Integro-differential Equations Research Articles

On a Hybrid Class of $p$-Laplacian Initial Value Problems with Modified Mittag-Leffler Kernel

In this work, we establish key results on the existence theory for a category of initial value problems (IVPs) involving hybrid fractional integro-differential equations (HFIDEs) with a $p$-Laplacian operator, utilizing the modified Mittag-Leffler kernel. By employing Krasnoselskii and Banach fixed point theorems (FPTs), we determine the conditions required for the existence of solutions. Additionally, we examine the Hyers-Ulam (H-U) stability of the problem. Lastly, we present an example to confirm our theoretical results.

An Algorithm for the Solution of Integro-Fractional Differential Equations with a Generalized Symmetric Singular Kernel

This work studies an integro-fractional differential equation (I-FrDE) with a generalized symmetric singular kernel. The scientific approach in this study was to transform the integro-differential equation (I-DE) into a mixed integral equation (MIE) with an Able kernel in fractional time and a generalized symmetric singular kernel in position. Additionally, the authors first set conditions on the singular kernels, whether related to time or position, and then transform the integral equation into an integral operator. Secondly, the solution is unique, which is proven by means of fixed-point theorems. In combination with the solution rules, the convergence of the solution is studied, and the error equation resulting from the solution is a stable error-integral influencer equation. Next, to solve this MIE, the authors apply a special technique to separate the variables and produce an integral equation in position with coefficients, in the form of an integral operator in time. As the most effective technique for resolving singular integral equations, the Toeplitz matrix method (TMM) is utilized to convert the integral equation into an algebraic system for the purpose of solving the position problem. The existence of a solution to the linear algebraic system in Banach space is then demonstrated. Lastly, certain applications where the functions of the generalized symmetric kernel are cubic or exponential and it assumes the logarithmic, Cauchy, or Carleman form are discussed. In each case, Maple 18 is also used to compute the error estimate.

A Beneficial Numerical Approach to Solve Systems of Linear Integro-Differential Equations

The system of linear Fredholm-Volterra integro-differential equations (FVIDEs) has been solved in this paper by an improved approximation method. Generalised Bernstein polynomials and collocation points have been used to construct the theory of the method. The aim of the technique is to reduce systems of integro-differential equations into an algebraic matrix equation, which corresponds to a linear algebraic equation system, by means of Bernstein polynomials. In order to analyse the applicability of the method, some illustrative examples have also been considered. It has been shown that the proposed method is faster and more effective than the others when comparing the numerical results.

Optimizing linear/non-linear Volterra-type integro-differential equations with Runge–Kutta 2 and 4 for time efficiency

A novel non-adaptative time-efficient (TE) numerical scheme based on the Runge–Kutta (RK) algorithm is devised for solving linear and non-linear Volterra-type integro-differential equations (VTIDE) characterized by a specific class of convolution memory, K(t,s)=aκ(t−s), where a,κ∈R. This kernel is commonly encountered in viscoelasticity and various applied sciences. The integration governing the convolution term is elegantly reformulated, allowing for the implementation of a backstage integration within the RK scheme’s main body. This approach circumvents the need for full integration of the convolution during each iteration, thereby significantly reducing computational time to O(Nt) for Nt iterations. Furthermore, this formulation facilitates the adoption of an implicit scheme by selecting appropriate methods and data stencils. We demonstrate this concept using an implicit trapezoidal method applied to a linear VTIDE, accompanied by stability analyses. Additionally, a complex VTIDE is constructed featuring nonlinearities both within and outside the convolutions, as well as a derivative-of-dependent-variable integrant. This setup enables the synergy of differentiation, integration, and RK schemes to generate data for intricate VTIDE types. Consequently, both the scheme and the equation exhibit uniqueness denoted as TE-RK and TE-RK-VTIDE. We illustrate how this development simplifies the implementation of implicit schemes and the calculation of stability regions concerning the convolution. The TE-RK scheme is tested on VTIDE using RK2 and RK4, yielding plots that exhibit strong agreement, thus validating our approach. Furthermore, we successfully apply the TE-RK scheme to a well-known nonlinear logistic equation. Plots for various step sizes are generated alongside corresponding error graphs, demonstrating consistent trends in error behavior.

Least Squares Approximation Method for Estimation of Volterra Fractional Integro-differential equation using Hermite Polynomial as the basis functions

This research work utilizes the least squares approximation method to estimate approximate solutions for fractional-order integro-differential equations, with Hermite polynomials as basis functions. The process begins by assuming an approximate solution of degree N, which is then substituted into the fractional-order integro-differential equation under investigation. After evaluating the integral, the equation is rearranged to isolate one side, allowing the application of the least squares method. Three examples were solved using this approach. In Example 1, the numerical results for α=0.9 and α=0.8 were compared to the exact solution for α=1. In Examples 2 and 3, the results for α=1.9 and α=1.8 were compared to the exact solution for α=2. These comparisons showed favorable alignment with the exact solutions. The numerical results and graphical illustrations demonstrate the validity, competence, and accuracy of the proposed method.

Two mixed virtual element formulations for parabolic integro-differential equations with nonsmooth initial data

This article presents and examines two distinctive approaches to the mixed virtual element method (VEM) applied to parabolic integro-differential equations (PIDEs) with non-smooth initial data. In the first part of the paper, we introduce and analyze a mixed virtual element scheme for PIDE that eliminates the need for the resolvent operator. Through the introduction of a novel projection involving a memory term, coupled with the application of energy arguments and the repeated use of an integral operator, this study establishes optimal L2-error estimates for the two unknowns p and σ. Furthermore, optimal error estimates are derived for the standard mixed formulation with a resolvent kernel. The paper offers a comprehensive analysis of the VEM, encompassing both formulations.

Existence and uniqueness of the solution to initial and inverse problems for integro-differential heat equations with fractional load

This work is devoted to the unique solvability of the direct and inverse problems for a multidimensional heat equation with a fractional load in Holder spaces. In the problem under consideration, the loaded term is in the form of a fractional integral operator for the time variable. We prove the existence and uniqueness of the solution to these problems by the contraction mapping theorem and the theory of integral equations. For more information see https://ejde.math.txstate.edu/Volumes/2024/64/abstr.html

Solution of the Vector Three-Dimensional Inverse Problem on an Inhomogeneous Dielectric Hemisphere Using a Two-Step Method

This work is devoted to the development and implementation of a two-step method for solving the vector three-dimensional inverse diffraction problem on an inhomogeneous dielectric scatterer having the form of a hemisphere characterized by piecewise constant permittivity. The original boundary value problem for Maxwell’s equations is reduced to a system of integro-differential equations. An integral formulation of the vector inverse diffraction problem is proposed and the uniqueness of the solution of the first-kind integro-differential equation in special function classes is established. A two-step method for solving the vector inverse diffraction problem on the hemisphere is developed. Unlike traditional approaches, the two-step method for solving the inverse problem is non-iterative and does not require knowledge of the exact initial approximation. Consequently, there are no issues related to the convergence of the numerical method. The results of calculations of approximate solutions to the inverse problem are presented. It is shown that the two-step method is an efficient approach to solving vector problems in near-field tomography.

Renormalization-Group Evolution for the Three-Particle B-Meson Soft Function.

We determine for the first time the renormalization-group equation for the three-particle B-meson soft function dictating the nonperturbative strong interaction dynamics of the long-distance penguin contributions to the double radiative B-meson decays. The distinctive feature of the ultraviolet renormalization of this fundamental soft function consists in the pattern of mixing positive into negative support for an arbitrary initial condition. The exact solution to this integrodifferential evolution equation is then derived with the Laplace transform technique, allowing for the model-independent extraction of the asymptotic behavior at large and small partonic momenta.

AM3: An Open-source Tool for Time-dependent Lepto-hadronic Modeling of Astrophysical Sources

We present the Astrophysical Multimessenger Modeling (AM 3 ) software. AM 3 is a documented open-source software (source code at https://gitlab.desy.de/am3/am3; user guide and documentation at https://am3.readthedocs.io/en/latest/) that efficiently solves the coupled integro-differential equations describing the temporal evolution of the spectral densities of particles interacting in astrophysical environments, including photons, electrons, positrons, protons, neutrons, pions, muons, and neutrinos. The software has been extensively used to simulate the multiwavelength and neutrino emission from active galactic nuclei (including blazars), gamma-ray bursts, and tidal disruption events. The simulations include all relevant nonthermal processes, namely synchrotron emission, inverse Compton scattering, photon–photon annihilation, proton–proton and proton–photon pion production, and photo-pair production. The software self-consistently calculates the full cascade of primary and secondary particles, including nonlinear feedback processes and predictions in the time domain. It also allows the user to track separately the particle densities produced by means of each distinct interaction process, including the different hadronic channels. With its efficient hybrid solver combining analytical and numerical techniques, AM 3 combines efficiency and accuracy at a user-adjustable level. We describe the technical details of the numerical framework and present three examples of applications to different astrophysical environments.

Convergence of a refined iterative method and its application to fractional Volterra–Fredholm integro-differential equations

Convergence of a refined iterative method and its application to fractional Volterra–Fredholm integro-differential equations

On Impulsive Nonlocal Nonlinear Fuzzy Integro-Differential Equations in Banach Space

The aim of this article is to investigate the existence, uniqueness and other qualitative properties of the solution of first-order nonlocal impulsive nonlinear fuzzy integro-differential equations in Banach space by using the concept of fuzzy numbers whose values are normal, upper semicontinuous, compact, and convex. The result is attained by utilizing a modified version of the Banach contraction principle. We offer an example as an application of the results.

Radiative-conductive heat transfer dynamics in dissipative dispersive anisotropic media

Abstract We develop a self-consistent theoretical formalism to model the dynamics of heat transfer in dissipative, dispersive, anisotropic nanoscale media, such as metamaterials. We employ our envelope dyadic Green’s function method to solve Maxwell’s macroscopic equations for the propagation of fluctuating electromagnetic fields in these media. We assume that the photonic radiative heat transfer mechanism in these media is complemented by dynamic phononic mechanisms of heat storage and conduction, accounting for effects of local heat generation. By employing the Poynting theorem and the fluctuation-dissipation theorem, we derive novel closed-form expressions for the radiative heat flux and the coupling term of photonic and phononic subsystems, which contains the heating rate and the radiative heat power contributions. We apply our formalism to the paraxial heat transfer in uniaxial media and present relevant closed-form expressions. By considering a Gaussian transverse temperature profile, we also obtain and solve a system of integro-differential heat diffusion equations to model the paraxial heat transfer in uniaxial reciprocal media. By applying the developed analytical model to radiative-conductive heat tranfer in nanolayered media constructed by layers of silica and germanium, we compute the temperature profiles for the three first orders of expansion and the total temperature profile as well. The results of this research can be of interest in areas of science and technology related to thermophotovoltaics, energy harvesting, radiative cooling, and thermal management at micro- and nanoscale.

A novel explicit fast numerical scheme for the Cauchy problem for integro-differential equations with a difference kernel and its application

The present study focuses on designing a second-order novel explicit fast numerical scheme for the Cauchy problem incorporating memory associated with an evolutionary equation, where the integral term's kernel is a discrete difference operator. The Cauchy problem under consideration is related to a real finite-dimensional Hilbert space and includes a self-adjoint operator that is both positive and definite. We introduce a transformative technique for converting the Cauchy problem incorporating memory, into a local evolutionary system of equations by approximating the difference kernel using the sum of exponentials (SoE) approach. A second-order explicit scheme is then constructed to solve the local system. We thoroughly investigate the stability of this explicit scheme, and present the necessary conditions for the stability of the scheme. Moreover, we extended our investigation to encompass time-fractional diffusion-wave equations (TFDWEs) involving a fractional Caputo derivative with an order ranging between (1,2). Initially, we transform the main TFDWE model into a new model that incorporates the fractional Riemann-Liouville integral. Subsequently, we expand the applicability of our idea to develop an explicit fast numerical algorithm for approximating the model. The stability properties of this fast scheme for solving TFDWEs are assessed. Numerical simulations including a two-dimensional Cauchy problem as well as one-dimensional and two-dimensional TFDWE models are provided to validate the accuracy and experimental order of convergence of the schemes.

Stability and boundedness of solutions to a certain third-order neutral integro-differential equation with constant delay

In this work, we examine a third-order nonlinear neutral integro-differential equation with constant delay. By building a Lyapunov functional, we obtain some sufficient criteria that ensure the asymptotic stability and boundedness of solutions for an analyzed equation. We present two examples to demonstrate the applicability of our conclusions, which extend and improve several well-known results in the literature.

Fractional Brownian motion in confining potentials: non-equilibrium distribution tails and optimal fluctuations

Abstract At long times, a fractional Brownian particle in a confining external potential reaches a non-equilibrium (non-Boltzmann)
steady state. Here we consider scale-invariant power-law potentials $V(x)\sim |x|^m$, where $m>0$, and employ the optimal fluctuation method (OFM) to determine the large-$|x|$ tails of the steady-state probability distribution $\mathcal{P}(x)$ of the particle position. The calculations involve finding the optimal (that is, the most likely) path of the particle, which determines these tails, via a minimization of the exact action functional for this system, which has recently become available. Exploiting dynamical scale invariance of the model in conjunction with the OFM ansatz, we establish the large-$|x|$ tails of $\ln \mathcal{P}(x)$ up to a dimensionless factor $\alpha(H,m)$, where $0<H<1$ is the Hurst exponent. We determine $\alpha(H,m)$ analytically (i) in the limits of $H\to 0$ and $H\to 1$, and (ii) for $m=2$ and arbitrary $H$, corresponding to the fractional Ornstein-Uhlenbeck (fOU) process. Our results for the fOU process are in agreement with the previously known exact $\mathcal{P}(x)$ and autocovariance. The form of the tails of $\mathcal{P}(x)$ yields exact conditions, in terms of $H$ and $m$, for the particle confinement in the potential.
For $H\neq 1/2$, the tails encode the non-equilibrium character of the steady state distribution, and we observe violation of time reversibility of the system except for $m=2$. To compute the optimal paths and the factor $\alpha(H,m)$ for arbitrary permissible $H$ and $m$, one needs to solve an (in general nonlinear) integro-differential equation. To this end we develop a specialized numerical iteration algorithm which
accounts analytically for an intrinsic cusp singularity of the optimal paths for $H<1/2$.


Study of a non-linear Volterra integro-differential equation with a non-linear unknown source term

In this paper, we propose a new class of integro-differential nonlinear Volterra equations with a non-linear unknown source term. Our study focuses on both theoretical analysis and numerical approximation of solutions to these equations. In the theoretical analysis partition, we define the specific assumptions on the integral kernel's characteristics, on the source term function and also on the derivative of the solutions of equations, under which the Krasnoselskii's fixed point theorem can be applied and in order to ensures the existence and uniqueness of solutions to the proposed integro-differential equations. In the numerical approximation partition, we use a well-known numerical technique for solving integral equations called Nyström method. This method allows us to get an approximate solutions of the proposed equations. Furthermore, we provide some illustrative examples and according to the numerical method described in this paper we approach them, where the comparison results between the exact and approximate solutions of these examples confirm the effectiveness of the numerical Nyström method for approaching the solutions of this class of integro-differential equations.

Single equation approach for linear Fredholm integrodifferential equations

Our primary objective is to develop an innovative approach for the numerical so lution of the Fredholm linear integrodifferential equation. Our overarching goal is to significantly enhance computational efficiency and minimize memory space utilization which is very important in the case of large integration intervals. To begin this en deavor, we establish sufficient conditions that guarantee the existence and uniqueness of the solution. Our novel method is grounded in the single equation approach, which consists in using the variable transformation represented as v(x) = v(a) + ∫ ︁ x a v(t)dt, completed by the trapezoidal rule. This transformation plays a pivotal role in convert ing our equation into an algebraic system, thereby reducing the number of equations and unknowns in the discrete system. Underlying these developments is a fundamental requirement ensuring the existence and uniqueness of the solution. Leveraging this, we formulate theorems that establish the convergence of the approximate solution, ensuring consistency between analytical and numerical investigations. Ultimately, we conduct a comparative analysis between our newly introduced technique and the older method. This comparison serves to highlight the superior computational efficiency and reduced storage space consumption offered by our innovative digital framework. Наша главная цель — значительно повысить вычислительную эффективность и минимизировать использование памяти, что очень важно в случае больших интервалов интегрирования. Сначала определяем достаточные условия, которые гарантируют существование и единственность решения. Наш новый метод основан на подходе с одним уравнением, он заключается в использовании преобразования переменной, представленного как v(x) = v(a) + ∫ ︁ x a v(t)dt, дополненного правилом трапеции. Оно играет ключевую роль в преобразовании нашего уравнения в алгебраическую систему, тем самым уменьшая количество уравнений и неизвестных в дискретной системе. В основе этих разработок лежит фундаментальное требование, гарантирующее существование и единственность решения. Используя это, мы формулируем теоремы, которые устанавливают сходимость приближенного решения, обеспечивая согласованность между аналитическими и численными исследованиями. В конечном итоге мы проводим сравнительный анализ между нашей новой введенной техникой и старым методом. Это сравнение служит для того, чтобы подчеркнуть существенную вычислительную эффективность и сокращенное использование памяти на диске, предлагаемые нашей инновационной цифровой структурой.

Adaptive hyperbolic-cross-space mapped Jacobi method on unbounded domains with applications to solving multidimensional spatiotemporal integrodifferential equations

In this paper, we develop a new adaptive hyperbolic-cross-space mapped Jacobi (AHMJ) method for solving multidimensional spatiotemporal integrodifferential equations in unbounded domains. By devising adaptive techniques for sparse mapped Jacobi spectral expansions defined in a hyperbolic cross space, our proposed AHMJ method can efficiently solve various spatiotemporal integrodifferential equations such as the anomalous diffusion model with reduced numbers of basis functions. Our analysis of the AHMJ method gives a uniform upper error bound for solving a class of spatiotemporal integrodifferential equations, leading to effective error control.

Solutions for non-autonomous fractional integrodifferential equations with delayed force term

Solutions for non-autonomous fractional integrodifferential equations with delayed force term