Integro-differential Equations Research Articles

The hyperbolic multi-term time fractional integro-differential equation with generalized Caputo derivative and error estimate in $$L_{p,\gamma ,\upsilon }$$ space

The hyperbolic multi-term time fractional integro-differential equation with generalized Caputo derivative and error estimate in $$L_{p,\gamma ,\upsilon }$$ space

Dirichlet problem for semilinear partial integro-differential equations: the method of orthogonal projection

AbstractWe study the Dirichlet problem for semilinear equations on general open sets with measure data on the right-hand side and irregular boundary data. For this purpose we develop the classical method of orthogonal projection. We treat in a unified form equations with operators belonging to the broad class of integro-differential operators associated with symmetric regular Dirichlet forms.

Adaptive time stepping for the two-time integro-differential Kadanoff-Baym equations

Adaptive time stepping for the two-time integro-differential Kadanoff-Baym equations

Wavelet-based collocation methods for solving multi-dimensional integro-differential systems

This paper presents a comprehensive investigation of wavelet-based collocation methods for solving multi-dimensional integro-differential systems. The research addresses the challenging problem of finding numerical solutions to integro-differential equations (IDEs) that are essential in various scientific and engineering applications. We propose novel numerical approximations for infinitely dimensional problems, including fractional Cattaneo-Rayleigh waves and nonlinear diffusion problems defined on unbounded domains. The study introduces a unified computational strategy that directly applies to IDEs without discretization, particularly focusing on the integer and fractional-order Cattaneo-Rayleigh model of thermoelasticity in one-dimensional zonal regions. Through numerical experiments, we demonstrate that our wavelet collocation approach achieves high efficiency and superior accuracy compared to traditional methods. The results show particularly strong performance in handling systems with differential physical singularities, functional physical singularities, and vector-valued physical singularities. Our approach provides a valuable toolbox for addressing complex modeling challenges in biology, mechanical signal processing, and digital communications where partial integro-differential equations naturally arise.

On a class of evolution equations in Lipschitz subdomains of Riemannian manifolds

We study the existence, uniqueness, analyticity, and continuous dependence of mild solutions for a class of evolutionary integrodifferential equations with free boundary conditions on L p -spaces in Lipschitz subdomains of Riemannian manifolds. Our strategy is to use the Laplace transform theory to ensure the existence of an analytic resolvent to the Hodge–Laplacian in the space of all ℓ-differential forms with p-th power integrable coefficients.

Solvability of some integro-differential equations with the logarithmic Laplacian

We address the existence in the sense of sequences of solutions for a certain integro-differential-type problem involving the logarithmic Laplacian. The argument is based on the fixed point technique when such equation contains the operator without the Fredholm property. It is established that, under the reasonable technical conditions, the convergence in L 1 ( R d ) of the integral kernels yields the existence and convergence in L 2 ( R d ) of the solutions.

Numerical Study of Complex Heat Transfer and Aerodynamics in a Tube Furnace with a Flat Fuel Combustion Mode

The object of the study is a tubular furnace for steam reforming of natural gas. The channel walls are formed by a refractory lining and a tubular screen with a known temperature. The chamber volume is filled with a selectively radiating, absorbing and weakly scattering medium of gaseous fuel combustion products. The research method is based on the numerical solution of a system of two-dimensional integro-differential equations of radiative gas dynamics, closed by a two-parameter model of turbulent convection. It is shown that when burners are located on the side walls of the radiation chamber, a significant restructuring of the heat flux distribution occurs. A decrease in the chamber width is accompanied by an increase in both radiant and convective heat fluxes to the tubular screen.

Stability of complement value problems for p-Lévy operators

We set up a general framework tailor-made to solve complement value problems governed by symmetric nonlinear nonlocal integro-differential p-Lévy operators. A prototypical example of integro-differential p-Lévy operators is the well-known fractional p-Laplace operator. Our main focus is on nonlinear integro-differential equations in the presence of Dirichlet, Neumann and Robin conditions and we show well-posedness results. Several results are new even for the fractional p-Laplace operator but we develop the approach for general translation-invariant nonlocal operators. We also bridge the gap from nonlocal to local, by showing that solutions to the local Dirichlet and Neumann boundary value problems associated with p-Laplacian are strong limits of the nonlocal ones.

Numerical solution based on the Haar wavelet collocation method for partial integro-differential equations of Volterra type

In this paper, a numerical investigation of a class of parabolic Volterra integro-differential equations has been carried out. Basically, the finite difference method associated with the Haar wavelet collocation technique is pursued. The main idea relies on semi-discretizing the parabolic-VIDEs by considering the finite differences in time and approximating the integral term within the trapezoidal rule. Afterwards, using a uniform mesh, the spatial derivative is approximated by the Haar wavelet method. Stability and convergence of the proposed approach are established in the frame of Sobolev space. Numerical examples are used to illustrate the effectiveness of this approach under different test scenarios. The error analysis, using both L 2 and L ∞ norms, shows that the method has high efficiency computationally with a great deal of accuracy. It is observed that these numerical results are in excellent agreement with the analytical solution. In addition, the methods and results obtained in this study are compared with those reported in recent literature.

A new linearized ADI compact difference method on graded meshes for a nonlinear 2D and 3D PIDE with a WSK

In this work, a new linearized alternating direction implicit (ADI) compact difference method (CDM) is proposed for solving nonlinear two-dimensional (2D) and three-dimensional (3D) partial integrodifferential equation (PIDE) with a weakly singular kernel (WSK). The time derivative is treated by Crank-Nicolson (CN) method and the Riemann-Liouville (R-L) integral by product integration (PI) rule on graded meshes. The linear interpolation combining with Taylor formula is applied in time to deal with nonlinear term v∇v in interval (t0,t1), and linear interpolation concerning two previous time points is employed to deal with v∇v in intervals (tn−1,tn),n≥2. A linearized semi-discrete scheme is obtained, which can achieve second-order convergence in time. Then via introducing two kinds of compact difference operators to discretize the spatial derivatives. To improve the computing efficiency, we construct a ADI compact difference method. It is the first time that the ADI compact difference method is applied for the nonlinear 2D and 3D PIDE with a WSK. The advantage of our proposed scheme is that it not only has second-order accuracy in time and fourth-order accuracy in space, but also fast computational speed, just by solving the linear coupled equations for tridiagonal matrices. In addition, we prove the existence, uniqueness and convergence of 2D scheme. Four numerical examples in 2D and 3D are present to demonstrate our proposed method.

Existence and stability results for $\psi$-Caputo fractional integrodifferential equations with delay

Existence and stability results for $\psi$-Caputo fractional integrodifferential equations with delay

МЕТОД МОНТЕ-КАРЛО ДЛЯ РАСЧЕТА ХАРАКТЕРИСТИК СВЕТОВЫХ ПОЛЕЙ В МОРСКОЙ ВОДЕ

The development of numerical methods for solving the integro-differential radiation transfer equation remains a relevant task. Among them, we can highlight the Monte Carlo method, which is in demand in various niches of modern ocean optics. The purpose of this work is a clear and concise presentation of the basics of the forward Monte Carlo method of light fields modeling in seawater, accompanied by a detailed description of its software implementation. The basics of the method are described, the procedures for choosing the type of interaction, the mean free path and the direction of photon motion are described. A simple case is considered, corresponding to an infinitely distant point source of unpolarized light, the absence of atmospheric influence, a smooth air-seawater interface, and the absence of stratification of inherent optical properties. In this case, realistic values of the absorption and scattering coefficients were used, calculated in accordance with the Case 1 model for a chlorophyll concentration of 1 μg/L, and a strongly elongated Henyey-Greenstein phase function with the parameter g = 0.95. The Fresnel reflection of light from the air-seawater interface was taken into account. The relative errors in the values of the diffuse attenuation coefficient for downward irradiance K d and the diffuse reflectance R, calculated in the spectral range of 400–700 nm using 106 photons, in comparison with the HydroLight results were 1.5 % and 0.4 %, respectively. Spectral calculation on one core of a 2017 Intel Core i5-8250U mobile processor in MATLAB takes 6 minutes. An assessment of the choice of the optimal number of photons required to obtain the desired quantities with a given accuracy was made. The implemented method is useful for becoming familiar with the basic principles used to numerically solve the radiative transfer equation in seawater using statistical methods and is used in the “Ocean Optics” course, taught by the author to 4th year students of the Department of Thermohydromechanics of the Ocean at MIPT.

Unveiling the dynamics of plasma dilution in medical science through analytical and numerical approaches via fractional integro-differential equations

Unveiling the dynamics of plasma dilution in medical science through analytical and numerical approaches via fractional integro-differential equations

Optimization of the integro-differential hyperbolic model

We consider a model of processes in viscoelastic media. The force arising inside the body may depend not only on the current values but also on the entire time history of the motion (for example, polymers, emulsions, and suspensions exhibit memory properties). Such processes often lead to the necessity of studying hyperbolic integro-differential equations with Volterra integral term. The main goal is to prove the existence and uniqueness of generalized solutions to the corresponding initial-boundary value problem, as well as to investigate the existence of optimal control for systems described by these models. The main results are obtained using the method of a priori inequalities in negative norms. In particular, we justify inequalities in negative norms for the integro-differential operator in certain functional spaces. The work begins with a short description of relevant results with similar formulations and referencing works with physical justification of the model. Afterwards, we describe the problem statement, constraints on equation parameters, and functional spaces used for the investigation. Below are the main results, in particular definitions of generalized solutions and properties on the well-posed of the initial-boundary value problem as well as the existence of optimal control. Control is carried out through the right-hand side of the equation using a certain special operator. It is possible to examine specific examples of such control operators (illustrating various control mechanisms) and the corresponding control spaces. By utilizing proven inequalities and relying on general results of S.I. Lyashko from the theory of a priori inequalities in negative norms, we establish sufficient conditions for the existence of optimal control. In particular, we impose the restrictions on the admissible set of controls and the quality criterion.

Solving fractional integro-differential equations with delay and relaxation impulsive terms by fixed point techniques

This paper presents a systematic approach to investigating the existence of solutions for fractional integro-differential equation systems incorporating delay and relaxation impulsive terms. By employing suitable definitions of fractional derivatives, we establish physically interpretable boundary conditions. To account for abrupt state changes, impulsive conditions are integrated into the model. The system is transformed into an equivalent integral equation, facilitating the application of Banach and Schaefer fixed-point theorems to prove the existence and uniqueness of solutions. The practical applicability of our findings is demonstrated through an illustrative example.

Calculating a perturbation of a plasma layer by an electric field

The paper presents the results of solving a boundary value problem for a system of two integro-differential equations that simulate the action of an external electric field on a plasma layer. This system is an implication of the Boltzmann–Maxwell equations, and the physical meaning of the sought functions is the strength of a self-consistent electric field and perturbation of the electron distribution density. The solution of the problem is constructed using the theories of Fourier transform of generalized functions and singular integral equations with the Cauchy kernel. The dependence of the solution on the frequency of the external field is studied.

Numerical solutions for second-order neutral volterra integro-differential equations: Stability analysis and finite difference method

This work deals with the initial-value problem for a second-order neutral Volterra integro-differential equation. First, we give the stability inequality indicating stability of the problem with respect to the right-side and initial conditions. Further, we develop a finite difference method that uses for differential part second difference derivative, for the integral part appropriate composite trapezoidal and midpoint rectangle rules followed by second-order accurate difference quantities at intermediate points. Error estimate for the approximate solution is established. In support of theoretical results, numerical results are performed by employing the proposed numerical technique.

Numerical Treatment of the Coupled Fredholm Integro-Differential Equations by Compact Finite Difference Method

Numerical Treatment of the Coupled Fredholm Integro-Differential Equations by Compact Finite Difference Method

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.