Integro-differential Equations Research Articles

Existence and Controllability Results for Integrodifferential Equations with State-Dependent Nonlocal Conditions via Fractional Power Operators in Fréchet Spaces

Existence and Controllability Results for Integrodifferential Equations with State-Dependent Nonlocal Conditions via Fractional Power Operators in Fréchet Spaces

Bridging the gap between models based on ordinary, delayed, and fractional differentials equations through integral kernels

Evolution equations with convolution-type integral operators have a history of study, yet a gap exists in the literature regarding the link between certain convolution kernels and new models, including delayed and fractional differential equations. We demonstrate, starting from the logistic model structure, that classical, delayed, and fractional models are special cases of a framework using a gamma Mittag-Leffler memory kernel. We discuss and classify different types of this general kernel, analyze the asymptotic behavior of the general model, and provide numerical simulations. A detailed classification of the memory kernels is presented through parameter analysis. The fractional models we constructed possess distinctive features as they maintain dimensional balance and explicitly relate fractional orders to past data points. Additionally, we illustrate how our models can reproduce the dynamics of COVID-19 infections in Australia, Brazil, and Peru. Our research expands mathematical modeling by presenting a unified framework that facilitates the incorporation of historical data through the utilization of integro-differential equations, fractional or delayed differential equations, as well as classical systems of ordinary differential equations.

The Way to Construct Innovative Methods for Solving Initial-Value Problem of the Volterra Integro-Differential Equation

Mathematical models for many problems in the natural sciences are often simplified to solving initial-value problems (IVPs) for the Volterra integro-differential equations (VIDE). Numerical methods of a multistep type are typically used to solve these problems. It is known that in some cases, the multi-step method (MSM) is applied to solving the IVPs of both ordinary differential equations (ODEs) and VIDE encountered in solving some problems in mathematical biology. Here, to solve such problems by combining different methods, some modifications of established methods were developed, and it was demonstrated that these methods outperform the existing ones. As is known, one of the main issues in solving the aforementioned problems is determining the reliability of calculating values using the known mathematicalstatistical models (MSMs). In this regard, some experts utilize the predictor-corrector method. Having highlighted the disadvantages of this method, the proposal is to develop an innovative approach and assess the errors that may arise when applying this method to solve various problems. Here, the IVPs for the VIDE of the first order are primarily investigated. To illustrate the benefits of the innovative methods proposed here, we discuss the use of simple numerical methods to solve some common examples.

About a new nonlinear integro-differential equation of Volterra-Strum-Liouville type with a weakly singular kernel

This present paper proposes an analytical and numerical study of a new integro-differential Volterra-Strum-Liouville equation of the second type, having a weak singularity. We provide conditions that guarantee the existence and uniqueness of the solution of the nonlinear problem. Then, we develop a numerical technique using the production integration method. The numerical application shows the efficiency of the proposed procedure.

Nonlinear Volterra integro-differential equations incorporating a delay term using Picard iterated method

Nonlinear Volterra integro-differential equations incorporating a delay term using Picard iterated method

A fully discrete scheme on piecewise-equidistant mesh for singularly perturbed delay integro-differential equations

. This paper chiefly takes into account the singularly perturbed delay Volterra-Fredholm integro-differential equations by numerically. In this context, firstly, priori estimates are given and a new discretization is constructed on piecewise-equidistant mesh by using interpolating quadrature rules [2] and composite integration formulas. Then, the convergence analysis and stability bounds of the presented method are discussed. Finally, numerical results are demonstrated with two test problems.

The solution of a complex integro differential equation in the right half-plane via an infinite plate weakened by a curvilinear hole

The solution of a complex integro differential equation in the right half-plane via an infinite plate weakened by a curvilinear hole

An α-robust error analysis of the [formula omitted] scheme for time-fractional integro-differential initial-boundary value problems

An initial-boundary value problem governed by the time-fractional integro-differential equation Dtαu−κItβΔu−λΔu=f is considered, where (x,t)∈Ω×(0,T]; here Dtα is a Caputo temporal derivative of order α∈(0,1), Itβ is a Riemann-Liouville temporal integral of order β∈(0,1), and Ω is a rectangular-type domain in Rd for some d≥1. The positive constants κ and λ and the function f are given data. A numerical scheme is constructed by employing the L1‾ scheme for the Caputo derivative, an averaged product integration rule for the Riemann-Liouville integral, and a standard discretisation (3 points in each coordinate direction) of Δu. The scheme satisfies a discrete comparison principle, which is used to establish an error bound for the computed solution that is O(N−2) in time on graded temporal meshes with N points. This result is α-robust: it does not blow up as α→1−. Numerical results show the sharpness of our theoretical results.

Hyers–Ulam–Rassias Stability of Nonlinear Implicit Higher-Order Volterra Integrodifferential Equations from above on Unbounded Time Scales

In this paper, we present sufficient conditions for Hyers–Ulam-Rassias stability of nonlinear implicit higher-order Volterra-type integrodifferential equations from above on unbounded time scales. These new sufficient conditions result by reducing Volterra-type integrodifferential equations to Volterra-type integral equations, using the Banach fixed point theorem, and by applying an appropriate Bielecki type norm, the Lipschitz type functions, where Lipschitz coefficient is replaced by unbounded rd-continuous function.

On the Controllability of Coupled Nonlocal Partial Integrodifferential Equations Using Fractional Power Operators

In this research paper, we investigate the controllability in the α-norm of a coupled system of integrodifferential equations with state-dependent nonlocal conditions in generalized Banach spaces. We establish sufficient conditions for the system’s controllability using resolvent operator theory introduced by Grimmer, fractional power operators, and fixed-point theorems associated with generalized measures of noncompactness for condensing operators in vector Banach spaces. Finally, we present an application example to validate the proposed methodology in this research.

Iterative addition of parallel non-local effects to full wave ICRF finite element models in axisymmetric tokamak plasmas

The current response of a hot magnetized plasma to a radio-frequency wave is non-local, turning the electromagnetic wave equation into an integro-differential equation. Non-local physics gives rise to wave physics and absorption processes not observed in local media. Furthermore, non-local physics alters wave propagation and absorption properties of the plasma. In this work, an iterative method that accounts for parallel non-local effects in 2D axisymmetric tokamak plasmas is developed, implemented, and verified. The iterative method is based on the finite element method and Fourier decomposition, with the advantage that this numerical scheme can describe non-local effects while using a high-fidelity antenna and wall representation, as well as limiting memory usage. The proposed method is implemented in the existing full wave solver FEMIC and applied to a minority heating scenario in ITER to quantify how parallel non-local physics affect wave propagation and dissipation in the ion cyclotron range of frequencies (ICRF). The effects are then compared to a reduced local plane wave model, both verifying the physics implemented in the model, as well as estimating how well a local plane wave approximation performs in scenarios with high single pass damping. Finally, the new version of FEMIC is benchmarked against the ICRF code TORIC.

Mixed Hilfer and Caputo Fractional Riemann–Stieltjes Integro-Differential Equations with Non-Separated Boundary Conditions

In this paper, we investigate a sequential fractional boundary value problem which contains a combination of Hilfer and Caputo fractional derivative operators and non-separated boundary conditions. We establish the existence of a unique solution via Banach’s fixed point theorem, while by applying Leray–Schauder’s nonlinear alternative, we prove an existence result. Finally, examples are provided to demonstrate the results obtained.

Optimal stopping of conditional McKean–Vlasov jump diffusions

The purpose of this paper is to study the optimal stopping problem of conditional McKean–Vlasov (mean-field) stochastic differential equations with jumps (conditional McKean–Vlasov jump diffusions, for short). We obtain sufficient variational inequalities for a function to be the value function of such a problem and for a stopping time to be optimal.The key is that we combine the conditional McKean–Vlasov equation with the associated stochastic Fokker–Planck partial integro-differential equation for the conditional law of the state. This leads to a Markovian system which can be handled by using a version of a Dynkin formula.Our verification result is illustrated by finding the optimal time to sell in a market with common noise and jumps.

The Inverse of Exact Renormalization Group Flows as Statistical Inference

We build on the view of the Exact Renormalization Group (ERG) as an instantiation of Optimal Transport described by a functional convection–diffusion equation. We provide a new information-theoretic perspective for understanding the ERG through the intermediary of Bayesian Statistical Inference. This connection is facilitated by the Dynamical Bayesian Inference scheme, which encodes Bayesian inference in the form of a one-parameter family of probability distributions solving an integro-differential equation derived from Bayes’ law. In this note, we demonstrate how the Dynamical Bayesian Inference equation is, itself, equivalent to a diffusion equation, which we dub Bayesian Diffusion. By identifying the features that define Bayesian Diffusion and mapping them onto the features that define the ERG, we obtain a dictionary outlining how renormalization can be understood as the inverse of statistical inference.

Utilizing Cubic B-Spline Collocation Technique for Solving Linear and Nonlinear Fractional Integro-Differential Equations of Volterra and Fredholm Types

Fractional integro-differential equations (FIDEs) of both Volterra and Fredholm types present considerable challenges in numerical analysis and scientific computing due to their complex structures. This paper introduces a novel approach to address such equations by employing a Cubic B-spline collocation method. This method offers a robust and systematic framework for approximating solutions to the FIDEs, facilitating precise representations of complex phenomena. Within this research, we establish the mathematical foundations of the proposed scheme, elucidate its advantages over existing methods, and demonstrate its practical utility through numerical examples. We adopt the Caputo definition for fractional derivatives and conduct a stability analysis to validate the accuracy of the method. The findings showcase the precision and efficiency of the scheme in solving FIDEs, highlighting its potential as a valuable tool for addressing a wide array of practical problems.

SKdV, SmKdV flows and their supersymmetric gauge-Miura transformations

The construction of Integrable Hierarchies in terms of zero curvature representation provides a systematic construction for a series of integrable non-linear evolution equations (flows) which shares a common affine Lie algebraic structure. The integrable hierarchies are then classified in terms of a decomposition of the underlying affine Lie algebra $\hat \lie $ into graded subspaces defined by a grading operator $Q$. In this paper we shall discuss explicitly the simplest case of the affine $\hat {sl}(2)$ Kac-Moody algebra within the principal gradation given rise to the KdV and mKdV hierarchies and extend to supersymmetric models. It is known that the positive mKdV sub-hierachy is associated to some positive odd graded abelian subalgebra with elements denoted by $E^{(2n+1)}$. Each of these elements in turn, defines a time evolution equation according to time $t=t_{2n+1}$. An interesting observation is that for negative grades, the zero curvature representation allows both, even or odd sub-hierarchies. In both cases, the flows are non-local leading to integro-differential equations. Whilst positive and negative odd sub-hierarchies admit zero vacuum solutions, the negative even admits strictly non-zero vacuum solutions. Soliton solutions can be constructed by gauge transforming the zero curvature from the vacuum into a non trivial configuration (dressing method). Inspired by the dressing transformation method, we have constructed a gauge-Miura transformation mapping mKdV into KdV flows. Interesting new results concerns the negative grade sector of the mKdV hierarchy in which a double degeneracy of flows (odd and its consecutive even) of mKdV are mapped into a single odd KdV flow. These results are extended to supersymmetric hierarchies based upon the affine $\hat {sl}(2,1)$ super-algebra.

The chemostat reactor: A stability analysis and model predictive control

This contribution develops the model predictive control for an unstable chemostat reactor. Initially, we analyze the system’s model — a nonlinear first-order hyperbolic partial integro-differential equation (PIDE) — and carry the model linearization around an unstable operating condition. Employing the structure-preserving Cayley–Tustin transformation, we obtain a discrete-time model representation of the continuous model. Subsequently, we solve the operator Ricatti equations in the discrete-time setting to derive a full state feedback controller that stabilizes the closed-loop and design a Luenberger observer for state reconstruction given the system output measures. Finally, we formulate a dual-mode MPC ensuring constraint satisfaction and optimality, integrating the gain-based unconstrained full-state feedback optimal control obtained from the Ricatti equation. This dual-mode strategy describes an optimization problem where the predictive controller acts only if constraints become active within the control horizon. Simulation studies validate the controller performance, where the MPC only takes action if the constraints are predicted to be active within the control horizon while also guaranteeing closed-loop stabilization under only output feedback. This type of controller can be easily implemented with other control strategies and significantly decreases the computational costs of solving the optimal control problems when compared to other MPC approaches.

Application of Laplace Transform Method for Solving Weakly-Singular Integro-Differential Equations

Application of Laplace Transform Method for Solving Weakly-Singular Integro-Differential Equations

Monotone Iterative Technique for a Kind of Nonlinear Fourth-Order Integro-Differential Equations and Its Application

In this paper, we consider the existence and iterative approximation of solutions for a class of nonlinear fourth-order integro-differential equations (IDEs) with Navier boundary conditions. We first prove the existence and uniqueness of analytical solutions for a linear fourth-order IDE, which has rich applications in engineering and physics, and then we establish a maximum principle for the corresponding operator. Based upon the maximum principle, we develop a monotone iterative technique in the presence of lower and upper solutions to obtain iterative solutions for the nonlocal nonlinear problem under certain conditions. Some examples are presented to illustrate the main results.

New retarded nonlinear integral inequalities of Gronwall–Bellman–Pachpatte type and their applications

Abstract The goal of the present article is to offer several new retarded nonlinear inequalities of Gronwall, Bellman, and Pachpatte kind for a class of integral and integro-differential equations. These inequalities generalize and provide new formulations of some well-known results in the mathematical framework of integral and differential inequalities that have been derived currently as well as in earlier times. These results can be utilized to investigate diverse aspects, both qualitative and quantitative, of a class of aforementioned equations. We propose a few applications to ensure the effectiveness of these inequalities.