System Of Integro-differential Equations Research Articles

Solution of a Boundary Value Problem for a System of Integro-Differential Equations Arising in a Model of Plasma Physics

Solution of a Boundary Value Problem for a System of Integro-Differential Equations Arising in a Model of Plasma Physics

Correction to: Linear Boundary-Value Problems for Systems of Integrodifferential Equations with Degenerate Kernel. Resonance Case for a Weakly Perturbed Boundary-Value Problem

Correction to: Linear Boundary-Value Problems for Systems of Integrodifferential Equations with Degenerate Kernel. Resonance Case for a Weakly Perturbed Boundary-Value Problem

Pattern Formation in Mesic Savannas

We analyze a spatially extended version of a well-known model of forest-savanna dynamics, which presents as a system of nonlinear partial integro-differential equations, and study necessary conditions for pattern-forming bifurcations. Homogeneous solutions dominate the dynamics of the standard forest-savanna model, regardless of the length scales of the various spatial processes considered. However, several different pattern-forming scenarios are possible upon including spatial resource limitation, such as competition for water, soil nutrients, or herbivory effects. Using numerical simulations and continuation, we study the nature of the resulting patterns as a function of system parameters and length scales, uncovering subcritical pattern-forming bifurcations and observing significant regions of multistability for realistic parameter regimes. Finally, we discuss our results in the context of extant savanna-forest modeling efforts and highlight ongoing challenges in building a unifying mathematical model for savannas across different rainfall levels.

Well-posedness and exponential stability of nonlinear Maxwell equations for dispersive materials with interface

In this paper we consider an abstract Cauchy problem for a Maxwell system modeling electromagnetic fields in the presence of an interface between optical media. The electric polarization is in general time-delayed and nonlinear, turning the macroscopic Maxwell equations into a system of nonlinear integro-differential equations. Within the framework of evolutionary equations, we obtain well-posedness in function spaces exponentially weighted in time and of different spatial regularity and formulate various conditions on the material functions, leading to exponential stability on a bounded domain.

A novel study for solving systems of nonlinear fractional integral equations

In this study, we explore the solution of a nonlinear system of fractional integro-differential equations based on the operational matrix method. We have modified the operational matrix method to accommodate such systems and have streamlined the resulting algebraic system in a practical manner. Instead of dealing with a system of equations, we have simplified the problem to finding the solution for a set of nonlinear equations. Here, m represents the number of block pulse functions, which is typically a large positive integer. This approach significantly reduces computational costs, especially for large values of m. We provide proofs for both the existence and uniqueness of the solution for this system. Additionally, we investigate two types of stability: Ulam–Hyers stability and generalized Ulam–Hyers stability. We have evaluated the effectiveness of the proposed method by applying it to three different problems and comparing our results with existing literature. The outcomes of these tests highlight the efficiency and efficacy of our proposed method.

Symmetry in a multi-strain epidemiological model with distributed delay as a general cross-protection period and disease enhancement factor

Important biological features of viral infectious diseases caused by multiple agents with interacting strain dynamics continue to pose challenges for mathematical modelling development. Motivated by dengue fever epidemiology, we study a system of Integro-Differential Equations (IDE) considering strain structure of pathogens. Knowing that complex dynamics observed in dengue models are driven by the combination of two biological features, the temporary cross-immunity (TCI) and disease enhancement via the antibody-dependent enhancement process (ADE), our IDE system incorporates the TCI with a general time delay term, and the ADE effect by a constant factor to differentiate the susceptibility of individuals experiencing a primary or a secondary infection. Aiming at analysing the effect of the symmetry on dengue serotypes in the IDE framework, a detailed qualitative analysis of the model is performed and the instability of the coexistence steady state is shown using the perturbation theory approach. Numerical simulations identify the bifurcation structures and confirm the stability analysis. Results for the symmetric and asymmetric models are discussed.

Parameter Estimation in a System of Integro-Differential Equations With Time-Delay

This brief presents a technique for estimating the parameters of a system of integro-differential equations with a time-delay component using some given samples. The considered system is reduced to a set of linear equations by expanding the involved functions in the triangular orthogonal polynomials. While the unknowns (system parameters) can now be obtained by solving the linear equations, however, it is an overdetermined system that is usually inconsistent. In addition, there is no guarantee that the estimated parameters will predict the states nicely outside the time range from which the samples are given. Hence, the upper bound on the error (in the state prediction) is minimized, which depends only on the system parameters and is obtained using the Parseval-Plancherel identity, Cauchy-Schwarz inequality, and the Mean Value theorem. Consequently, an optimization problem is set up to minimize the accumulated error subject to the set of linear equations. The proposed technique is validated numerically where the estimated parameters are found to be close to the true values.

Fast time-domain model for the preliminary design of a wave power farm

This study presents a novel, fast time-domain model developed for an array of interacting point-absorber wave energy converters. The model is validated using experimental wave tank data. The point-absorbers, based on Uppsala University’s design, are arranged in a symmetric grid and interact with scattered and radiated waves while constrained to the heave motion. The model employs linear potential flow theory to solve the hydrodynamic coefficients in the frequency domain and employs Cummins’ formulation to solve the equations of motion in the time domain. Modeling an array of wave energy converters in the time domain yields a system of integro-differential equations, featuring convolution terms in the excitation and radiation forces. This implies that past waves radiated by the body continue to impact future dynamics. Irregular long-crested waves, generated from the Bretschneider spectrum, serve as the incident waves for the study. The model’s accuracy in capturing the dynamics and power absorption of the farm is demonstrated through validation against experimental data from a 1:10 scaled prototype of a six-point-absorber array. Despite inherent differences between the experimental and numerical set-ups, the model accurately represents the farm’s behavior. Furthermore, an efficiency test reveals that the numerical scheme approximates the performance of wave power farms comprising 6, 12, 24, 48, and 96 interacting devices within a maximum computational time of 20 s. Overall, this research presents a novel and accurate time-domain model for analyzing an array of point-absorber wave energy converters. The model’s ability to capture the dynamics and power absorption, along with its efficiency in simulating larger wave power farms, make it a valuable tool for the preliminary design stage.

A fractional matter sector for general relativity

In this work, we construct a fractional matter sector for general relativity. In particular, we propose a suitable fractional anisotropy function relating both the tangential and radial pressure of a spherically symmetric fluid based on the Grünwald–Letnikov fractional derivative. The system is closed by implementing the polytropic equation of state for the radial pressure. We solve the system of integro-differential equations by Euler’s method and explore the behavior of the physical quantities, namely, the normalized density energy, the normalized mass function, and the compactness.

Periodic Solutions for an Impulsive System of Fractional Order Integro-Differential Equations with Maxima

Periodic Solutions for an Impulsive System of Fractional Order Integro-Differential Equations with Maxima

A Novel Vieta–Fibonacci Projection Method for Solving a System of Fractional Integrodifferential Equations

In this paper, a new approach for numerically solving the system of fractional integrodifferential equations is devised. To approximate the issue, we employ Vieta–Fibonacci polynomials as basis functions and derive the projection method for Caputo fractional order for the first time. An efficient transformation reduces the problem to a system of two independent equations. Solving two algebraic equations yields an approximate solution to the problem. The proposed method’s efficiency and accuracy are validated. We demonstrate the existence of the solution to the approximate problem and conduct an error analysis. Numerical tests reinforce the interpretations of the theory.

Threshold dynamics of an age-structured vaccinated epidemic model with both direct and indirect routes of infections

Age-structured epidemic models play a crucial role in the study of epidemiological modeling. Motivated by this, in this article, we propose an epidemic model with age since infection and vaccination age of individuals, a coupled system of PDE and integro-differential equations (IDE). Here, two contagious routes, (a) direct human-to-human contact and (b) indirect environmentally contaminated surfaces or objects are considered. First, we establish the well-posedness of the model, followed by the basic reproduction number (R0) and the role of the threshold value of R0 in the asymptotic profile of the solution semi-flow is established. We observe the global stability of the disease-free steady state for R0<1, the uniform persistence of the disease and the existence of the endemic steady state for R0>1. This endemic steady state is also globally asymptotically stable for R0>1. We have further analyzed the influence of vaccination age and age since infection in the threshold parameter R0. Our analysis shows that the threshold parameter R0 does not depend explicitly on vaccination age, but it strictly decreases with the natural depletion rate of the contaminated environment. Finally, the model is discretized using the finite difference method to illustrate our theoretical results numerically.

The Fuzzy Differential Transform Method for the Solution of the System of Fuzzy Integro-Differential Equations Arising in Biological Model

This article deals with the implementation of fuzzy differential transform method for solving a system of nonlinear fuzzy integro-differential equations. This system appears in a model of biological species living together. Though the differential transform method is an iterative method, the current approach reduces this model to a set of nonlinear algebraic equations due to its delay terms. The basic definitions and theorems are first presented. The applicability and accuracy of the current methodologies have been demonstrated through the discussion of a few exemplary situations.

Fast time-domain model for an array of interactive point-absorbers

&#x0D; &#x0D; &#x0D; A fast time-domain model is developed for an array of six interacting point-absorber wave energy converters, based on the design originated at Uppsala University. The point-absorbers are placed in a symmetric grid, were each row contains one pair. The devices interact by scattered and radiated waves, while they are restricted to move in one degree of freedom, heave. Under the assumption of linear potential flow theory, the hydrodynamic coefficients for the excitation and radiation forces are obtained using an analytical multiple-scattering method. The equations of motion are solved directly in the time domain following the Cummins’ formulation. Modelling an array of wave energy converters in the time domain comes down to solving a system of integro-differential equations, were convolution terms appear in the computation of the excitation and radiation forces. In the majority of wave farm models, frequency-domain approaches are used to solve the equations of motion, since time-domain models are more computationally demanding and significantly more challenging to develop. This is not only because of the numerical integration involved, but especially due to the computation of the convolution term accounting for the radiated water waves on the free-surface, implying that waves radiated by the body in the past continue to affect the dynamics in the future. Regardless the computational effort associated with time-domain approaches, their use is required for realistic control applications and complex device dynamics, like non-linearities due to the power take-off configuration. In particular the non-linear effects that arise during the wave energy conversion chain are treated as time-varying coefficients within the system of differential equations describing the motion. Input to the numerical scheme are irregular, long-crested waves, obtained from the Bretschneider spectrum, corresponding to four, different sea states. It is of high interest to study whether the linear numerical model simulates accurately the performance of each interactive surface buoy in response to the irregular waves. Therefore, the numerical results for the full array configuration are compared with experimental data. At this point we emphasize that there are not a lot of experimental works considering arrays of point- absorbers, due to the complexity and costs associated with the task. Therefore, finding a set of data to validate the array model is not trivial. The experimental results we use were carried out in the COAST Lab at Plymouth University, UK, corresponding to a 1:10 scaled prototype of an array of point-absorbers. The set-up consists of six ellipsoid floats free to move in six degrees of freedom and connected via ropes and pulleys to individual power take-off systems. Despite the highly non-linear effects in the physical experiment, the free motion of the buoys in all directions, and the power take-off configuration, the numerical scheme is able to accurately capture the heaving motion of the buoys and their power absorption.&#x0D; &#x0D; &#x0D;

Haar wavelets method for solving class of coupled systems of linear fractional Fredholm integro-differential equations

In this paper, firstly, the “ Haar wavelet method ” is used to give approximate solutions for coupled systems of linear fractional Fredholm integro-differential equations. Moreover, we consider the fractional derivative to be described in the Caputo sense. The general idea of this technique is simply based on reducing this kinds of coupled systems into systems of algebraic equations which are easily to deal with and solve. Also, Laplace transform operator is included to develop a sophisticated approach which we called “ Laplace Haar wavelet method ” as an adjustment to “ Haar wavelet method ” to reduce the error and computational time. We provide illustrative examples to confirm validity, efficiency, accuracy, and applicability of the proposed methods.

Convergence analysis and numerical implementation of projection methods for solving classical and fractional Volterra integro-differential equations

In this article, we discuss the convergence analysis of the classical first-order and fractional-order Volterra integro-differential equations of the second kind with a smooth kernel by reducing them into a system of fractional Fredholm integro-differential equations (FFIDEs). For that, we first reformulate the given equation into a system of fractional Volterra integro-differential equations and then transform it into the system of FFIDEs using a simple transformation. We develop a general framework of the newly defined iterated Galerkin method for the reduced system of equations and investigate the existence and uniqueness of the approximate solutions in the given Banach space. We provide the error estimates and convergence analysis for the iterated Galerkin approximate solutions in the supremum norm without any limiting conditions. Further, we provide the superconvergence results for classical first-order and fractional-order Volterra integro-differential equations by proposing a general framework of multi-Galerkin and iterated multi-Galerkin methods for the reduced system of equations. Moreover, we prove that the order of convergence of the proposed methods increases theoretically and numerically with the increasing order of the fractional derivatives. Finally, numerical implementations and illustrative examples are provided to demonstrate our theoretical aspects.

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