Linear Integro-differential Equations Research Articles

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

Efficient numerical solution of linear Fredholm integro-differential equations via backward finite difference and Nyström methods

Efficient numerical solution of linear Fredholm integro-differential equations via backward finite difference and Nyström methods

Study of a class of fractional order non linear neutral abstract Volterra integro-differential equations with deviated arguments

In this paper, we give new results on the study of a class of fractional order non linear neutral abstract Volterra integro-differential equations with deviated arguments, considered in a Banach space. Using the method suggested by Hernández et al. (2010), which is based on the theory of resolvent operators for integral equations and fixed point theorems, we prove the main results on the existence and uniqueness of the mild solution of this type of problems which have not been studied before. We also give an example to which our main results are applicable.

Euler wavelets method for optimal control problems of fractional integro-differential equations

This research introduces an efficient direct method for finding the numerical solution to optimal control problems involving linear and nonlinear fractional integro-differential equations using orthonormal Euler wavelets. Using orthonormal Euler wavelets, fractional integral, and product operational matrices have been obtained. By using these fractional operational matrices and collocation points, optimal control problems of fractional Volterra integro-differential equations have been reduced to a system of linear or non-linear equations. We establish the convergence analysis and error bound of the proposed method. Orthonormal Euler wavelets have been utilized to solve numerical examples and determine the approximate value of the cost function by approximating state and control functions. This has been done in order to verify the effectiveness of the proposed numerical technique.

Hypersingular integro-differential equation with quadratic functions in coefficients

A new linear integro-differential equation of order n ≥ 3, given on a closed curve located in the complex plane, is investigated. Integrals in the equation are understood in the sense of the finite part according to Hadamard. A characteristic feature of the equation is the presence of quadratic functions of a special kind in its coefficients. The equation is reduced first to the boundary value problem of linear conjugation for analytical functions. In the case of its solvability, two linear differential equations should be further solved in the domains of the complex plane with some additional conditions for the solution. All conditions for the solvability of the original equation are explicitly specified. When they are executed, the desired solution is constructed in a closed form. An example is given.

Asymptotical convergence of solutions of boundary value problems for singularly perturbed higher-order integro-differential equations

The article considers a two-point boundary value problem for a linear integro-differential equation of the n + m order with small parameters for m higher derivatives, provided that the roots of the additional characteristic equation are negative. The aim of the study is to obtain asymptotic estimates of the solution, to find out the asymptotic behavior of solutions in the vicinity of points where additional conditions are set, as well as to construct a degenerate problem, the solution of which tend to the solution of the initial perturbed boundary value problem. The Cauchy function and boundary functions of a boundary value problem for a singularly perturbed homogeneous differential equation are constructed, and their asymptotic estimates are obtained. Using the Cauchy functions and boundary functions, an analytical formula for solutions to the boundary value problem is obtained. A theorem on an asymptotic estimate for the solution of the considered boundary value problem is proved. The asymptotic behavior of the solution with respect to a small parameter and the order of growth of its derivatives are established. It is shown that the solution of the boundary value problem under consideration at the left end of this segment has the phenomenon of an initial jump and the order of this jump is determined. A modified degenerate boundary value problem containing initial jumps of the solution and the integral term is constructed.

Rapid variable-step computation of dynamic convolutions and Volterra-type integro-differential equations: RK45 Fehlberg, RK4

We introduce a novel, time-efficient adaptive Runge-Kutta computational scheme tailored for systematically solving linear and nonlinear Volterra-type Integro-Differential Equations (VTIDEs). This scheme is particularly effective for equations featuring a specific class of convolution memory, K(t,s)=aκ(t−s) within convolution integrals, where a,κ∈R. Such equations frequently arise in fields like viscoelasticity, population dynamics, epidemiology, control systems, financial mathematics, and neuroscience, where current states are influenced by historical data, especially in dynamical systems within physics. Our proposed scheme demonstrates remarkable computational efficiency, achieving a cost-effective computation time of O(Nt) for Nt iterations, a significant improvement over existing schemes for VTIDE, which typically exhibit complexities of O(Ntlog⁡Nt) and O(Nt(log⁡Nt))2. The key feature of our scheme lies in its ability to elegantly and efficiently handle convoluted integrals of varying complexity such as those including nonlinearities and derivatives. Specifically, our approach eliminates the need for backstage integration, allowing the main Runge-Kutta scheme to operate efficiently with only a few data points at each time step. This not only saves time but also prevents potential data storage issues while rendering numerical implementation straightforward and tractable. Moreover, this article introduces flexible discretizations for integration and differentiation, accommodating unequally spaced data points. We implement a fast RK45-Fehlberg algorithm (TE-RK45-Fehlberg), which we use to solve three examples of nonlinear VTIDE presented in this study. We further demonstrate the effectiveness of our approach by using an example, where we employ a novel parabolic step size generator function to efficiently generate step sizes for a computationally fast RK4 scheme (Flex-RK4). For all examples tackled using TE-RK45-Fehlberg and Flex-RK4, we generate plots that compare numerical data with exact solutions, providing insight into the accuracy and reliability of our approach. We also show how the scheme provides straightforward ways of transforming it into implicit forms and obtaining stability curves.

Approximating Higher Order Linear Fredholm Integro-Differential Equations by an Efficient Adomian Decomposition Method

Approximating Higher Order Linear Fredholm Integro-Differential Equations by an Efficient Adomian Decomposition Method

Numerical treatment of linear Volterra integro differential equations using variational iteration algorithm with collocation

To numerically solve the linear Volterra integro-differential equation, this study employs fourth-kind Chebyshev polynomials and the variational iteration algorithm with collocation, which is a combination of the variational iteration strategy and collocation technique. By applying fourth-kind Chebyshev polynomials to the variational iteration method with collocation for solving Volterra integro-differential equations, mathematical problems with a broad range of multidisciplinary applications are addressed, and numerical techniques that produce more accurate and efficient results are developed. The recommended method is then used, and the fourth-kind Chebyshev polynomials generated for the given integro-differential equation serve as the trial functions for the approximation. As a result, the suggested method's significance probably goes beyond a particular equation or application, as it contributes to the larger field of mathematical modeling and numerical analysis. Research methods employing a variational iteration algorithm with collocation aim to provide general techniques that can be applied to a wide range of problems. Additionally, numerical examples were provided to highlight the applicability and dependability of the proposed methodology. The mathematical computations were carried out using the Maple 18 software.

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.

Hölder estimates for viscosity solutions of nonlocal equations with variable-order fractional Laplace term

We consider a class of linear integro-differential equations with variable-order fractional Laplacian. Under sharp assumptions on the variable-order α(x,y), we prove the local Hölder continuity of bounded viscosity solutions. In particular, for the continuous right-hand side f, we show that weak solutions are viscosity solutions. Furthermore, we prove a comparison principle for viscosity solutions when the function f is strictly away from zero.

Development of integrated radial basis function Kriging interpolation for linear and nonlinear parabolic integro-differential equations

In this study, we explore linear and nonlinear parabolic integro-differential equations in one and two dimensions. We employ a semi-implicit scheme to discretize the temporal variable and discretize the spatial variable using an integrated radial basis function based on the moving Kriging interpolation (MKI) method. Unlike the global integrated radial basis function (IRBF) method, our proposed approach is particularly well-suited for addressing large-scale problems. Moreover, issues such as the selection of shape parameters leading to matrices with high condition numbers can be mitigated by devising an algorithm to determine optimal shape parameters, resulting in lower condition numbers. To assess the accuracy of the proposed method, we utilize two criteria, namely, L2 and H1. We compare the computed results obtained from the IRBF-MKI method with those from two grid and finite element methods. To demonstrate the accuracy and efficiency of our approach, we consider irregular areas and scattered Halton points in the two-dimensional case.

Bounded solutions and exponential stability for linear integro-differential equations of Volterra type

For the following scalar functional–differential equation ẍ(t)+∫t0tG(t,s)ẋ(s)ds+∫t0tH(t,s)x(s)ds=f(t),t≥t0,we obtain sufficient conditions for boundedness of all solutions and uniform exponential stability for the homogeneous equation. As examples, equations with singular kernels and equations with bounded delays ẍ(t)+∫g(t)tG(t,s)ẋ(s)ds+∫h(t)tH(t,s)x(s)ds=f(t),t≥t0,are considered.

Stability of solutions of systems of Volterra integral equations

In this paper, we propose new sufficient conditions for stability of solutions of systems of Volterra linear integral equations and systems of linear integro-differential Volterra equations. Solution stability conditions for systems of Volterra linear integral equations are studied with perturbed right hand sides of the equations. These new sufficient conditions are expressed in terms of logarithmic norms of matrices consisting of coefficients of equations and derivatives of their kernels. For Volterra integro-differential equations we also provide sufficient conditions for stability of solutions when the initial conditions are perturbed.

Laguerre Collocation Approach of Caputo Fractional Fredholm-Volterra Integro-Differential Equations

This paper discusses the linear fractional Fredholm-Volterra integro-differential equations (IDEs) considered in the Caputo sense. For this purpose, Laguerre polynomials have been used to construct an approximation method to obtain the solutions of the linear fractional Fredholm-Volterra IDEs. By this approximation method, the IDE has been transformed into a linear algebraic equation system using appropriate collocation points. In addition, a novel and exact matrix expression for the Caputo fractional derivatives of Laguerre polynomials and an associated explicit matrix formulation has been established for the first time in the literature. Furthermore, a comparison between the results of the proposed method and those of methods in the literature has been provided by implementing the method in numerous examples.

Cubic spline quasi-interpolation operator to numerically solve integro-differential equations with weakly singular kernels

In this paper we use cubic spline quasi-interpolant operator to numerically address a class of linear integro-differential equations with weakly singular kernel. As stated in Pedas and Tamme (2006), the exact solution of this equation lacks the desired level of smoothness and belongs to a particular function space. Then, in the first part of this paper, we analyze the approximation properties of the cubic spline quasi-interpolant operator in this particular space. Subsequently, we use these results in the analysis of the quasi-collocation method used to solve an integro-differential equation with weakly singular kernel. Also, some numerical tests are provided to confirm the theoretical results.

Solving Linear Volterra-Fredholm Integro-Differential Equations using Chebyshev-Galerkin with Error Estimation

The semi-analytic method based on the Galerkin technique with Chebyshev basis is presented in this paper to solve the linear integro-differential equations in the Volterra-Fredholm type. To facilitate, the use of the Chebyshev-Galerkin technique, new theorems and lemmas are established. Different numerical examples are introduced to demonstrate how the proposed method has efficacy and is easier to apply for this problem type. Some issues’ domains don’t match the Chebyshev’s domain. So, how to adequate the solution domain is presented. The error estimate is computed to prove the method's applicability and accuracy of the proposed method for any problem, especially in the case of the exact solution is not easy to determine.

Conformable Triple Sumudu Transform with Applications

One of the important topics in applied mathematics is the topic of integral transformations, due to their importance in electrical engineering applications, including communications in particular, and other sciences. In this work, one of the most important transformations in its three dimensions was presented, which is the triple Sumudu transform, including solving some real-life applications of physics, some of which have not been solved using such an integral transform before. In this work, we extend the Sumudu transform formula to the conformable fractional order, as well as other interesting and significant rules. The general analytical solution of a singular and nonlinear conformable fractional differential equation based on the conformable fractional Sumudu transform is also presented in this paper. The general solutions of several linear and nonhomogeneous conformable fractional differential equations can be obtained using the method we’ve proposed. As a result, our results reveal that our proposed method is an efficient one that can be used for solving all conformable fractional differential equations. The relationship between the Sumudu integral transform and other important and recently proposed integral transforms are also discussed. Finally, the triple Sumudu transform is used to solve boundary value problems, such as the heat equation with boundary values. The triple Sumudu integral transform is also used to solve linear partial integro-differential equations. The transform capability to handle such equations has been proven via its utilization in three applications.

Fejér-Quadrature Collocation Algorithm for Solving Fractional Integro-Differential Equations via Fibonacci Polynomials

In this article, we introduce a novel spectral algorithm utilizing Fibonacci polynomials to numerically solve both linear and nonlinear integro-differential equations with fractional-order derivatives. Our approach employs a quadrature-collocation method, transforming complex equations and associated conditions into systems of linear or nonlinear algebraic equations. The solutions to these equations, involving unknown coefficients, provide accurate numerical approximations for the original fractional-order equations. To validate the method, we present numerical examples illustrating its robustness and versatility. Comparative analyses with available analytical solutions affirm the reliability and accuracy of our algorithm, establishing its practical utility in addressing fractional-order integro-differential equations. This research contributes to computational mathematics and spectral methods, offering a promising tool for diverse scientific and engineering challenges.

High-degree polynomial approximations for solving linear integral, integro-differential, and ordinary differential equations

High-degree polynomial approximations for solving linear integral, integro-differential, and ordinary differential equations