersome technicalities.

Asymptotic behavior of the solution of the boundary value problem for a singularly perturbed system of the integro-differential equations

In this paper, we study the asymptotic behavior of solutions to the boundary value problem for singularly perturbed systems of integro-differential equations. The aim of the work is to obtain an analytical formula, an asymptotic estimate of the solution of a boundary value problem, and to determine the asymptotic behavior of the solution by a smaller parameter at the starting point. The boundary value problem given in the paper is reduced to a boundary value problem posed in a singularly perturbed integral-differential equation of mixed type with respect to a fast variable. The Cauchy function, boundary functions and Green’s function of a singularly perturbed homogeneous differential equation are obtained, and their asymptotic estimates are also determined. With the help of these constructed functions, an analytical formula and an asymptotic estimate of this solution of the boundary value problem are obtained. The asymptotic behavior of the solution with respect to a small parameter is determined and the order of growth of its derivatives at the left point of a given segment is shown. It is established that the solution of the boundary value problem under consideration has an initial jump of zero order at the initial point.

On the stability, integrability and boundedness analysis of systems of integro-differential equations with time-delay

On the stability, integrability and boundedness analysis of systems of integro-differential equations with time-delay

Numerical solution of the system of second-order integro-differential equations using non-classical double sinc method

This paper presents a new numerical technique based on the sinc-collocation approximation for solving of the system of second-order integro-differential equations with initial boundary conditions. The novelty of the approach consists on the usage of the weight functions in the traditional sinc-expansion. As it is known, the sinc basis functions are not differentiable at zero, so we modified the basis functions into a non-classical basis which is differentiable with zero derivative at the initial point. The properties of sinc-collocation are used to reduce the system of integro-differential equations into a system of algebraic equations. Furthermore, exponential convergence of the method is investigated which demonstrate its applicability in achieving the good accuracy. By solving some examples and comparing the results with existing methods the efficiency is examined.

Absolute Stability and Synchronization in Second-Order Neural Fields with Conduction Delays

We consider a class of neural field models represented by a second-order nonlinear system of integro-differential equations with space-dependent delays. Such a system models interaction of populations of neurons, each with a continuum description. We justify the existence and uniqueness of solution for the system in a suitable function space. Global existence and boundedness of solutions for the system are confirmed. Two methodologies, the comparison argument and sequential contracting, are developed to establish sufficient conditions for absolute stability and synchronization among different populations of the system. Finally, we present some numerical examples to demonstrate the theoretical results.

An immunity-structured SEIRS epidemic model with variable infectivity and spatial heterogeneity

A mathematical model is proposed for the spread of an epidemic disease of agedependent infectivity through an asexual population with spatial heterogeneity, assuming that some individuals recover from the disease with temporary immunity, another part recover with permanent immunity, and the last part recover with no immunity. The demographic changes such as births and deaths due to natural causes and the chronological age of individuals are not taken into account. The model is based on a system of partial integro-differential equations including a differential equation to describe the evolution of individuals who have recovered with temporary immunity. The existence and uniqueness of the globally defined solution is proved, and its long-time behaviour is studied.

Nonexistence results for a system of nonlinear fractional integro-differential equations

UDC 517.9 We investigate the nonexistence of (nontrivial) global solutions for a system of nonlinear fractional equations. Each equation involves n fractional derivatives, a subfirst-order ordinary derivative, and a nonlinear source term. The fractional derivatives are of the Caputo type of order between 0 and 1. The nonlinear sources have the form of the convolution of a function of state with (possibly singular) kernel. We generalize the results available in the literature, in particular, the results obtained by Mennouni and Youkana [Electron. J. Different. Equat., <strong>152</strong>, 1–15 (2017)] and Ahmad and Tatar [Turkish J. Math., <strong>43</strong>, 2715–2730 (2019)].

Approximate Method for Solving System of Linear Fredholm Fractional Integro-Differential Equations Using Least Squares Method and Lauguerre Polynomials

Approximate Method for Solving System of Linear Fredholm Fractional Integro-Differential Equations Using Least Squares Method and Lauguerre Polynomials

Computer simulations and mode-coupling theory of glass-forming confined hard-sphere fluids.

We present mode-coupling theory (MCT) results for densely packed hard-sphere fluids confined between two parallel walls and compare them quantitatively to computer simulations. The numerical solution of MCT is calculated using the full system of matrix-valued integro-differential equations. We investigate several dynamical properties of supercooled liquids including scattering functions, frequency-dependent susceptibilities, and mean-square displacements. Close to the glass transition, we find quantitative agreement between the coherent scattering function predicted from theory and that evaluated from simulations, which enables us to make quantitative statements on caging and relaxation dynamics of the confined hard-sphere fluid.

Thermal Stressed State of a Bimaterial with Interface Crack Filled with a Compressible Liquid

We formulate the problem of thermoelasticity for a bimaterial with interface crack filled with a liquid with regard for the pressure and heat conductivity of the liquid. We consider the case of partial crack closure on the edges under the action of a compressive load and a heat flux. The problem is reduced to a system of nonlinear singular integrodifferential equations for a temperature jump between the crack faces and its opening displacement. The length of the contact zone is determined from the condition of smooth closure of crack faces at the points separating the open and closed parts of the crack. We analyze the changes in the crack length and height and in the pressure of liquid depending on the compressive loads, density of the heat flux, and its direction. We also determine the dependence of the mode-II stress intensity factor on the level of pressure and the thermal conductivity coefficient of liquid.

Stochastic theory of ferroelectric domain structure formation dominated by quenched disorder

A self-consistent stochastic model of domain structure formation in a uniaxial ferroelectric, quenched from a high-temperature paraelectric phase to a low-temperature ferroelectric phase, is developed with an account of the applied electric field and the feedback effect via local depolarization fields. Both polarization and field components are considered as Gauss random variables. A system of integro-differential equations for correlation functions of all involved variables is derived and solved analytically and numerically. Phase diagram in terms of the average value and dispersion of polarization reveals different possible equilibrium states and available final single-domain and multi-domain states. The time-dependent evolution of the average polarization and dispersion discloses a bifurcation behavior and the temperature-dependent value of the electric field, deciding between the single-domain and multi-domain final states, which can be interpreted as the coercive field. Analytical and numerical results for the time-dependent correlation length and correlation functions exhibit plausible agreement with available experimental data.

Optimal Interruption of P. vivax Malaria Transmission Using Mass Drug Administration

Plasmodium vivax is the most geographically widespread malaria-causing parasite resulting in significant associated global morbidity and mortality. One of the factors driving this widespread phenomenon is the ability of the parasites to remain dormant in the liver. Known as ‘hypnozoites’, they reside in the liver following an initial exposure, before activating later to cause further infections, referred to as ‘relapses’. As around 79–96% of infections are attributed to relapses from activating hypnozoites, we expect it will be highly impactful to apply treatment to target the hypnozoite reservoir (i.e. the collection of dormant parasites) to eliminate P. vivax. Treatment with radical cure, for example tafenoquine or primaquine, to target the hypnozoite reservoir is a potential tool to control and/or eliminate P. vivax. We have developed a deterministic multiscale mathematical model as a system of integro-differential equations that captures the complex dynamics of P. vivax hypnozoites and the effect of hypnozoite relapse on disease transmission. Here, we use our multiscale model to study the anticipated effect of radical cure treatment administered via a mass drug administration (MDA) program. We implement multiple rounds of MDA with a fixed interval between rounds, starting from different steady-state disease prevalences. We then construct an optimisation model with three different objective functions motivated on a public health basis to obtain the optimal MDA interval. We also incorporate mosquito seasonality in our model to study its effect on the optimal treatment regime. We find that the effect of MDA interventions is temporary and depends on the pre-intervention disease prevalence (and choice of model parameters) as well as the number of MDA rounds under consideration. The optimal interval between MDA rounds also depends on the objective (combinations of expected intervention outcomes). We find radical cure alone may not be enough to lead to P. vivax elimination under our mathematical model (and choice of model parameters) since the prevalence of infection eventually returns to pre-MDA levels.

Исключение интегрального члена в уравнениях одной эредитарной системы, связанной с задачей гидромагнитного динамо

В работе изучается двумерная система интегро-дифференциальных уравнений, которая является простейшей эредитарной моделью двумодового гидромагнитного динамо. Учет пространственной и временной нелокальности взаимодействий в динамо-системах сейчас активно исследуется. В маломодовых приближениях уравнений динамо можно рассматривать только временную нелокальность, т.е. эредитарность (память). Память в исследуемой системе реализована в виде обратной связи, распределенной по всем прошлым состояниям системы. Обратная связь представлена с помощью интегрального члена типа свертки от квадратичной комбинации фазовых переменных с ядром достаточно общего вида. Этот член моделирует подавление турбулентного генератора поля (α-эффекта) квадратичной формой от фазовых переменных. В реальных динамо-системах такое подавление обеспечивается силой Лоренца. Основной результат работы – доказательство возможности исключения интегрального члена для одного класса ядер. Такие ядра являются решениями однородного линейного дифференциального уравнения с постоянными коэффициентами. Доказано, что исследуемую интегро-дифференциальную систему можно заменить дифференциальной системой большей размерности с подходящими начальными условиями на дополнительные фазовые переменные. Если ядро является решением уравнения n-го порядка, то размерность системы может достигать 3n−2 и зависит от начальных условий, которым удовлетворяет ядро. В работе используются классические методы теории дифференциальных уравнений. Приводятся примеры динамических систем, возникающих при некоторых ядрах в результате исключения интегрального члена. Результаты работы можно использовать для верификации вычислительных алгоритмов и программных кодов, разработанных для решения интегро-дифференциальных уравнений.We study a two-dimensional system of integro-differential equations, which is the simplest hereditary model of a two-mode hydromagnetic dynamo. Accounting for the spatial and temporal nonlocality of interactions in dynamo systems is currently being actively studied. In the low-mode approximations of the dynamo equations, one can consider only temporal nonlocality, i.e. heredity (memory). Memory in the system under study is implemented in the form of feedback distributed over all past states of the system. The feedback is represented by a convolution-type integral term of a quadratic combination of phase variables with a fairly general kernel. This term models the quenching of the turbulent field generator (α-effect) by a quadratic form in phase variables. In real dynamo systems, such quenchingn is provided by the Lorentz force. The main result of the work is a proof of the possibility of eliminating the integral term for one class of kernels. Such kernels are solutions of a homogeneous linear differential equation with constant coefficients. It is proved that the studed integro-differential system can be replaced by a higher-dimensional differential system with suitable initial conditions for additional phase variables. If the kernel is a solution to an n-order equation, then the dimension of the system can reach 3n−2 and depends on the initial conditions that the kernel satisfies. The work uses classical methods of the theory of differential equations. Examples are given of dynamical systems that arise for some kernels as a result of the elimination of the integral term. The results of the work can be used to verify computational algorithms and program codes developed for solving integro-differential equations.

A stochastic-local volatility model with [formula omitted] jumps for pricing derivatives

We introduce a new mixed model unifying local volatility, pure stochastic volatility, and Le´vy type of jumps in this paper. Our model framework allows the pure stochastic volatility and the jump intensity to be functions of fast (and slowly) varying stochastic processes and the local volatility to be given in a constant elasticity of variance type of parametric form. We use asymptotic analysis to derive a system of partial integro-differential equations for the prices of European derivatives and use Fourier analysis to obtain an explicit formula for the prices. Our result is an extension of a stochastic-local volatility model from no jumps to Le´vy jumps and an extension of a multiscale stochastic volatility model with Le´vy jumps from the zero elasticity of variance to the non-zero elasticity of variance. We find that our model outperforms two benchmark models in view of fitting performance when the time-to-maturities of European options are relatively short and outperforms one benchmark model in terms of pricing time cost.

Numerical solution of system of second-order integro-differential equations using nonclassical sinc collocation method

In this paper, a nonclassical sinc collocation method is constructed for the numerical solution of systems of second-order integro-differential equations of the Volterra and Fredholm types. The novelty of the approach is based on using the new nonclassical weight function for sinc method instead of the classic ones. The sinc collocation method based on nonclassical weight functions is used to reduce the system of integro-differential equations to a system of algebraic equations. Furthermore, the convergence of the method is proposed theoretically, showing that the method converges exponentially. By solving some examples, including problems with a non-smooth solution, the results are compared with other existing results to demonstrate the efficiency of the new method.