Integro-differential Equations Research Articles

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.

Radial Kernels Collocation Method for the Solution of Volterra Integro-Differential Equations

Radial kernel interpolation is an advanced method in approximation theory for the construction of higher order accurate interpolants for scattered data up to higher dimensional spaces. In this manuscript, we formulate a radial kernel collocation approach for solving problems involving the Volterra integro-differential equations using two radial kernels: The Generalized Multi-quadrics and the linear Laguerre-Gaussians. This was achieved by simplifying the Volterra integral problem's solution to an algebraic system of equations. The impact of the shape parameter present in every kernel on the method's accuracy is examined and found to be significant. Two examples were used to illustrate the process; the numerical results are shown as tables and graphs. MATLAB 2018a was employed in the process.

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

Existence and stability of mild solutions to non autonomous impulsive stochastic neutral integrodifferential equations

Existence and stability of mild solutions to non autonomous impulsive stochastic neutral integrodifferential equations

Representations of the solutions for volterra integro-differential equations in hilbert spaces

Representations of the solutions for volterra integro-differential equations in hilbert spaces

RESULTS ON IMPULSIVE ψ-CAPUTO FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS WITH BOUNDARY CONDITIONS

In this research article, we investigate the existence and uniqueness of solutions for a type of boundary value problems involving fractional integro-differential equations with the ψ-Caputo fractional derivative. Our method utilizes several classical fixed point theorems, such as the Banach contraction principle and Krasnoselskii's fixed point theorem. We establish our main results through theoretical analysis and present a specific case study to illustrate the practical significance of our findings.

Existence of mild solutions for non-instantaneous impulsive ξ-Caputo fractional integro-differential equations

The aim of this paper is to investigate the existence of mild solutions for a nonlocal ξ-Caputo fractional non-instantaneous impulses semilinear integro-differential equation in a Banach space. The proofs are based on some fixed point theorems for condensing maps. As an application, an example is given to illustrate our theoretical results.

Numerical computational technique for solving Volterra integro-differential equations of the third kind using meshless collocation method

The primary goal of this study is to give an approximate algorithm for solving Volterra integro-differential equations (VIDEs) of the third kind using meshless collocation techniques. The basic framework of the novel approach is based on a collocation scheme and radial basis functions (RBFs) created on scattered points. This technique requires no background approximation cells, and the algorithm is powerful, has greater stability, and does not require much computer memory. This approach represents the solution of VIDEs of the third kind by interpolating the RBFs based on the Gauss–Legendre quadrature formula. The problem is reduced to a system of algebraic equations that can be easily solved. A description of the technique for the proposed equations is provided. Furthermore, the error analysis of this scheme is examined. A few numerical experiments are presented to prove the reliability and precision of the suggested approach for solving VIDEs of the third kind. Certain problems were compared with analytical solutions, the moving least squares method, and other methods to prove the effectiveness and applicability of the approach described.

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.

Numerical solutions of stochastic delay integro-differential equations by block pulse functions

This paper presents an efficient numerical method for solving nonlinear stochastic delay integro-differential equations based on block pulse functions. Firstly, the equation is transformed into an algebraic system by the integral delay operator matrixes of block pulse functions. Then, error analysis is conducted on the method. Finally, some numerical examples are provided to validate the method. This work provides numerical solutions for the stochastic delay integro-differential equations by global approximation method. This method has the advantages of simple calculation and higher error accuracy.

Iterative Method of Determining Stress Intensity Coefficients Under Dynamic Loading of the Crack System

An elastic isotropic body in a state of plane deformation, which contains a system of randomly placed cracks under the action of a dynamic (harmonic) loading, is considered. The authors set the problem of determining the stress field around the cracks under the conditions of their wave interaction. The solution method is based on the introduction of displacements in the body in the form of a superposition of discontinuous solutions of the equations of motion constructed for each crack. With this in mind, the initial problem is reduced to a system of singular integro-differential equations with respect to unknown displacement jumps on the crack surfaces. To solve this system, a new iterative method, which involves solving a set of independent integro-differential equations that differ only in their right-hand parts at each iteration, is proposed. For the zero approximation, solutions that correspond to individual cracks under the action of dynamic loading are chosen. Such a new approach allows to avoid the difficulties associated with the need to solve systems of integro-differential equations of large dimensions that arise when traditional methods are used. Based on the results of the iterations, formulas for calculating the stress intensity coefficients for each crack were obtained. In the partial case of four cracks, a good agreement between the results obtained during the direct solution of the system of eight integro-differential equations by the mechanical quadrature method and the results obtained by the iterative method was established. In general, numerical examples demonstrate the convergence and stability of the proposed method in the case of systems with a fairly large number of densely located cracks. The influence of the interaction between cracks on the stress intensity factor (SIF) value under dynamic loading conditions was studied. An important and new result for fracture mechanics is the detection of the absolute maximum of the normal stresses at certain frequencies of the oscillating normal loading. The number of interacting cracks and the configuration of the crack system itself affect the values of the frequencies at which SIF reaches a maximum and the maximum values. These maximum values significantly (by several times) exceed the SIF values of single cracks under a similar loading. At the same time, under conditions of static or low-frequency loading, it is possible to reduce the SIF values compared to the SIF for individual cracks. When cracks are sheared, the values of the tangential stresses have a tendency to decrease with increasing frequency, and their values do not significantly differ from the values of the tangential stress for an individual crack.

СТІЙКІСТЬ ЗАЛІЗОБЕТОННИХ СТІЙОК ПРИ КОРОТКОЧАСНІЙ ТА ТРИВАЛІЙ ДІЇ НАВАНТАЖЕННЯ

The behavior of hinged support at the ends of the rack with initial bending under the action of load is considered. Spring rods are arranged symmetrically. The rack has an initial bend, compressed by forces constant in time. All real elements have one or another imperfections in the form of technological bends, therefore, they begin to bend from the very beginning of the load. The load, when it is exceeded even by an infinitesimally small amount, there is a loss of stability of this type of deformation, is called critical. In the calculation of stability under the long-term action of external forces, it is necessary to determine the load, at which the rate of movement in time monotonically decays. Solving the problem in such a setting is acceptable for systems, the development of movements of which in time leads to a change in the stress state. This condition for a compressed rack is fulfilled only in the presence of initial imperfections (initial bending, off-center application of compressive force, etc.). When solving the problems of the theory of stability, taking into account the creep of the material plays an important role. Creep can be limited in time or unlimited. When solving the problem of the stability of racks with initial imperfections, made of a material that has creep and reinforced with elastic rods, the following assumptions are made: 1) the hypothesis of flat sections is considered valid; 2) the deformations of the creeping material and the elastic rods at the points of contact are the same; 3) the modulus of deformations during stretching and compression are equal; 4) the creep material works in the stretched zone without the appearance of cracks. The relationship between deformations and elasticities in the material of the rack is established by a formula based on the linear relationship between deformations and elasticities. The creep rate of concrete is based on the hereditary theory of aging. In the work, an integro-differential equation was obtained - the equation of slow motion of the rod, expressions were obtained for the study of bends in any at what point in time, the formula for determining the critical force with long-term load action is derived.

Synthesis of uniformly distributed optimal control with nonlinear optimization of oscillatory processes

In the article the problem of synthesizing uniformly distributed optimal control for nonlinear optimization of oscillatory processes described by integro-differential partial differential equations with the Volterra integral operator was explored. The study was conducted according to the Bellman-Egorov scheme and an algorithm for constructing a uniformly distributed optimal control in the form of a functional from the state of the controlled process was developed. Sufficient conditions for the solvability of the synthesis problem in nonlinear optimization were established.

Existence of extremal solutions for a class of fractional integro-differential equations

In the study the existence of solutions of a class of fractional integro-differential equations with boundary conditions was considered. The main tool, we employ, is the conventional monotone iterative technique, which is highly effective method to examine the quantitative and qualitative characteristics of various nonlinear problems. This technique produces monotone sequences whose iterations are unique solutions of the certain linear problems. These bounds converge uniformly to the maximal solutions of the given problems. Some types of coupled solutions are considered to obtain the claim of the main results under suitable conditions.

Numerical implementation of solving a boundary value problem with parameter for Fredholm integro-differential equation

The boundary value problem with parameter for Fredholm integro-differential equation with degenerate kernel is investigated in this paper. The aim of the paper is to establish the solvability conditions, to construct analytical and numerical solutions of the considered problem. The basis for achieving the goal is the ideas of Dzhumabayev parameterization method, classical numerical methods of solving Cauchy problems and numerical integration techniques. A problem with parameters is obtained by introducing an additional parameter and a new unknown function. A system of equations with respect to parameters is compiled according to the initial data of the considered equation and boundary conditions. The unknown function is found as a solution of the Cauchy problem for the ordinary differential equation. The equivalence of the original problem and the problem with parameters, the conditions of unique solvability are established and the formula for finding an analytical solution is obtained. Test examples of finding analytical and approximate solutions of the original problem are given.

An optimization-based equilibrium measure describes fixed points of non-equilibrium dynamics: application to edge of chaos

Abstract Understanding neural dynamics is a central topic in machine learning, non-linear physics and neuroscience. However, the dynamics is non-linear, stochastic and particularly non-gradient, i.e., the driving force can not be written as gradient of a potential. These features make analytic studies very challenging. The common tool is the path integral approach or dynamical mean-field theory, but the drawback is that one has to solve the integro-differential or dynamical mean-field equations, which is computationally expensive and has no closed form solutions in general. From the aspect of associated Fokker-Planck equation, the steady state solution is generally unknown. Here, we treat searching for the steady states as an optimization problem, and construct an approximate potential related to the speed of the dynamics, and find that searching for the ground state of this potential is equivalent to running an approximate stochastic gradient dynamics or Langevin dynamics. Only in the zero temperature limit, the distribution of the original steady states can be achieved. The resultant stationary state of the dynamics follows exactly the canonical Boltzmann measure. Within this framework, the quenched disorder intrinsic in the neural networks can be averaged out by applying the replica method, which leads naturally to order parameters for the non-equilibrium steady states. Our theory reproduces the well-known result of edge-of-chaos, and further the order parameters characterizing the continuous transition are derived, and the order parameters are explained as fluctuations and responses of the steady states. Our method thus opens the door to analytically study the steady state landscape of the deterministic or stochastic high dimensional dynamics.

An RBF Method for Time Fractional Jump-Diffusion Option Pricing Model under Temporal Graded Meshes

This paper explores a numerical method for European and American option pricing under time fractional jump-diffusion model in Caputo scene. The pricing problem for European options is formulated using a time fractional partial integro-differential equation, whereas the pricing of American options is described by a linear complementarity problem. For European option, we present nonuniform discretization along time and the radial basis function (RBF) method for spatial discretization. The stability and convergence analysis of the discrete scheme are carried out in the case of European options. For American option, the operator splitting method is adopted which split linear complementary problem into two simple equations. The numerical results confirm the accuracy of the proposed method.

Operational Matrices of Genocchi Polynomials for Solving High-Order Linear Fredholm Integro-Differential Equations

Operational Matrices of Genocchi Polynomials for Solving High-Order Linear Fredholm Integro-Differential Equations